3.596 \(\int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=317 \[ -\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {b^2 \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d \left (a^2+b^2\right )^2} \]

[Out]

b^(3/2)*(5*a^2+b^2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(3/2)/(a^2+b^2)^2/d+1/2*(a^2-2*a*b-b^2)*arctan(
-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/2*(a^2-2*a*b-b^2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2
+b^2)^2/d*2^(1/2)-1/4*(a^2+2*a*b-b^2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)+1/4*(a^2
+2*a*b-b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)+b^2*tan(d*x+c)^(1/2)/a/(a^2+b^2)/d
/(a+b*tan(d*x+c))

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Rubi [A]  time = 0.50, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3569, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {b^2 \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2),x]

[Out]

-(((a^2 - 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 - 2*a*b - b^
2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) + (b^(3/2)*(5*a^2 + b^2)*ArcTan[(Sqrt[b]*
Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)^2*d) - ((a^2 + 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*
x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2 + 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + T
an[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + (b^2*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 3534

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3653

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx &=\frac {b^2 \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\frac {1}{2} \left (2 a^2+b^2\right )-a b \tan (c+d x)+\frac {1}{2} b^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac {b^2 \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {a \left (a^2-b^2\right )-2 a^2 b \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )^2}+\frac {\left (b^2 \left (5 a^2+b^2\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 a \left (a^2+b^2\right )^2}\\ &=\frac {b^2 \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {2 \operatorname {Subst}\left (\int \frac {a \left (a^2-b^2\right )-2 a^2 b x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {\left (b^2 \left (5 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{2 a \left (a^2+b^2\right )^2 d}\\ &=\frac {b^2 \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (b^2 \left (5 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}\\ &=\frac {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}+\frac {b^2 \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}\\ &=\frac {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {b^2 \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}\\ &=-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {b^2 \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 0.62, size = 166, normalized size = 0.52 \[ \frac {-\frac {\sqrt [4]{-1} a \left ((a+i b)^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+(a-i b)^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {b^{3/2} \left (5 a^2+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\tan (c+d x)}}{a+b \tan (c+d x)}}{a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2),x]

[Out]

((b^(3/2)*(5*a^2 + b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) - ((-1)^(1/4)*a*((
a + I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 +
b^2) + (b^2*Sqrt[Tan[c + d*x]])/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)

