3.407 \(\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=391 \[ \frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\sqrt {b} \left (-3 a^3 B+5 a^2 A b+a b^2 B+A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d \left (a^2+b^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.87, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3609, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\sqrt {b} \left (5 a^2 A b-3 a^3 B+a b^2 B+A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d \left (a^2+b^2\right )^2}+\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)
- ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2
*d) + (Sqrt[b]*(5*a^2*A*b + A*b^3 - 3*a^3*B + a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*
(a^2 + b^2)^2*d) - ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d
*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c +
d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + (b*(A*b - a*B)*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 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n
+ 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(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 {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx &=\frac {b (A b-a B) \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 A+A b^2+a b B\right )-a (A b-a B) \tan (c+d x)+\frac {1}{2} b (A b-a B) \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 (A b-a B) \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 A-A b^2+2 a b B\right )-a \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )^2}+\frac {\left (b \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\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 (A b-a B) \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 A-A b^2+2 a b B\right )-a \left (2 a A b-a^2 B+b^2 B\right ) 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 \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\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 (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (b \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\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 {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\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 (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\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 {\sqrt {b} \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\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 (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\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 (A-B)-b^2 (A-B)+2 a b (A+B)\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 (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\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 (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\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 {\sqrt {b} \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\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 (A-B)-b^2 (A-B)+2 a b (A+B)\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 (A-B)-b^2 (A-B)+2 a b (A+B)\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 (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\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 (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\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 (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\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 (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\sqrt {b} \left (5 a^2 A b+A b^3-3 a^3 B+a b^2 B\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 (A-B)-b^2 (A-B)+2 a b (A+B)\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 (A-B)-b^2 (A-B)+2 a b (A+B)\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 (A b-a B) \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 = 1.15, size = 204, normalized size = 0.52 \[ \frac {\frac {\sqrt [4]{-1} \left (-a (a+i b)^2 (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-a (a-i b)^2 (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {\sqrt {b} \left (-3 a^3 B+5 a^2 A b+a b^2 B+A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )}+\frac {b (A b-a B) \sqrt {\tan (c+d x)}}{a+b \tan (c+d x)}}{a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.37, size = 1136, normalized size = 2.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.93, size = 356, normalized size = 0.91 \[ -\frac {\frac {4 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2} - B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a b}} + \frac {4 \, {\left (B a b - A b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{4} + a^{2} b^{2} + {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )} - \frac {2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(3*B*a^3*b - 5*A*a^2*b^2 - B*a*b^3 - A*b^4)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^5 + 2*a^3*b^2 +
a*b^4)*sqrt(a*b)) + 4*(B*a*b - A*b^2)*sqrt(tan(d*x + c))/(a^4 + a^2*b^2 + (a^3*b + a*b^3)*tan(d*x + c)) - (2*
sqrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*s
qrt(2)*((A + B)*a^2 - 2*(A - B)*a*b - (A + B)*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqr
t(2)*((A - B)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*
((A - B)*a^2 + 2*(A + B)*a*b - (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 = 33.53, size = 17494, normalized size = 44.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(- (((((((((256*B*b^3*(2*a^4 - b^4 + a^2*b^2))/d - 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((
4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2)
)*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(
1/2))/4 - (64*B^2*a*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*b^2))/(d^2*(a^2 + b^2)^2))*((4*
(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/
4 + (32*B^3*a*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*b^2))/(d^3*(a^2 + b^2)^3))*((4*(-B^4*d^4*(a^4 + b^4 - 6*
a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*b^3*tan(c + d
*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)
^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (8*B^5*b^3*(9*a^4 - b^4))/(d^
5*(a^2 + b^2)^4))*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6
*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*
a^6*b^2*d^4))^(1/2))/4 + (log(- (((((((((256*B*b^3*(2*a^4 - b^4 + a^2*b^2))/d - 128*b^3*tan(c + d*x)^(1/2)*(a^
