3.588 \(\int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx\)

Optimal. Leaf size=250 \[ \frac {(a+b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a+b) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a-b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {(a-b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d \left (a^2+b^2\right )}-\frac {2}{a d \sqrt {\tan (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.41, antiderivative size = 250, 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} \[ -\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d \left (a^2+b^2\right )}+\frac {(a+b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a+b) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a-b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {(a-b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {2}{a d \sqrt {\tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]

[Out]

((a + b)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a + b)*ArcTan[1 + Sqrt[2]*Sqrt[Ta
n[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2
+ b^2)*d) - ((a - b)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) + ((a - b)
*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - 2/(a*d*Sqrt[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}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx &=-\frac {2}{a d \sqrt {\tan (c+d x)}}-\frac {2 \int \frac {\frac {b}{2}+\frac {1}{2} a \tan (c+d x)+\frac {1}{2} b \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a}\\ &=-\frac {2}{a d \sqrt {\tan (c+d x)}}-\frac {2 \int \frac {\frac {a b}{2}+\frac {1}{2} a^2 \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {1+\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 {2}{a d \sqrt {\tan (c+d x)}}-\frac {4 \operatorname {Subst}\left (\int \frac {\frac {a b}{2}+\frac {a^2 x^2}{2}}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a \left (a^2+b^2\right ) d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{a \left (a^2+b^2\right ) d}\\ &=-\frac {2}{a d \sqrt {\tan (c+d x)}}+\frac {(a-b) \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 ) d}-\frac {\left (2 b^3\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 ) d}-\frac {(a+b) \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 ) d}\\ &=-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right ) d}-\frac {2}{a d \sqrt {\tan (c+d x)}}-\frac {(a-b) \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 ) d}-\frac {(a-b) \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 ) d}-\frac {(a+b) \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 ) d}-\frac {(a+b) \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 ) d}\\ &=-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right ) d}-\frac {(a-b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2}{a d \sqrt {\tan (c+d x)}}-\frac {(a+b) \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 ) d}+\frac {(a+b) \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 ) d}\\ &=\frac {(a+b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right ) d}-\frac {(a-b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a-b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2}{a d \sqrt {\tan (c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 0.52, size = 131, normalized size = 0.52 \[ \frac {-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2 \left (a^2+b^2\right )}{a \sqrt {\tan (c+d x)}}-(-1)^{3/4} (a+i b) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+\sqrt [4]{-1} (b+i a) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]

[Out]