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fricas [B]  time = 12.30, size = 14311, normalized size = 45.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(2)*((a^15 + 5*a^13*b^2 + 9*a^11*b^4 + 5*a^9*b^6 - 5*a^7*b^8 - 9*a^5*b^10 - 5*a^3*b^12 - a*b^14)*d
^5*cos(d*x + c)^2 + 2*(a^14*b + 6*a^12*b^3 + 15*a^10*b^5 + 20*a^8*b^7 + 15*a^6*b^9 + 6*a^4*b^11 + a^2*b^13)*d^
5*cos(d*x + c)*sin(d*x + c) + (a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12 + a*
b^14)*d^5)*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7
- 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 +
38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 +
28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*(1/((a^8 + 4*a^6
*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(3/4)*arctan(((a^16 - 20*a^12*b^4 - 64*a^10*b^6 - 90*a^8*b^8 - 64*a^
6*b^10 - 20*a^4*b^12 + b^16)*d^4*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 +
28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 +
4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - sqrt(2)*((a^18 + 7*a^16*b^2 + 20*a^14*b^4 + 28*a^12*b^6 + 14*
a^10*b^8 - 14*a^8*b^10 - 28*a^6*b^12 - 20*a^4*b^14 - 7*a^2*b^16 - b^18)*d^7*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^
4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12
+ 8*a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*(a^13*b + 6*a^11
*b^3 + 15*a^9*b^5 + 20*a^7*b^7 + 15*a^5*b^9 + 6*a^3*b^11 + a*b^13)*d^5*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 1
2*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*
a^2*b^14 + b^16)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^
5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 -
12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(((a^12 - 10*a^10*b^2 + 15*a^8*b^4 + 52*a^6*b^6 + 15*a^4*b^8
- 10*a^2*b^10 + b^12)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) + sqrt(2)
*(2*(a^13*b - 10*a^11*b^3 + 15*a^9*b^5 + 52*a^7*b^7 + 15*a^5*b^9 - 10*a^3*b^11 + a*b^13)*d^3*sqrt(1/((a^8 + 4*
a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) + (a^10 - 13*a^8*b^2 + 50*a^6*b^4 - 50*a^4*b^6 + 13*
a^2*b^8 - b^10)*d*cos(d*x + c))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 +
2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))
/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*
a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4) + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*sin(d*x + c))/cos(
d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(3/4) + sqrt(2)*((a^22 + a^20*b^2 - 21*a^1
8*b^4 - 85*a^16*b^6 - 134*a^14*b^8 - 70*a^12*b^10 + 70*a^10*b^12 + 134*a^8*b^14 + 85*a^6*b^16 + 21*a^4*b^18 -
a^2*b^20 - b^22)*d^7*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4
+ 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*(a^17*b - 20*a^13*b^5 - 64*a^11*b^7 - 90*a^9*b^9 - 64*a^7*b^11 - 20*a^5
*b^13 + a*b^17)*d^5*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4
+ 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a
^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^
8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(s
in(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(3/4))/(a^8 - 12*a^6*b^2 +
38*a^4*b^4 - 12*a^2*b^6 + b^8)) + 4*sqrt(2)*((a^15 + 5*a^13*b^2 + 9*a^11*b^4 + 5*a^9*b^6 - 5*a^7*b^8 - 9*a^5*
b^10 - 5*a^3*b^12 - a*b^14)*d^5*cos(d*x + c)^2 + 2*(a^14*b + 6*a^12*b^3 + 15*a^10*b^5 + 20*a^8*b^7 + 15*a^6*b^
9 + 6*a^4*b^11 + a^2*b^13)*d^5*cos(d*x + c)*sin(d*x + c) + (a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 +
15*a^5*b^10 + 6*a^3*b^12 + a*b^14)*d^5)*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^
9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8
)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6
+ b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 +
b^16)*d^4))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(3/4)*arctan(-((a^16 - 20*a^12*b^4 - 64
*a^10*b^6 - 90*a^8*b^8 - 64*a^6*b^10 - 20*a^4*b^12 + b^16)*d^4*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^
6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14
+ b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + sqrt(2)*((a^18 + 7*a^16*b^2 + 2
0*a^14*b^4 + 28*a^12*b^6 + 14*a^10*b^8 - 14*a^8*b^10 - 28*a^6*b^12 - 20*a^4*b^14 - 7*a^2*b^16 - b^18)*d^7*sqrt
((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b
^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b
^8)*d^4)) + 2*(a^13*b + 6*a^11*b^3 + 15*a^9*b^5 + 20*a^7*b^7 + 15*a^5*b^9 + 6*a^3*b^11 + a*b^13)*d^5*sqrt((a^8
- 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 +
56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*
(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4
*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(((a^12 - 10*a^10*b^2 + 15*a^8*
b^4 + 52*a^6*b^6 + 15*a^4*b^8 - 10*a^2*b^10 + b^12)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8
)*d^4))*cos(d*x + c) - sqrt(2)*(2*(a^13*b - 10*a^11*b^3 + 15*a^9*b^5 + 52*a^7*b^7 + 15*a^5*b^9 - 10*a^3*b^11 +
a*b^13)*d^3*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) + (a^10 - 13*a^8*b^2 +
50*a^6*b^4 - 50*a^4*b^6 + 13*a^2*b^8 - b^10)*d*cos(d*x + c))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 +
b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4
*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(sin(d*x + c)/cos(d*x +
c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4) + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2
*b^6 + b^8)*sin(d*x + c))/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(3/4) - sqrt
(2)*((a^22 + a^20*b^2 - 21*a^18*b^4 - 85*a^16*b^6 - 134*a^14*b^8 - 70*a^12*b^10 + 70*a^10*b^12 + 134*a^8*b^14
+ 85*a^6*b^16 + 21*a^4*b^18 - a^2*b^20 - b^22)*d^7*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a
^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4
))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*(a^17*b - 20*a^13*b^5 - 64*a^11*b^7 - 90*
a^9*b^9 - 64*a^7*b^11 - 20*a^5*b^13 + a*b^17)*d^5*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^
16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4)
))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3
*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b
^4 - 12*a^2*b^6 + b^8))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^
4))^(3/4))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)) + sqrt(2)*((a^7 + a^5*b^2 - a^3*b^4 - a*b^6)*d*