2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/
(d^4*(a^2 + b^2)^4))^(1/2))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*
d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (64*B^2*a*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*b^2)
)/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/
(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*b^2))/(d^3*(a^2 + b^2)^3))*(
-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/
2))/4 - (16*B^4*b^3*tan(c + d*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(d^4*(a^2 + b^2)^4))*(-(4*(-B
^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 -
(8*B^5*b^3*(9*a^4 - b^4))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 60
8*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a
^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log(- (((((((((256*B*b^3*(2*a^4 - b^4 + a^2*b^2))/d +
128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a
*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B
^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^
6 - 29*a^2*b^4 + 19*a^4*b^2))/(d^2*(a^2 + b^2)^2))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b
^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*b
^2))/(d^3*(a^2 + b^2)^3))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2
)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*b^3*tan(c + d*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(d^
4*(a^2 + b^2)^4))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(
a^2 + b^2)^4))^(1/2))/4 - (8*B^5*b^3*(9*a^4 - b^4))/(d^5*(a^2 + b^2)^4))*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d
^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/
(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - log(- (((((((((256*B*b^3
*(2*a^4 - b^4 + a^2*b^2))/d + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^4 + b^4 -
6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))*(-(4*(-B^4*d^4*(a^4 +
b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*b^2
*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*d^4*(a^4 + b^4 -
6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a*b^2*(a^6
+ 13*b^6 - 45*a^2*b^4 + 39*a^4*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 1
6*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*b^3*tan(c + d*x)^(1/2)*(9*a^6 - 3*
b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(d^4*(a^2 + b^2)^4))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a
*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (8*B^5*b^3*(9*a^4 - b^4))/(d^5*(a^2 + b^2)^4))*(-
((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) - 1
6*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^
4))^(1/2) + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2
*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*A*b^2*(2*b^6 - a^6 + 9
*a^2*b^4 + 6*a^4*b^2))/(a*d))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b
*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*A^2*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))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3
*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*A^3*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))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2
)^4))^(1/2))/4 + (16*A^4*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))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))
^(1/2))/4 + (16*A^5*b^6*(5*a^2 + b^2))/(a*d^5*(a^2 + b^2)^4))*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4
*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(a^8*d^4 +
b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(
a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2
)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*A*b^2*(2*b^6 - a^6 + 9*a^2*b^4 + 6*a^4*b^2))/(a*d))*(-(4*(-A^4*d^4*(a^4 +
b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*A^2*b^2*t
an(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))*(-(4*(-A^4*d^4*(a^4
+ b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*A^3*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))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)
^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^4*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))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/
2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^5*b^6*(5*a^2 + b^2))/(a*d^5*(a
^2 + b^2)^4))*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^
2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6
*b^2*d^4))^(1/2))/4 - log((16*A^5*b^6*(5*a^2 + b^2))/(a*d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2
)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d
^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*A*b^2*(2*b^6 - a^6 + 9*a^2*b^4 + 6*a^4*b^2))/(a*d))*((4*(-A^4*d^4*(a^4 +
b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*A^2*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))*((4*(-A^4*d^4*(a^4
+ b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*A^3*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))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^
(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^4*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))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2)
- 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4
- 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(1
6*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - log((16*A^5*b^6*(5*a^2 + b
^2))/(a*d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^4
+ b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*A*b^2*(2*
b^6 - a^6 + 9*a^2*b^4 + 6*a^4*b^2))/(a*d))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2
- 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*A^2*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))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d
^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*A^3*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))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)
/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^4*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))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^
4*(a^2 + b^2)^4))^(1/2))/4)*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 +
192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 +