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

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fricas [B]  time = 10.89, size = 7748, normalized size = 30.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(2)*((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*cos(d*x + c)^2 - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b
^6)*d^5)*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))
)/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))
*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(-((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^
2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2
)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^
4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d
^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2
+ b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt
(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5
*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*c
os(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d
^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4
- 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((a^10*b + 3*
a^8*b^3 + 2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4
*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^
5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 +
b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(
sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4)) + 4*sqrt(2)*((a^7
+ 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*cos(d*x + c)^2 - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5)*sqrt((a^4 +
2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 +
b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4))^(3/4)*arctan(((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6
*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((a^8*b + 4*a^6*b^3
+ 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^
8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*sqrt((a^4 - 2*a^2*b
^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a
^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2
*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6
)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4
+ 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2
+ b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin
(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((a^10*b + 3*a^8*b^3 + 2*a^6*b^5 -
2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8
)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((a^4 - 2*a^2*b^
2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^
3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x +
c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4)) + 8*(a^2 + b^2)*sqrt(sin(d*x + c)/cos(d
*x + c))*cos(d*x + c)*sin(d*x + c) + sqrt(2)*((a^3 + a*b^2)*d*cos(d*x + c)^2 - (a^3 + a*b^2)*d - 2*((a^4*b + a
^2*b^3)*d^3*cos(d*x + c)^2 - (a^4*b + a^2*b^3)*d^3)*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a^4 + 2*a^2*b
^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(
1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4
)*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*co
s(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*
b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((
a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*((a^3 + a*b
^2)*d*cos(d*x + c)^2 - (a^3 + a*b^2)*d - 2*((a^4*b + a^2*b^3)*d^3*cos(d*x + c)^2 - (a^4*b + a^2*b^3)*d^3)*sqrt
(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^
4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((a^6 - a^4*b
^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4
+ a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*s
qrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2
*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 +
b^4)*sin(d*x + c))/cos(d*x + c)) + 2*(b^2*cos(d*x + c)^2 - b^2)*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)))/((a^3 + a*
b^2)*d*cos(d*x + c)^2 - (a^3 + a*b^2)*d), 1/4*(4*sqrt(2)*((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*cos(d*x +
c)^2 - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5)*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*
d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6
*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(-((a^8 + 2*a^6*b^2 -
2*a^2*b^6 - b^8)*d^4*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(
1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt((a^4 -
2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))
- (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2
*b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)
*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos
(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b
^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a
^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4))^(3/4) - sqrt(2)*((a^10*b + 3*a^8*b^3 + 2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7*sqrt((a^4 - 2
*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) -
(a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*
b^6 + b^8)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b
^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(
a^4 - 2*a^2*b^2 + b^4)) + 4*sqrt(2)*((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^5*cos(d*x + c)^2 - (a^7 + 3*a^5*b
^2 + 3*a^3*b^4 + a*b^6)*d^5)*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*
a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*
a^2*b^6 + b^8)*d^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*arctan(((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^4*s
qrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 +
b^4)*d^4)) + sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^7 + 3*a^5*b^2 + 3
*a^3*b^4 + a*b^6)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqr
t((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a
^2*b^2 + b^4))*sqrt(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) -
sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2
*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 +
2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*
d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + sq
rt(2)*((a^10*b + 3*a^8*b^3 + 2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d^7*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - (a^9 + 2*a^7*b^2 - 2*
a^3*b^6 - a*b^8)*d^5*sqrt((a^4 - 2*a^2*b^2 + b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt
((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^
2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(a^4 - 2*a^2*b^2 + b^4)
) + 8*(a^2 + b^2)*sqrt(sin(d*x + c)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + sqrt(2)*((a^3 + a*b^2)*d*cos(d*x
+ c)^2 - (a^3 + a*b^2)*d - 2*((a^4*b + a^2*b^3)*d^3*cos(d*x + c)^2 - (a^4*b + a^2*b^3)*d^3)*sqrt(1/((a^4 + 2*
a^2*b^2 + b^4)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((a^6 - a^4*b^2 - a^2*b^4
+ b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3
*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*
a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^
4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x
+ c))/cos(d*x + c)) - sqrt(2)*((a^3 + a*b^2)*d*cos(d*x + c)^2 - (a^3 + a*b^2)*d - 2*((a^4*b + a^2*b^3)*d^3*cos
(d*x + c)^2 - (a^4*b + a^2*b^3)*d^3)*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a
^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*(1/((a^4 + 2*a^2
*b^2 + b^4)*d^4))^(1/4)*log(((a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^2*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x
+ c) - sqrt(2)*((a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3*sqrt(1/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - (a
^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c))*sqrt((a^4 + 2*a^2*b^2 + b^4 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(1
/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(a^4 - 2*a^2*b^2 + b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^4 + 2*a^2*b^2
+ b^4)*d^4))^(1/4) + (a^4 - 2*a^2*b^2 + b^4)*sin(d*x + c))/cos(d*x + c)) - 8*(b^2*cos(d*x + c)^2 - b^2)*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))))/((a^3 + a*b^2)*d*cos(d*x + c)^2 - (a^3 + a*b^2)*d)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate(1/((b*tan(d*x + c) + a)*tan(d*x + c)^(3/2)), x)

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maple [A]  time = 0.20, size = 317, normalized size = 1.27 \[ -\frac {2 b^{3} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right )}{d a \left (a^{2}+b^{2}\right ) \sqrt {a b}}-\frac {2}{a d \sqrt {\tan \left (d x +c \right )}}-\frac {b \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{4 d \left (a^{2}+b^{2}\right )}-\frac {b \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {b \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {a \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )}{4 d \left (a^{2}+b^{2}\right )}-\frac {a \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {a \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )}{2 d \left (a^{2}+b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x)

[Out]

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

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maxima [A]  time = 0.41, size = 185, normalized size = 0.74 \[ -\frac {\frac {8 \, b^{3} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a + b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a - b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a - b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{2} + b^{2}} + \frac {8}{a \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/4*(8*b^3*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^3 + a*b^2)*sqrt(a*b)) + (2*sqrt(2)*(a + b)*arctan(1/2*s
qrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a + b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c
)))) - sqrt(2)*(a - b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a - b)*log(-sqrt(2)*sqrt(
tan(d*x + c)) + tan(d*x + c) + 1))/(a^2 + b^2) + 8/(a*sqrt(tan(d*x + c))))/d

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mupad [B]  time = 5.33, size = 3829, normalized size = 15.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))),x)