cos(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*d*cos(d*x + c)*sin(d*x + c) + (a^5*b^2 + 2*a^3*b^4 + a*b^6)*d
+ 4*((a^10*b - 2*a^6*b^5 + a^2*b^9)*d^3*cos(d*x + c)^2 + 2*(a^9*b^2 + a^7*b^4 - a^5*b^6 - a^3*b^8)*d^3*cos(d*
x + c)*sin(d*x + c) + (a^8*b^3 + a^6*b^5 - a^4*b^7 - a^2*b^9)*d^3)*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^
2*b^6 + b^8)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 -
2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a
^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4)*log((
(a^12 - 10*a^10*b^2 + 15*a^8*b^4 + 52*a^6*b^6 + 15*a^4*b^8 - 10*a^2*b^10 + b^12)*d^2*sqrt(1/((a^8 + 4*a^6*b^2
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) + sqrt(2)*(2*(a^13*b - 10*a^11*b^3 + 15*a^9*b^5 + 52*a^7*b^7
+ 15*a^5*b^9 - 10*a^3*b^11 + a*b^13)*d^3*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*
x + c) + (a^10 - 13*a^8*b^2 + 50*a^6*b^4 - 50*a^4*b^6 + 13*a^2*b^8 - b^10)*d*cos(d*x + c))*sqrt((a^8 + 4*a^6*b
^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqr
t(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)
)*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4) + (a^8 - 12*
a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*((a^7 + a^5*b^2 - a^3*b^4 - a*b
^6)*d*cos(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*d*cos(d*x + c)*sin(d*x + c) + (a^5*b^2 + 2*a^3*b^4 + a*
b^6)*d + 4*((a^10*b - 2*a^6*b^5 + a^2*b^9)*d^3*cos(d*x + c)^2 + 2*(a^9*b^2 + a^7*b^4 - a^5*b^6 - a^3*b^8)*d^3*
cos(d*x + c)*sin(d*x + c) + (a^8*b^3 + a^6*b^5 - a^4*b^7 - a^2*b^9)*d^3)*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4
+ 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*
b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8
- 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4)
*log(((a^12 - 10*a^10*b^2 + 15*a^8*b^4 + 52*a^6*b^6 + 15*a^4*b^8 - 10*a^2*b^10 + b^12)*d^2*sqrt(1/((a^8 + 4*a^
6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) - sqrt(2)*(2*(a^13*b - 10*a^11*b^3 + 15*a^9*b^5 + 52*a
^7*b^7 + 15*a^5*b^9 - 10*a^3*b^11 + a*b^13)*d^3*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*
cos(d*x + c) + (a^10 - 13*a^8*b^2 + 50*a^6*b^4 - 50*a^4*b^6 + 13*a^2*b^8 - b^10)*d*cos(d*x + c))*sqrt((a^8 + 4
*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d
^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6
+ b^8))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4) + (a^8
- 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*sin(d*x + c))/cos(d*x + c)) + (5*a^2*b^3 + b^5 + (5*a^4*b - 4*a
^2*b^3 - b^5)*cos(d*x + c)^2 + 2*(5*a^3*b^2 + a*b^4)*cos(d*x + c)*sin(d*x + c))*sqrt(-b/a)*log(-(6*a*b*cos(d*x
+ c)*sin(d*x + c) - (a^2 - b^2)*cos(d*x + c)^2 - b^2 + 4*(a^2*cos(d*x + c)^2 - a*b*cos(d*x + c)*sin(d*x + c))
*sqrt(-b/a)*sqrt(sin(d*x + c)/cos(d*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b
^2)) + 4*((a^3*b^2 + a*b^4)*cos(d*x + c)^2 + (a^2*b^3 + b^5)*cos(d*x + c)*sin(d*x + c))*sqrt(sin(d*x + c)/cos(
d*x + c)))/((a^7 + a^5*b^2 - a^3*b^4 - a*b^6)*d*cos(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*d*cos(d*x + c
)*sin(d*x + c) + (a^5*b^2 + 2*a^3*b^4 + a*b^6)*d), 1/4*(4*sqrt(2)*((a^15 + 5*a^13*b^2 + 9*a^11*b^4 + 5*a^9*b^6
- 5*a^7*b^8 - 9*a^5*b^10 - 5*a^3*b^12 - a*b^14)*d^5*cos(d*x + c)^2 + 2*(a^14*b + 6*a^12*b^3 + 15*a^10*b^5 + 2
0*a^8*b^7 + 15*a^6*b^9 + 6*a^4*b^11 + a^2*b^13)*d^5*cos(d*x + c)*sin(d*x + c) + (a^13*b^2 + 6*a^11*b^4 + 15*a^
9*b^6 + 20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12 + a*b^14)*d^5)*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b
^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*
b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt((a^8 - 12*a^6*b^2 + 38*
a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^
4*b^12 + 8*a^2*b^14 + b^16)*d^4))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(3/4)*arctan(((a^1
6 - 20*a^12*b^4 - 64*a^10*b^6 - 90*a^8*b^8 - 64*a^6*b^10 - 20*a^4*b^12 + b^16)*d^4*sqrt((a^8 - 12*a^6*b^2 + 38
*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a
^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - sqrt(2)*((a
^18 + 7*a^16*b^2 + 20*a^14*b^4 + 28*a^12*b^6 + 14*a^10*b^8 - 14*a^8*b^10 - 28*a^6*b^12 - 20*a^4*b^14 - 7*a^2*b
^16 - b^18)*d^7*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56
*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4
*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*(a^13*b + 6*a^11*b^3 + 15*a^9*b^5 + 20*a^7*b^7 + 15*a^5*b^9 + 6*a^3*b^11 + a
*b^13)*d^5*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10
*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 +
4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6
*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(((a^12 - 1
0*a^10*b^2 + 15*a^8*b^4 + 52*a^6*b^6 + 15*a^4*b^8 - 10*a^2*b^10 + b^12)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b
^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) + sqrt(2)*(2*(a^13*b - 10*a^11*b^3 + 15*a^9*b^5 + 52*a^7*b^7 + 15*a^5
*b^9 - 10*a^3*b^11 + a*b^13)*d^3*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) +
(a^10 - 13*a^8*b^2 + 50*a^6*b^4 - 50*a^4*b^6 + 13*a^2*b^8 - b^10)*d*cos(d*x + c))*sqrt((a^8 + 4*a^6*b^2 + 6*a^
4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(si
n(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4) + (a^8 - 12*a^6*b^2 +
38*a^4*b^4 - 12*a^2*b^6 + b^8)*sin(d*x + c))/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8
)*d^4))^(3/4) + sqrt(2)*((a^22 + a^20*b^2 - 21*a^18*b^4 - 85*a^16*b^6 - 134*a^14*b^8 - 70*a^12*b^10 + 70*a^10*
b^12 + 134*a^8*b^14 + 85*a^6*b^16 + 21*a^4*b^18 - a^2*b^20 - b^22)*d^7*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 1
2*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*
a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*(a^17*b - 20*a^13*b^5
- 64*a^11*b^7 - 90*a^9*b^9 - 64*a^7*b^11 - 20*a^5*b^13 + a*b^17)*d^5*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12
*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a
^2*b^14 + b^16)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5
- 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 1
2*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 +
4*a^2*b^6 + b^8)*d^4))^(3/4))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)) + 4*sqrt(2)*((a^15 + 5*a^13