96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) + (atan(((((((8*(160*B^3*a^7*b^4*d^2 - 24*B^3*a^5*b^6*d^2 - 128*B^3*a^
3*b^8*d^2 + 4*B^3*a^9*b^2*d^2 + 52*B^3*a*b^10*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6
*b^2*d^5) + (((16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^10*d^2 - 88*B^2*a^5*b^8*d^2 + 40*B^2*a^7*b^6*d^2 + 84*B^2*a
^9*b^4*d^2 + 4*B^2*a^11*b^2*d^2 + 68*B^2*a*b^12*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a
^6*b^2*d^4) + (((8*(320*B*a^6*b^9*d^4 - 96*B*a^2*b^13*d^4 - 32*B*b^15*d^4 + 480*B*a^8*b^7*d^4 + 288*B*a^10*b^5
*d^4 + 64*B*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (4*tan(c + d*
x)^(1/2)*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*
a^7*b^2*d^2))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 -
288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 +
4*a^6*b^2*d^4)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^
4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2
+ a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a
^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^
6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*
a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 +
4*a^7*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(3*B^4*b^9 - 3*B^4*a^2*b^7 + 17*B^4*a^4*b^5 - 9*B^4*a^6*b^3))/(a^8*d
^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^
9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*1i)/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*
b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)) - (((((8*(160*B^3*a^7*b^4*d^2 - 24*B^3*a^5*b^6*d^2 - 128*B^3*a^3*b^8
*d^2 + 4*B^3*a^9*b^2*d^2 + 52*B^3*a*b^10*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*
d^5) - (((16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^10*d^2 - 88*B^2*a^5*b^8*d^2 + 40*B^2*a^7*b^6*d^2 + 84*B^2*a^9*b^
4*d^2 + 4*B^2*a^11*b^2*d^2 + 68*B^2*a*b^12*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^
2*d^4) - (((8*(320*B*a^6*b^9*d^4 - 96*B*a^2*b^13*d^4 - 32*B*b^15*d^4 + 480*B*a^8*b^7*d^4 + 288*B*a^10*b^5*d^4
+ 64*B*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (4*tan(c + d*x)^(1
/2)*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b
^2*d^2))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a
^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6
*b^2*d^4)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b -
6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*
b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^
2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2
+ 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b
^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^
7*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*(3*B^4*b^9 - 3*B^4*a^2*b^7 + 17*B^4*a^4*b^5 - 9*B^4*a^6*b^3))/(a^8*d^4 +
b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2
+ a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*1i)/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d
^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))/((((((8*(160*B^3*a^7*b^4*d^2 - 24*B^3*a^5*b^6*d^2 - 128*B^3*a^3*b^8*d^2
+ 4*B^3*a^9*b^2*d^2 + 52*B^3*a*b^10*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5)
+ (((16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^10*d^2 - 88*B^2*a^5*b^8*d^2 + 40*B^2*a^7*b^6*d^2 + 84*B^2*a^9*b^4*d^2
+ 4*B^2*a^11*b^2*d^2 + 68*B^2*a*b^12*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4
) + (((8*(320*B*a^6*b^9*d^4 - 96*B*a^2*b^13*d^4 - 32*B*b^15*d^4 + 480*B*a^8*b^7*d^4 + 288*B*a^10*b^5*d^4 + 64*
B*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (4*tan(c + d*x)^(1/2)*(
-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^
2))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b
^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*
d^4)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^
2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d
^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a
*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*
a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^
2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2
*d^2)) - (16*tan(c + d*x)^(1/2)*(3*B^4*b^9 - 3*B^4*a^2*b^7 + 17*B^4*a^4*b^5 - 9*B^4*a^6*b^3))/(a^8*d^4 + b^8*d
^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*
b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a
^5*b^4*d^2 + 4*a^7*b^2*d^2)) - (16*(B^5*b^7 - 9*B^5*a^4*b^3))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d
^5 + 4*a^6*b^2*d^5) + (((((8*(160*B^3*a^7*b^4*d^2 - 24*B^3*a^5*b^6*d^2 - 128*B^3*a^3*b^8*d^2 + 4*B^3*a^9*b^2*d
^2 + 52*B^3*a*b^10*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d
*x)^(1/2)*(20*B^2*a^3*b^10*d^2 - 88*B^2*a^5*b^8*d^2 + 40*B^2*a^7*b^6*d^2 + 84*B^2*a^9*b^4*d^2 + 4*B^2*a^11*b^2
*d^2 + 68*B^2*a*b^12*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((8*(320*B*a
^6*b^9*d^4 - 96*B*a^2*b^13*d^4 - 32*B*b^15*d^4 + 480*B*a^8*b^7*d^4 + 288*B*a^10*b^5*d^4 + 64*B*a^12*b^3*d^4))/
(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (4*tan(c + d*x)^(1/2)*(-4*(B^2*b^5 + 9*B
^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*(32*b^1
7*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12
*b^5*d^4 - 32*a^14*b^3*d^4))/((a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)*(a^9*d^2 + a
*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d
^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^
2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*
b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a
^7*b^2*d^2)))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2
+ 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)) + (16*tan(
c + d*x)^(1/2)*(3*B^4*b^9 - 3*B^4*a^2*b^7 + 17*B^4*a^4*b^5 - 9*B^4*a^6*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^
4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b
^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^
7*b^2*d^2))))*(-4*(B^2*b^5 + 9*B^2*a^4*b - 6*B^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2
+ 4*a^7*b^2*d^2))^(1/2)*1i)/(2*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)) - (atan
(((((16*tan(c + d*x)^(1/2)*(A^4*b^11 + 7*A^4*a^2*b^9 + 11*A^4*a^4*b^7 - 27*A^4*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) + (((16*(24*A^3*a^2*b^11*d^2 - 2*A^3*b^13*d^2 + 196*A^3*a^
4*b^9*d^2 + 120*A^3*a^6*b^7*d^2 - 50*A^3*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*(16*A*a*b^16*d^4 + 136*A*a^3*b^14*d^4 + 432*A*a^5*b^12*d^4 + 680*A*a^7*b^10*d^4 +