[Out]

atan(((-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((tan
(c + d*x)^(1/2)*(512*a^8*b^8*d^7 - 448*a^10*b^6*d^7 + 128*a^12*b^4*d^7 + 64*a^14*b^2*d^7) - (-1i/(4*(b^2*d^2 -
a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(512*a^9*b
^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*a^8*b^9*d^8 - 640*a^10*b^7*d^8 + 256*a^
12*b^5*d^8 + 384*a^14*b^3*d^8))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) + 128*a^7*b^8*d^6 - 32*a^11*b
^4*d^6 - 32*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*a^7*b^7*d^5 - 32*a^9*b^5*d^5))*1i + (-1i/(4*(b^2*d^2 - a^2*
d^2 + a*b*d^2*2i)))^(1/2)*((-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((tan(c + d*x)^(1/2)*(512*a^8*b^8*
d^7 - 448*a^10*b^6*d^7 + 128*a^12*b^4*d^7 + 64*a^14*b^2*d^7) - (-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2
)*(tan(c + d*x)^(1/2)*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 5
12*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*a^8*b^9*d^8 + 640*a^10*b^7*d^8 - 256*a^12*b^5*d^8 - 384*a^14*b^3*d^8
))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - 128*a^7*b^8*d^6 + 32*a^11*b^4*d^6 + 32*a^13*b^2*d^6) + t
an(c + d*x)^(1/2)*(64*a^7*b^7*d^5 - 32*a^9*b^5*d^5))*1i)/((-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((-
1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((tan(c + d*x)^(1/2)*(512*a^8*b^8*d^7 - 448*a^10*b^6*d^7 + 128*
a^12*b^4*d^7 + 64*a^14*b^2*d^7) - (-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(-1i/(4
*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^
3*d^9) - 512*a^8*b^9*d^8 - 640*a^10*b^7*d^8 + 256*a^12*b^5*d^8 + 384*a^14*b^3*d^8))*(-1i/(4*(b^2*d^2 - a^2*d^2
+ a*b*d^2*2i)))^(1/2) + 128*a^7*b^8*d^6 - 32*a^11*b^4*d^6 - 32*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*a^7*b^7
*d^5 - 32*a^9*b^5*d^5)) - (-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*((-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d
^2*2i)))^(1/2)*((tan(c + d*x)^(1/2)*(512*a^8*b^8*d^7 - 448*a^10*b^6*d^7 + 128*a^12*b^4*d^7 + 64*a^14*b^2*d^7)
- (-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*(tan(c + d*x)^(1/2)*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i
)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*a^8*b^9*d^8 + 640*a
^10*b^7*d^8 - 256*a^12*b^5*d^8 - 384*a^14*b^3*d^8))*(-1i/(4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2) - 128*a^7
*b^8*d^6 + 32*a^11*b^4*d^6 + 32*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*a^7*b^7*d^5 - 32*a^9*b^5*d^5))))*(-1i/(
4*(b^2*d^2 - a^2*d^2 + a*b*d^2*2i)))^(1/2)*2i + (log(((-1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(((tan(
c + d*x)^(1/2)*(512*a^8*b^8*d^7 - 448*a^10*b^6*d^7 + 128*a^12*b^4*d^7 + 64*a^14*b^2*d^7) - ((-1/(b^2*d^2*1i -
a^2*d^2*1i + 2*a*b*d^2))^(1/2)*((tan(c + d*x)^(1/2)*(-1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2)*(512*a^9*
b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/2 + 512*a^8*b^9*d^8 + 640*a^10*b^7*d^8 - 25
6*a^12*b^5*d^8 - 384*a^14*b^3*d^8))/2)*(-1/(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - 128*a^7*b^8*d^6 +
32*a^11*b^4*d^6 + 32*a^13*b^2*d^6))/2 + tan(c + d*x)^(1/2)*(64*a^7*b^7*d^5 - 32*a^9*b^5*d^5))*(-1/(b^2*d^2*1i
- a^2*d^2*1i + 2*a*b*d^2))^(1/2))/2 - log((-1/(4*(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2)))^(1/2)*((tan(c + d*x)