*b^2 + 9*a^11*b^4 + 5*a^9*b^6 - 5*a^7*b^8 - 9*a^5*b^10 - 5*a^3*b^12 - a*b^14)*d^5*cos(d*x + c)^2 + 2*(a^14*b +
6*a^12*b^3 + 15*a^10*b^5 + 20*a^8*b^7 + 15*a^6*b^9 + 6*a^4*b^11 + a^2*b^13)*d^5*cos(d*x + c)*sin(d*x + c) + (
a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12 + a*b^14)*d^5)*sqrt((a^8 + 4*a^6*b^
2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt
(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))
*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*
a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 +
b^8)*d^4))^(3/4)*arctan(-((a^16 - 20*a^12*b^4 - 64*a^10*b^6 - 90*a^8*b^8 - 64*a^6*b^10 - 20*a^4*b^12 + b^16)*d
^4*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 7
0*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*
b^6 + b^8)*d^4)) + sqrt(2)*((a^18 + 7*a^16*b^2 + 20*a^14*b^4 + 28*a^12*b^6 + 14*a^10*b^8 - 14*a^8*b^10 - 28*a^
6*b^12 - 20*a^4*b^14 - 7*a^2*b^16 - b^18)*d^7*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 +
8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*sq
rt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + 2*(a^13*b + 6*a^11*b^3 + 15*a^9*b^5 + 20*a^7*b^7
+ 15*a^5*b^9 + 6*a^3*b^11 + a*b^13)*d^5*sqrt((a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^
14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4)))*sqrt((
a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*
b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a
^2*b^6 + b^8))*sqrt(((a^12 - 10*a^10*b^2 + 15*a^8*b^4 + 52*a^6*b^6 + 15*a^4*b^8 - 10*a^2*b^10 + b^12)*d^2*sqrt
(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) - sqrt(2)*(2*(a^13*b - 10*a^11*b^3 + 15
*a^9*b^5 + 52*a^7*b^7 + 15*a^5*b^9 - 10*a^3*b^11 + a*b^13)*d^3*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8)*d^4))*cos(d*x + c) + (a^10 - 13*a^8*b^2 + 50*a^6*b^4 - 50*a^4*b^6 + 13*a^2*b^8 - b^10)*d*cos(d*x + c)
)*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*
b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^
4 - 12*a^2*b^6 + b^8))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4
))^(1/4) + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*sin(d*x + c))/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(3/4) - sqrt(2)*((a^22 + a^20*b^2 - 21*a^18*b^4 - 85*a^16*b^6 - 134*a^14
*b^8 - 70*a^12*b^10 + 70*a^10*b^12 + 134*a^8*b^14 + 85*a^6*b^16 + 21*a^4*b^18 - a^2*b^20 - b^22)*d^7*sqrt((a^8
- 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 +
56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4))*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d
^4)) + 2*(a^17*b - 20*a^13*b^5 - 64*a^11*b^7 - 90*a^9*b^9 - 64*a^7*b^11 - 20*a^5*b^13 + a*b^17)*d^5*sqrt((a^8
- 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 5
6*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(
a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*
a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/(
(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(3/4))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8
)) + sqrt(2)*((a^7 + a^5*b^2 - a^3*b^4 - a*b^6)*d*cos(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*d*cos(d*x +
c)*sin(d*x + c) + (a^5*b^2 + 2*a^3*b^4 + a*b^6)*d + 4*((a^10*b - 2*a^6*b^5 + a^2*b^9)*d^3*cos(d*x + c)^2 + 2*
(a^9*b^2 + a^7*b^4 - a^5*b^6 - a^3*b^8)*d^3*cos(d*x + c)*sin(d*x + c) + (a^8*b^3 + a^6*b^5 - a^4*b^7 - a^2*b^9
)*d^3)*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^
2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*(1/((a^8 + 4*a^6*b^
2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4)*log(((a^12 - 10*a^10*b^2 + 15*a^8*b^4 + 52*a^6*b^6 + 15*a^4*b^8 -
10*a^2*b^10 + b^12)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) + sqrt(2)*
(2*(a^13*b - 10*a^11*b^3 + 15*a^9*b^5 + 52*a^7*b^7 + 15*a^5*b^9 - 10*a^3*b^11 + a*b^13)*d^3*sqrt(1/((a^8 + 4*a
^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) + (a^10 - 13*a^8*b^2 + 50*a^6*b^4 - 50*a^4*b^6 + 13*a
^2*b^8 - b^10)*d*cos(d*x + c))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2
*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/
(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2 + 6*a
^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4) + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*sin(d*x + c))/cos(d
*x + c)) - sqrt(2)*((a^7 + a^5*b^2 - a^3*b^4 - a*b^6)*d*cos(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*d*cos
(d*x + c)*sin(d*x + c) + (a^5*b^2 + 2*a^3*b^4 + a*b^6)*d + 4*((a^10*b - 2*a^6*b^5 + a^2*b^9)*d^3*cos(d*x + c)^
2 + 2*(a^9*b^2 + a^7*b^4 - a^5*b^6 - a^3*b^8)*d^3*cos(d*x + c)*sin(d*x + c) + (a^8*b^3 + a^6*b^5 - a^4*b^7 - a
^2*b^9)*d^3)*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4
+ 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a
^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*(1/((a^8 + 4*
a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4)*log(((a^12 - 10*a^10*b^2 + 15*a^8*b^4 + 52*a^6*b^6 + 15*a^4
*b^8 - 10*a^2*b^10 + b^12)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) - sq
rt(2)*(2*(a^13*b - 10*a^11*b^3 + 15*a^9*b^5 + 52*a^7*b^7 + 15*a^5*b^9 - 10*a^3*b^11 + a*b^13)*d^3*sqrt(1/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*cos(d*x + c) + (a^10 - 13*a^8*b^2 + 50*a^6*b^4 - 50*a^4*b^6
+ 13*a^2*b^8 - b^10)*d*cos(d*x + c))*sqrt((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8 - 4*(a^11*b + 3*a^9*b
^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^2*sqrt(1/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d
^4)))/(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^8 + 4*a^6*b^2
+ 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))^(1/4) + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*sin(d*x + c))
/cos(d*x + c)) + 4*(5*a^2*b^3 + b^5 + (5*a^4*b - 4*a^2*b^3 - b^5)*cos(d*x + c)^2 + 2*(5*a^3*b^2 + a*b^4)*cos(d
*x + c)*sin(d*x + c))*sqrt(b/a)*arctan((2*a^2*b*cos(d*x + c)^2*sin(d*x + c) + a*b^2*cos(d*x + c) + (a^3 - a*b^
2)*cos(d*x + c)^3)*sqrt(b/a)*sqrt(sin(d*x + c)/cos(d*x + c))/(2*a*b^2*cos(d*x + c)^3 - 2*a*b^2*cos(d*x + c) -
(b^3 + (a^2*b - b^3)*cos(d*x + c)^2)*sin(d*x + c))) + 4*((a^3*b^2 + a*b^4)*cos(d*x + c)^2 + (a^2*b^3 + b^5)*co
s(d*x + c)*sin(d*x + c))*sqrt(sin(d*x + c)/cos(d*x + c)))/((a^7 + a^5*b^2 - a^3*b^4 - a*b^6)*d*cos(d*x + c)^2
+ 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*d*cos(d*x + c)*sin(d*x + c) + (a^5*b^2 + 2*a^3*b^4 + a*b^6)*d)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