560*A*a^9*b^8*d^4 + 216*A*a^11*b^6*d^4 + 16*A*a^13*b^4*d^4 - 8*A*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) - (4*tan(c + d*x)^(1/2)*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*
b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*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^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*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)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d
^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b
^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*(36*A^2*a^3*b^12*d^2 + 316*A^2*a^5*b^10*d^2
+ 552*A^2*a^7*b^8*d^2 + 256*A^2*a^9*b^6*d^2 - 12*A^2*a^11*b^4*d^2 - 4*A^2*a^13*b^2*d^2 + 8*A^2*a*b^14*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))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2
*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*
b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^
11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a
^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*
b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2)*1i)/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2
+ 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)) + (((16*tan(c + d*x)^(1/2)*(A^4*b^11 + 7*A^4*a^2*b^9 + 11*A^4*a^4*b^7 - 27*
A^4*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) - (((16*(24*A^3*a^2*b^1
1*d^2 - 2*A^3*b^13*d^2 + 196*A^3*a^4*b^9*d^2 + 120*A^3*a^6*b^7*d^2 - 50*A^3*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*(16*A*a*b^16*d^4 + 136*A*a^3*b^14*d^4 + 432*A*a
^5*b^12*d^4 + 680*A*a^7*b^10*d^4 + 560*A*a^9*b^8*d^4 + 216*A*a^11*b^6*d^4 + 16*A*a^13*b^4*d^4 - 8*A*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) + (4*tan(c + d*x)^(1/2)*(-4*(A^2
*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*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^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 +
4*a^9*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)))*(-4*(A^2*b^7 + 10*A^
2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(
4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(36*A^2*a
^3*b^12*d^2 + 316*A^2*a^5*b^10*d^2 + 552*A^2*a^7*b^8*d^2 + 256*A^2*a^9*b^6*d^2 - 12*A^2*a^11*b^4*d^2 - 4*A^2*a
^13*b^2*d^2 + 8*A^2*a*b^14*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))*(-4
*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b
^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(A^2*b^7 + 1
0*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2
))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5
+ 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2)*1i)/(4*(a^1
1*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))/((32*(A^5*a*b^8 + 5*A^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) + (((16*tan(c + d*x)^(1/2)*(A^4*b^11 + 7*A
^4*a^2*b^9 + 11*A^4*a^4*b^7 - 27*A^4*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) + (((16*(24*A^3*a^2*b^11*d^2 - 2*A^3*b^13*d^2 + 196*A^3*a^4*b^9*d^2 + 120*A^3*a^6*b^7*d^2 - 50*A^3*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*(16*A*a*b^16*d
^4 + 136*A*a^3*b^14*d^4 + 432*A*a^5*b^12*d^4 + 680*A*a^7*b^10*d^4 + 560*A*a^9*b^8*d^4 + 216*A*a^11*b^6*d^4 + 1
6*A*a^13*b^4*d^4 - 8*A*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)
- (4*tan(c + d*x)^(1/2)*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d
^2 + 6*a^7*b^4*d^2 + 4*a^9*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^11*d^2 + a^3*b^8*d^2 +
4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*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)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b
^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)) +
(16*tan(c + d*x)^(1/2)*(36*A^2*a^3*b^12*d^2 + 316*A^2*a^5*b^10*d^2 + 552*A^2*a^7*b^8*d^2 + 256*A^2*a^9*b^6*d^
2 - 12*A^2*a^11*b^4*d^2 - 4*A^2*a^13*b^2*d^2 + 8*A^2*a*b^14*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))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*
b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 +
4*a^9*b^2*d^2)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a
^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2
)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 +
4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)) - (((16*ta
n(c + d*x)^(1/2)*(A^4*b^11 + 7*A^4*a^2*b^9 + 11*A^4*a^4*b^7 - 27*A^4*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) - (((16*(24*A^3*a^2*b^11*d^2 - 2*A^3*b^13*d^2 + 196*A^3*a^4*b^9*d^2
+ 120*A^3*a^6*b^7*d^2 - 50*A^3*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*(16*A*a*b^16*d^4 + 136*A*a^3*b^14*d^4 + 432*A*a^5*b^12*d^4 + 680*A*a^7*b^10*d^4 + 560*A*a^9*
b^8*d^4 + 216*A*a^11*b^6*d^4 + 16*A*a^13*b^4*d^4 - 8*A*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) + (4*tan(c + d*x)^(1/2)*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11
*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*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^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*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)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b
^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6
*a^7*b^4*d^2 + 4*a^9*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(36*A^2*a^3*b^12*d^2 + 316*A^2*a^5*b^10*d^2 + 552*A^2*
a^7*b^8*d^2 + 256*A^2*a^9*b^6*d^2 - 12*A^2*a^11*b^4*d^2 - 4*A^2*a^13*b^2*d^2 + 8*A^2*a*b^14*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))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*
(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 +
4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a
^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2
+ 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2)))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 +
4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2))/(4*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*
d^2 + 4*a^9*b^2*d^2))))*(-4*(A^2*b^7 + 10*A^2*a^2*b^5 + 25*A^2*a^4*b^3)*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^
2 + 6*a^7*b^4*d^2 + 4*a^9*b^2*d^2))^(1/2)*1i)/(2*(a^11*d^2 + a^3*b^8*d^2 + 4*a^5*b^6*d^2 + 6*a^7*b^4*d^2 + 4*a
^9*b^2*d^2)) - (B*b*tan(c + d*x)^(1/2))/((a*d + b*d*tan(c + d*x))*(a^2 + b^2)) + (A*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(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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