^(1/2)*(512*a^8*b^8*d^7 - 448*a^10*b^6*d^7 + 128*a^12*b^4*d^7 + 64*a^14*b^2*d^7) - (-1/(4*(b^2*d^2*1i - a^2*d^
2*1i + 2*a*b*d^2)))^(1/2)*(tan(c + d*x)^(1/2)*(-1/(4*(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2)))^(1/2)*(512*a^9*b^
9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*a^8*b^9*d^8 - 640*a^10*b^7*d^8 + 256*a^1
2*b^5*d^8 + 384*a^14*b^3*d^8))*(-1/(4*(b^2*d^2*1i - a^2*d^2*1i + 2*a*b*d^2)))^(1/2) + 128*a^7*b^8*d^6 - 32*a^1
1*b^4*d^6 - 32*a^13*b^2*d^6) + tan(c + d*x)^(1/2)*(64*a^7*b^7*d^5 - 32*a^9*b^5*d^5))*(-1/(4*(b^2*d^2*1i - a^2*
d^2*1i + 2*a*b*d^2)))^(1/2) - 2/(a*d*tan(c + d*x)^(1/2)) - (atan((((tan(c + d*x)^(1/2)*(64*a^7*b^7*d^5 - 32*a^
9*b^5*d^5) + ((-a^3*b^5)^(1/2)*(32*a^11*b^4*d^6 - 128*a^7*b^8*d^6 + 32*a^13*b^2*d^6 + ((tan(c + d*x)^(1/2)*(51
2*a^8*b^8*d^7 - 448*a^10*b^6*d^7 + 128*a^12*b^4*d^7 + 64*a^14*b^2*d^7) - ((-a^3*b^5)^(1/2)*(512*a^8*b^9*d^8 +
640*a^10*b^7*d^8 - 256*a^12*b^5*d^8 - 384*a^14*b^3*d^8 + (tan(c + d*x)^(1/2)*(-a^3*b^5)^(1/2)*(512*a^9*b^9*d^9
+ 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(a^3*d*(a^2 + b^2))))/(a^3*d*(a^2 + b^2)))*(-a^3*b
^5)^(1/2))/(a^3*d*(a^2 + b^2))))/(a^3*d*(a^2 + b^2)))*(-a^3*b^5)^(1/2)*1i)/(a^3*d*(a^2 + b^2)) + ((tan(c + d*x
)^(1/2)*(64*a^7*b^7*d^5 - 32*a^9*b^5*d^5) + ((-a^3*b^5)^(1/2)*(128*a^7*b^8*d^6 - 32*a^11*b^4*d^6 - 32*a^13*b^2
*d^6 + ((tan(c + d*x)^(1/2)*(512*a^8*b^8*d^7 - 448*a^10*b^6*d^7 + 128*a^12*b^4*d^7 + 64*a^14*b^2*d^7) - ((-a^3
*b^5)^(1/2)*(256*a^12*b^5*d^8 - 640*a^10*b^7*d^8 - 512*a^8*b^9*d^8 + 384*a^14*b^3*d^8 + (tan(c + d*x)^(1/2)*(-
a^3*b^5)^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(a^3*d*(a^2 + b^2))
))/(a^3*d*(a^2 + b^2)))*(-a^3*b^5)^(1/2))/(a^3*d*(a^2 + b^2))))/(a^3*d*(a^2 + b^2)))*(-a^3*b^5)^(1/2)*1i)/(a^3
*d*(a^2 + b^2)))/(((tan(c + d*x)^(1/2)*(64*a^7*b^7*d^5 - 32*a^9*b^5*d^5) + ((-a^3*b^5)^(1/2)*(32*a^11*b^4*d^6
- 128*a^7*b^8*d^6 + 32*a^13*b^2*d^6 + ((tan(c + d*x)^(1/2)*(512*a^8*b^8*d^7 - 448*a^10*b^6*d^7 + 128*a^12*b^4*
d^7 + 64*a^14*b^2*d^7) - ((-a^3*b^5)^(1/2)*(512*a^8*b^9*d^8 + 640*a^10*b^7*d^8 - 256*a^12*b^5*d^8 - 384*a^14*b
^3*d^8 + (tan(c + d*x)^(1/2)*(-a^3*b^5)^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^1
5*b^3*d^9))/(a^3*d*(a^2 + b^2))))/(a^3*d*(a^2 + b^2)))*(-a^3*b^5)^(1/2))/(a^3*d*(a^2 + b^2))))/(a^3*d*(a^2 + b
^2)))*(-a^3*b^5)^(1/2))/(a^3*d*(a^2 + b^2)) - ((tan(c + d*x)^(1/2)*(64*a^7*b^7*d^5 - 32*a^9*b^5*d^5) + ((-a^3*
b^5)^(1/2)*(128*a^7*b^8*d^6 - 32*a^11*b^4*d^6 - 32*a^13*b^2*d^6 + ((tan(c + d*x)^(1/2)*(512*a^8*b^8*d^7 - 448*
a^10*b^6*d^7 + 128*a^12*b^4*d^7 + 64*a^14*b^2*d^7) - ((-a^3*b^5)^(1/2)*(256*a^12*b^5*d^8 - 640*a^10*b^7*d^8 -
512*a^8*b^9*d^8 + 384*a^14*b^3*d^8 + (tan(c + d*x)^(1/2)*(-a^3*b^5)^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9
- 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(a^3*d*(a^2 + b^2))))/(a^3*d*(a^2 + b^2)))*(-a^3*b^5)^(1/2))/(a^3*d*(a
^2 + b^2))))/(a^3*d*(a^2 + b^2)))*(-a^3*b^5)^(1/2))/(a^3*d*(a^2 + b^2))))*(-a^3*b^5)^(1/2)*2i)/(a^3*d*(a^2 + b
^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right ) \tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(d*x+c)**(3/2)/(a+b*tan(d*x+c)),x)

[Out]

Integral(1/((a + b*tan(c + d*x))*tan(c + d*x)**(3/2)), x)

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