integrate(1/((b*tan(d*x + c) + a)^2*sqrt(tan(d*x + c))), x)

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maple [B]  time = 0.25, size = 559, normalized size = 1.76 \[ \frac {a \left (\sqrt {\tan }\left (d x +c \right )\right ) b^{2}}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{4} \left (\sqrt {\tan }\left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2} a \left (a +b \tan \left (d x +c \right )\right )}+\frac {5 a \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right ) b^{2}}{d \left (a^{2}+b^{2}\right )^{2} \sqrt {a b}}+\frac {b^{4} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right )}{d \left (a^{2}+b^{2}\right )^{2} a \sqrt {a b}}+\frac {\sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) a^{2}}{4 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) b^{2}}{4 d \left (a^{2}+b^{2}\right )^{2}}-\frac {a b \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {a b \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {a b \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{d \left (a^{2}+b^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x)

[Out]

1/d*a/(a^2+b^2)^2*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))*b^2+1/d*b^4/(a^2+b^2)^2/a*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c)
)+5/d*a/(a^2+b^2)^2/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*b^2+1/d*b^4/(a^2+b^2)^2/a/(a*b)^(1/2)*a
rctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))+1/2/d/(a^2+b^2)^2*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2-1/2/d/(
a^2+b^2)^2*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^2+1/2/d/(a^2+b^2)^2*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+
c)^(1/2))*a^2-1/2/d/(a^2+b^2)^2*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^2+1/4/d/(a^2+b^2)^2*2^(1/2)*ln((
1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^2-1/4/d/(a^2+b^2)^2*2^(1/2)*
ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*b^2-1/2/d/(a^2+b^2)^2*a*b*
2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))-1/d/(a^2+b^2)^2*a*
b*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))-1/d/(a^2+b^2)^2*a*b*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))

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maxima [A]  time = 0.45, size = 277, normalized size = 0.87 \[ \frac {\frac {4 \, b^{2} \sqrt {\tan \left (d x + c\right )}}{a^{4} + a^{2} b^{2} + {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )} + \frac {4 \, {\left (5 \, a^{2} b^{2} + b^{4}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

1/4*(4*b^2*sqrt(tan(d*x + c))/(a^4 + a^2*b^2 + (a^3*b + a*b^3)*tan(d*x + c)) + 4*(5*a^2*b^2 + b^4)*arctan(b*sq
rt(tan(d*x + c))/sqrt(a*b))/((a^5 + 2*a^3*b^2 + a*b^4)*sqrt(a*b)) + (2*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(1/2*
sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 - 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqr
t(tan(d*x + c)))) + sqrt(2)*(a^2 + 2*a*b - b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*(
a^2 + 2*a*b - b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4))/d

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mupad [B]  time = 11.26, size = 8282, normalized size = 26.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(tan(c + d*x)^(1/2)*(a + b*tan(c + d*x))^2),x)

[Out]

