3.739 \(\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx\)

Optimal. Leaf size=222 \[ -\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)}+\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a d}-\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a d}+\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a d}-\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a d} \]

[Out]

((3/4 + (5*I)/4)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((3/4 + (5*I)/4)*ArcTan[1 + Sqrt[2]*S
qrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((5*I)/2)/(a*d*Sqrt[Cot[c + d*x]]) - 1/(2*d*Sqrt[Cot[c + d*x]]*(I*a + a*Co
t[c + d*x])) + ((3/8 - (5*I)/8)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) - ((3/8 - (5
*I)/8)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)

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Rubi [A]  time = 0.239181, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3673, 3552, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)}+\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a d}-\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a d}+\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a d}-\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a d} \]

Antiderivative was successfully verified.

[In]

Int[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])),x]

[Out]

((3/4 + (5*I)/4)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((3/4 + (5*I)/4)*ArcTan[1 + Sqrt[2]*S
qrt[Cot[c + d*x]]])/(Sqrt[2]*a*d) - ((5*I)/2)/(a*d*Sqrt[Cot[c + d*x]]) - 1/(2*d*Sqrt[Cot[c + d*x]]*(I*a + a*Co
t[c + d*x])) + ((3/8 - (5*I)/8)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d) - ((3/8 - (5
*I)/8)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a*d)

Rule 3673

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_.))^(p_.), x_Symbol] :> Dist
[d^(n*p), Int[(d*Cot[e + f*x])^(m - n*p)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x
] &&  !IntegerQ[m] && IntegersQ[n, p]

Rule 3552

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

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))} \, dx &=\int \frac{1}{\cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))} \, dx\\ &=-\frac{1}{2 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))}-\frac{\int \frac{\frac{5 i a}{2}-\frac{3}{2} a \cot (c+d x)}{\cot ^{\frac{3}{2}}(c+d x)} \, dx}{2 a^2}\\ &=-\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))}-\frac{\int \frac{-\frac{3 a}{2}-\frac{5}{2} i a \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))}-\frac{\operatorname{Subst}\left (\int \frac{\frac{3 a}{2}+\frac{5}{2} i a x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^2 d}\\ &=-\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))}-\frac{\left (\frac{3}{4}-\frac{5 i}{4}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a d}-\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a d}\\ &=-\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))}-\frac{\left (\frac{3}{8}+\frac{5 i}{8}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a d}-\frac{\left (\frac{3}{8}+\frac{5 i}{8}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a d}--\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\sqrt{2} a d}--\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\sqrt{2} a d}\\ &=-\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))}+\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a d}-\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a d}--\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a d}-\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a d}\\ &=\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a d}-\frac{\left (\frac{3}{4}+\frac{5 i}{4}\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a d}-\frac{5 i}{2 a d \sqrt{\cot (c+d x)}}-\frac{1}{2 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))}+\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a d}-\frac{\left (\frac{3}{8}-\frac{5 i}{8}\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a d}\\ \end{align*}

Mathematica [A]  time = 1.03766, size = 213, normalized size = 0.96 \[ -\frac{\sqrt{\cot (c+d x)} \csc (c+d x) \sec (c+d x) \left (10 i \sin (2 (c+d x))+8 \cos (2 (c+d x))+(3+5 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\cos (c+d x)+i \sin (c+d x))+(3-5 i) \sqrt{\sin (2 (c+d x))} \cos (c+d x) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )+(5+3 i) \sin (c+d x) \sqrt{\sin (2 (c+d x))} \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )-8\right )}{8 a d (\cot (c+d x)+i)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])),x]

[Out]

-(Sqrt[Cot[c + d*x]]*Csc[c + d*x]*Sec[c + d*x]*(-8 + 8*Cos[2*(c + d*x)] + (3 - 5*I)*Cos[c + d*x]*Log[Cos[c + d
*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + (3 + 5*I)*ArcSin[Cos[c + d*x] - Sin[c +
d*x]]*(Cos[c + d*x] + I*Sin[c + d*x])*Sqrt[Sin[2*(c + d*x)]] + (5 + 3*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqr
t[Sin[2*(c + d*x)]]]*Sin[c + d*x]*Sqrt[Sin[2*(c + d*x)]] + (10*I)*Sin[2*(c + d*x)]))/(8*a*d*(I + Cot[c + d*x])
)