(log((((((((((128*b^2*(2*b^6 - a^6 + 9*a^2*b^4 + 6*a^4*b^2))/(a*d) + 256*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a
^2 + b^2)^2*(1i/(d^2*(a*1i - b)^4))^(1/2))*(1i/(d^2*(a*1i - b)^4))^(1/2))/2 + (64*b^2*tan(c + d*x)^(1/2)*(2*b^
8 - a^8 + 5*a^2*b^6 + 67*a^4*b^4 - a^6*b^2))/(a*d^2*(a^2 + b^2)^2))*(1i/(d^2*(a*1i - b)^4))^(1/2))/2 - (32*b^5
*(25*a^6 + b^6 - 13*a^2*b^4 - 85*a^4*b^2))/(a^2*d^3*(a^2 + b^2)^3))*(1i/(d^2*(a*1i - b)^4))^(1/2))/2 + (16*b^5
*tan(c + d*x)^(1/2)*(b^6 - 27*a^6 + 7*a^2*b^4 + 11*a^4*b^2))/(a^2*d^4*(a^2 + b^2)^4))*(1i/(d^2*(a*1i - b)^4))^
(1/2))/2 + (16*b^6*(5*a^2 + b^2))/(a*d^5*(a^2 + b^2)^4))*(-1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*
d^2 - a^2*b^2*d^2*6i))^(1/2))/2 - atan(((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d
^2)))^(1/2)*((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d
^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i
- a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((16*(16*a*b^16*d^4 + 136*a^3*b^14*d^4 + 432*a^5*b^12*d^4 + 680*a^7*b
^10*d^4 + 560*a^9*b^8*d^4 + 216*a^11*b^6*d^4 + 16*a^13*b^4*d^4 - 8*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*
a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) - (16*tan(c + d*x)^(1/2)*(-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4
i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^1
1*d^4 - 160*a^10*b^9*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/(a^10*d^4 + a^2*b^8*d^4 + 4
*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)) + (16*tan(c + d*x)^(1/2)*(8*a*b^14*d^2 + 36*a^3*b^12*d^2 + 316*
a^5*b^10*d^2 + 552*a^7*b^8*d^2 + 256*a^9*b^6*d^2 - 12*a^11*b^4*d^2 - 4*a^13*b^2*d^2))/(a^10*d^4 + a^2*b^8*d^4
+ 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)) + (16*(24*a^2*b^11*d^2 - 2*b^13*d^2 + 196*a^4*b^9*d^2 + 120*
a^6*b^7*d^2 - 50*a^8*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5)) + (16
*tan(c + d*x)^(1/2)*(b^11 + 7*a^2*b^9 + 11*a^4*b^7 - 27*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*
a^6*b^4*d^4 + 4*a^8*b^2*d^4))*1i - (-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))
^(1/2)*((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d^2 +
b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^
3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((16*(16*a*b^16*d^4 + 136*a^3*b^14*d^4 + 432*a^5*b^12*d^4 + 680*a^7*b^10*d
^4 + 560*a^9*b^8*d^4 + 216*a^11*b^6*d^4 + 16*a^13*b^4*d^4 - 8*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b
^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) + (16*tan(c + d*x)^(1/2)*(-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a
^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d^4
- 160*a^10*b^9*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*
b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)) - (16*tan(c + d*x)^(1/2)*(8*a*b^14*d^2 + 36*a^3*b^12*d^2 + 316*a^5*b
^10*d^2 + 552*a^7*b^8*d^2 + 256*a^9*b^6*d^2 - 12*a^11*b^4*d^2 - 4*a^13*b^2*d^2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a
^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)) + (16*(24*a^2*b^11*d^2 - 2*b^13*d^2 + 196*a^4*b^9*d^2 + 120*a^6*b
^7*d^2 - 50*a^8*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5)) - (16*tan(
c + d*x)^(1/2)*(b^11 + 7*a^2*b^9 + 11*a^4*b^7 - 27*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b
^4*d^4 + 4*a^8*b^2*d^4))*1i)/((32*(a*b^8 + 5*a^3*b^6))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5
+ 4*a^8*b^2*d^5) + (-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4
*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3
*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*
a^2*b^2*d^2)))^(1/2)*((16*(16*a*b^16*d^4 + 136*a^3*b^14*d^4 + 432*a^5*b^12*d^4 + 680*a^7*b^10*d^4 + 560*a^9*b^
8*d^4 + 216*a^11*b^6*d^4 + 16*a^13*b^4*d^4 - 8*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*
b^4*d^5 + 4*a^8*b^2*d^5) - (16*tan(c + d*x)^(1/2)*(-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6
*a^2*b^2*d^2)))^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d^4 - 160*a^10*b^9
*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6
*b^4*d^4 + 4*a^8*b^2*d^4)) + (16*tan(c + d*x)^(1/2)*(8*a*b^14*d^2 + 36*a^3*b^12*d^2 + 316*a^5*b^10*d^2 + 552*a
^7*b^8*d^2 + 256*a^9*b^6*d^2 - 12*a^11*b^4*d^2 - 4*a^13*b^2*d^2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*
a^6*b^4*d^4 + 4*a^8*b^2*d^4)) + (16*(24*a^2*b^11*d^2 - 2*b^13*d^2 + 196*a^4*b^9*d^2 + 120*a^6*b^7*d^2 - 50*a^8
*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5)) + (16*tan(c + d*x)^(1/2)*
(b^11 + 7*a^2*b^9 + 11*a^4*b^7 - 27*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*
b^2*d^4)) + (-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d^
2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i
- a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*
d^2)))^(1/2)*((16*(16*a*b^16*d^4 + 136*a^3*b^14*d^4 + 432*a^5*b^12*d^4 + 680*a^7*b^10*d^4 + 560*a^9*b^8*d^4 +
216*a^11*b^6*d^4 + 16*a^13*b^4*d^4 - 8*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5
+ 4*a^8*b^2*d^5) + (16*tan(c + d*x)^(1/2)*(-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2
*d^2)))^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d^4 - 160*a^10*b^9*d^4 - 2
88*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4
+ 4*a^8*b^2*d^4)) - (16*tan(c + d*x)^(1/2)*(8*a*b^14*d^2 + 36*a^3*b^12*d^2 + 316*a^5*b^10*d^2 + 552*a^7*b^8*d
^2 + 256*a^9*b^6*d^2 - 12*a^11*b^4*d^2 - 4*a^13*b^2*d^2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*
d^4 + 4*a^8*b^2*d^4)) + (16*(24*a^2*b^11*d^2 - 2*b^13*d^2 + 196*a^4*b^9*d^2 + 120*a^6*b^7*d^2 - 50*a^8*b^5*d^2
))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5)) - (16*tan(c + d*x)^(1/2)*(b^11 +
7*a^2*b^9 + 11*a^4*b^7 - 27*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)
)))*(-1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*2i - log((((((((((128*b^
2*(2*b^6 - a^6 + 9*a^2*b^4 + 6*a^4*b^2))/(a*d) - 256*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(1i/(d^2
*(a*1i - b)^4))^(1/2))*(1i/(d^2*(a*1i - b)^4))^(1/2))/2 - (64*b^2*tan(c + d*x)^(1/2)*(2*b^8 - a^8 + 5*a^2*b^6
+ 67*a^4*b^4 - a^6*b^2))/(a*d^2*(a^2 + b^2)^2))*(1i/(d^2*(a*1i - b)^4))^(1/2))/2 - (32*b^5*(25*a^6 + b^6 - 13*
a^2*b^4 - 85*a^4*b^2))/(a^2*d^3*(a^2 + b^2)^3))*(1i/(d^2*(a*1i - b)^4))^(1/2))/2 - (16*b^5*tan(c + d*x)^(1/2)*
(b^6 - 27*a^6 + 7*a^2*b^4 + 11*a^4*b^2))/(a^2*d^4*(a^2 + b^2)^4))*(1i/(d^2*(a*1i - b)^4))^(1/2))/2 + (16*b^6*(
5*a^2 + b^2))/(a*d^5*(a^2 + b^2)^4))*(-1/(4*(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2
*6i)))^(1/2) - (atan((((5*a^2 + b^2)*(-a^3*b^3)^(1/2)*((16*tan(c + d*x)^(1/2)*(b^11 + 7*a^2*b^9 + 11*a^4*b^7 -
27*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) + ((5*a^2 + b^2)*((16*(
24*a^2*b^11*d^2 - 2*b^13*d^2 + 196*a^4*b^9*d^2 + 120*a^6*b^7*d^2 - 50*a^8*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 +
4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) + ((5*a^2 + b^2)*((16*tan(c + d*x)^(1/2)*(8*a*b^14*d^2 + 36*a^3
*b^12*d^2 + 316*a^5*b^10*d^2 + 552*a^7*b^8*d^2 + 256*a^9*b^6*d^2 - 12*a^11*b^4*d^2 - 4*a^13*b^2*d^2))/(a^10*d^
4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) + ((5*a^2 + b^2)*((16*(16*a*b^16*d^4 + 136*a^
3*b^14*d^4 + 432*a^5*b^12*d^4 + 680*a^7*b^10*d^4 + 560*a^9*b^8*d^4 + 216*a^11*b^6*d^4 + 16*a^13*b^4*d^4 - 8*a^
15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) - (8*tan(c + d*x)^(1/2)*
(5*a^2 + b^2)*(-a^3*b^3)^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d^4 - 160
*a^10*b^9*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/((a^7*d + a^3*b^4*d + 2*a^5*b^2*d)*(a^
10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*b^4*
d + 2*a^5*b^2*d)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*
b^4*d + 2*a^5*b^2*d)))*1i)/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d)) + ((5*a^2 + b^2)*(-a^3*b^3)^(1/2)*((16*tan(c
+ d*x)^(1/2)*(b^11 + 7*a^2*b^9 + 11*a^4*b^7 - 27*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4
*d^4 + 4*a^8*b^2*d^4) - ((5*a^2 + b^2)*((16*(24*a^2*b^11*d^2 - 2*b^13*d^2 + 196*a^4*b^9*d^2 + 120*a^6*b^7*d^2
- 50*a^8*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) - ((5*a^2 + b^2)*(
(16*tan(c + d*x)^(1/2)*(8*a*b^14*d^2 + 36*a^3*b^12*d^2 + 316*a^5*b^10*d^2 + 552*a^7*b^8*d^2 + 256*a^9*b^6*d^2