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Maple [C]  time = 0.239, size = 1897, normalized size = 8.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c)),x)

[Out]

-1/4/a/d*2^(1/2)*(cos(d*x+c)-1)*(-I*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(
1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/
2+1/2*I,1/2*2^(1/2))*cos(d*x+c)^2+I*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(
1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/
2+1/2*I,1/2*2^(1/2))-5*I*2^(1/2)*cos(d*x+c)+5*I*2^(1/2)*cos(d*x+c)^2+4*I*cos(d*x+c)^2*((cos(d*x+c)-1+sin(d*x+c
))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2
),1/2-1/2*I,1/2*2^(1/2))*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)+4*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c
))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+
c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)^2+(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))
^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)
-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(d*x+c)^2-5*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos
(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-
1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*cos(d*x+c)^2-4*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))
^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c
))/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)+(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/
2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-s
in(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(d*x+c)*sin(d*x+c)+4*I*cos(d*x+c)*sin(d*x+c)*((cos(d*x+
c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(
1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))-5*I*((cos(d*x+c)-1)/sin(d
*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*Ellipt
icF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*cos(d*x+c)*sin(d*x+c)-4*I*((cos(d*x+c)-1+sin(d*
x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*Ellipt
icPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))-4*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x
+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d
*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))-(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((c
os(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*EllipticPi((-(cos(d*x+c)-1-sin(d*x
+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))+5*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/s
in(d*x+c))^(1/2)*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c
))^(1/2),1/2*2^(1/2))+I*EllipticPi((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(d*
x+c)*sin(d*x+c)*((cos(d*x+c)-1)/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(cos(d*x+c)-1
-sin(d*x+c))/sin(d*x+c))^(1/2)-4*cos(d*x+c)*sin(d*x+c)*2^(1/2)+4*2^(1/2)*sin(d*x+c))*cos(d*x+c)^2*(cos(d*x+c)+
1)^2/(I*sin(d*x+c)+cos(d*x+c))/sin(d*x+c)^6/(cos(d*x+c)/sin(d*x+c))^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 1.48912, size = 1513, normalized size = 6.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/4*((a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/4*I/(a^2*d^2))*log(2*(2*(a*d*e^(2*I*d*x + 2*I*
c) - a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(1/4*I/(a^2*d^2)) + I*e^(2*I*d*x + 2
*I*c))*e^(-2*I*d*x - 2*I*c)) - (a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x + 2*I*c))*sqrt(1/4*I/(a^2*d^2))*log(-
2*(2*(a*d*e^(2*I*d*x + 2*I*c) - a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(1/4*I/(a
^2*d^2)) - I*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) - (a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x + 2*I*c))*
sqrt(-4*I/(a^2*d^2))*log(-((a*d*e^(2*I*d*x + 2*I*c) - a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*
c) - 1))*sqrt(-4*I/(a^2*d^2)) + 2*I)*e^(-2*I*d*x - 2*I*c)/(a*d)) + (a*d*e^(4*I*d*x + 4*I*c) + a*d*e^(2*I*d*x +
2*I*c))*sqrt(-4*I/(a^2*d^2))*log(((a*d*e^(2*I*d*x + 2*I*c) - a*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*
x + 2*I*c) - 1))*sqrt(-4*I/(a^2*d^2)) - 2*I)*e^(-2*I*d*x - 2*I*c)/(a*d)) - sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e
^(2*I*d*x + 2*I*c) - 1))*(9*e^(4*I*d*x + 4*I*c) - 8*e^(2*I*d*x + 2*I*c) - 1))/(a*d*e^(4*I*d*x + 4*I*c) + a*d*e
^(2*I*d*x + 2*I*c))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cot(d*x+c)**(5/2)/(a+I*a*tan(d*x+c)),x)

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )} \cot \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/cot(d*x+c)^(5/2)/(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate(1/((I*a*tan(d*x + c) + a)*cot(d*x + c)^(5/2)), x)