- 12*a^11*b^4*d^2 - 4*a^13*b^2*d^2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)
- ((5*a^2 + b^2)*((16*(16*a*b^16*d^4 + 136*a^3*b^14*d^4 + 432*a^5*b^12*d^4 + 680*a^7*b^10*d^4 + 560*a^9*b^8*d^
4 + 216*a^11*b^6*d^4 + 16*a^13*b^4*d^4 - 8*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*
d^5 + 4*a^8*b^2*d^5) + (8*tan(c + d*x)^(1/2)*(5*a^2 + b^2)*(-a^3*b^3)^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^
4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d^4 - 160*a^10*b^9*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3
*d^4))/((a^7*d + a^3*b^4*d + 2*a^5*b^2*d)*(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*
d^4)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*b^4*d + 2*a^
5*b^2*d)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d)))*1i)/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d)))
/((32*(a*b^8 + 5*a^3*b^6))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) + ((5*a^2
+ b^2)*(-a^3*b^3)^(1/2)*((16*tan(c + d*x)^(1/2)*(b^11 + 7*a^2*b^9 + 11*a^4*b^7 - 27*a^6*b^5))/(a^10*d^4 + a^2*
b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) + ((5*a^2 + b^2)*((16*(24*a^2*b^11*d^2 - 2*b^13*d^2 +
196*a^4*b^9*d^2 + 120*a^6*b^7*d^2 - 50*a^8*b^5*d^2))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5
+ 4*a^8*b^2*d^5) + ((5*a^2 + b^2)*((16*tan(c + d*x)^(1/2)*(8*a*b^14*d^2 + 36*a^3*b^12*d^2 + 316*a^5*b^10*d^2 +
552*a^7*b^8*d^2 + 256*a^9*b^6*d^2 - 12*a^11*b^4*d^2 - 4*a^13*b^2*d^2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^
4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) + ((5*a^2 + b^2)*((16*(16*a*b^16*d^4 + 136*a^3*b^14*d^4 + 432*a^5*b^12*d^4
+ 680*a^7*b^10*d^4 + 560*a^9*b^8*d^4 + 216*a^11*b^6*d^4 + 16*a^13*b^4*d^4 - 8*a^15*b^2*d^4))/(a^10*d^5 + a^2*b
^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) - (8*tan(c + d*x)^(1/2)*(5*a^2 + b^2)*(-a^3*b^3)^(1/2)
*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^11*d^4 - 160*a^10*b^9*d^4 - 288*a^12*b^7*d
^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/((a^7*d + a^3*b^4*d + 2*a^5*b^2*d)*(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b
^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d)))*(-a^3*b^3)^
(1/2))/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d))))/(2*(a^
7*d + a^3*b^4*d + 2*a^5*b^2*d)) - ((5*a^2 + b^2)*(-a^3*b^3)^(1/2)*((16*tan(c + d*x)^(1/2)*(b^11 + 7*a^2*b^9 +
11*a^4*b^7 - 27*a^6*b^5))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) - ((5*a^2 +
b^2)*((16*(24*a^2*b^11*d^2 - 2*b^13*d^2 + 196*a^4*b^9*d^2 + 120*a^6*b^7*d^2 - 50*a^8*b^5*d^2))/(a^10*d^5 + a^
2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) - ((5*a^2 + b^2)*((16*tan(c + d*x)^(1/2)*(8*a*b^14*
d^2 + 36*a^3*b^12*d^2 + 316*a^5*b^10*d^2 + 552*a^7*b^8*d^2 + 256*a^9*b^6*d^2 - 12*a^11*b^4*d^2 - 4*a^13*b^2*d^
2))/(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4) - ((5*a^2 + b^2)*((16*(16*a*b^16*
d^4 + 136*a^3*b^14*d^4 + 432*a^5*b^12*d^4 + 680*a^7*b^10*d^4 + 560*a^9*b^8*d^4 + 216*a^11*b^6*d^4 + 16*a^13*b^
4*d^4 - 8*a^15*b^2*d^4))/(a^10*d^5 + a^2*b^8*d^5 + 4*a^4*b^6*d^5 + 6*a^6*b^4*d^5 + 4*a^8*b^2*d^5) + (8*tan(c +
d*x)^(1/2)*(5*a^2 + b^2)*(-a^3*b^3)^(1/2)*(32*a^2*b^17*d^4 + 160*a^4*b^15*d^4 + 288*a^6*b^13*d^4 + 160*a^8*b^
11*d^4 - 160*a^10*b^9*d^4 - 288*a^12*b^7*d^4 - 160*a^14*b^5*d^4 - 32*a^16*b^3*d^4))/((a^7*d + a^3*b^4*d + 2*a^
5*b^2*d)*(a^10*d^4 + a^2*b^8*d^4 + 4*a^4*b^6*d^4 + 6*a^6*b^4*d^4 + 4*a^8*b^2*d^4)))*(-a^3*b^3)^(1/2))/(2*(a^7*
d + a^3*b^4*d + 2*a^5*b^2*d)))*(-a^3*b^3)^(1/2))/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d)))*(-a^3*b^3)^(1/2))/(2*(
a^7*d + a^3*b^4*d + 2*a^5*b^2*d))))/(2*(a^7*d + a^3*b^4*d + 2*a^5*b^2*d))))*(5*a^2 + b^2)*(-a^3*b^3)^(1/2)*1i)
/(a^7*d + a^3*b^4*d + 2*a^5*b^2*d) + (b^2*tan(c + d*x)^(1/2))/(a*(a*d + b*d*tan(c + d*x))*(a^2 + b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{2} \sqrt {\tan {\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)**(1/2)/(a+b*tan(d*x+c))**2,x)

[Out]

Integral(1/((a + b*tan(c + d*x))**2*sqrt(tan(c + d*x))), x)

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