3.746 \(\int \frac{\sqrt{\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=273 \[ \frac{5 \sqrt{\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}-\frac{\left (\frac{7}{32}+\frac{5 i}{32}\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{7}{32}+\frac{5 i}{32}\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (a \cot (c+d x)+i a)^2} \]

[Out]

((7/16 - (5*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) - ((7/16 - (5*I)/16)*ArcTan[1 + Sqr
t[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + Cot[c + d*x]^(5/2)/(6*d*(I*a + a*Cot[c + d*x])^3) + Cot[c + d*x]^(
3/2)/(3*a*d*(I*a + a*Cot[c + d*x])^2) + (5*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) - ((7/32 + (5*
I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d) + ((7/32 + (5*I)/32)*Log[1 + Sqrt[2
]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d)

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Rubi [A]  time = 0.468511, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3673, 3558, 3595, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{5 \sqrt{\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}-\frac{\left (\frac{7}{32}+\frac{5 i}{32}\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{7}{32}+\frac{5 i}{32}\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (a \cot (c+d x)+i a)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^3,x]

[Out]

((7/16 - (5*I)/16)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) - ((7/16 - (5*I)/16)*ArcTan[1 + Sqr
t[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + Cot[c + d*x]^(5/2)/(6*d*(I*a + a*Cot[c + d*x])^3) + Cot[c + d*x]^(
3/2)/(3*a*d*(I*a + a*Cot[c + d*x])^2) + (5*Sqrt[Cot[c + d*x]])/(8*d*(I*a^3 + a^3*Cot[c + d*x])) - ((7/32 + (5*
I)/32)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*d) + ((7/32 + (5*I)/32)*Log[1 + Sqrt[2
]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^3*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 3558

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

Rule 3595

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

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{\sqrt{\cot (c+d x)}}{(a+i a \tan (c+d x))^3} \, dx &=\int \frac{\cot ^{\frac{7}{2}}(c+d x)}{(i a+a \cot (c+d x))^3} \, dx\\ &=\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\int \frac{\cot ^{\frac{3}{2}}(c+d x) \left (-\frac{5 i a}{2}+\frac{11}{2} a \cot (c+d x)\right )}{(i a+a \cot (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (i a+a \cot (c+d x))^2}+\frac{\int \frac{\sqrt{\cot (c+d x)} \left (-12 i a^2+18 a^2 \cot (c+d x)\right )}{i a+a \cot (c+d x)} \, dx}{24 a^4}\\ &=\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{-15 i a^3+21 a^3 \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{48 a^6}\\ &=\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{15 i a^3-21 a^3 x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{24 a^6 d}\\ &=\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+-\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}+\frac{\left (\frac{7}{16}+\frac{5 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}\\ &=\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+-\frac{\left (\frac{7}{32}-\frac{5 i}{32}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}+-\frac{\left (\frac{7}{32}-\frac{5 i}{32}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}+-\frac{\left (\frac{7}{32}+\frac{5 i}{32}\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^3 d}+-\frac{\left (\frac{7}{32}+\frac{5 i}{32}\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^3 d}\\ &=\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac{\left (\frac{7}{32}+\frac{5 i}{32}\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{7}{32}+\frac{5 i}{32}\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^3 d}+-\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}\\ &=\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{7}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\cot ^{\frac{5}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{3 a d (i a+a \cot (c+d x))^2}+\frac{5 \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac{\left (\frac{7}{32}+\frac{5 i}{32}\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{7}{32}+\frac{5 i}{32}\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^3 d}\\ \end{align*}

Mathematica [A]  time = 1.23237, size = 235, normalized size = 0.86 \[ \frac{\cot ^{\frac{5}{2}}(c+d x) \csc (c+d x) \sec ^3(c+d x) \left (12 \sin (2 (c+d x))+21 \sin (4 (c+d x))-19 i \cos (4 (c+d x))-(21-15 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\cos (3 (c+d x))+i \sin (3 (c+d x)))-(15-21 i) \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )+(21+15 i) \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )+19 i\right )}{96 a^3 d (\cot (c+d x)+i)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cot[c + d*x]]/(a + I*a*Tan[c + d*x])^3,x]

[Out]

(Cot[c + d*x]^(5/2)*Csc[c + d*x]*Sec[c + d*x]^3*(19*I - (19*I)*Cos[4*(c + d*x)] + (21 + 15*I)*Cos[3*(c + d*x)]
*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]] + 12*Sin[2*(c + d*x)] - (21
- 15*I)*ArcSin[Cos[c + d*x] - Sin[c + d*x]]*Sqrt[Sin[2*(c + d*x)]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) - (
15 - 21*I)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]]*Sqrt[Sin[2*(c + d*x)]]*Sin[3*(c + d*x)] +
21*Sin[4*(c + d*x)]))/(96*a^3*d*(I + Cot[c + d*x])^3)

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Maple [C]  time = 0.451, size = 844, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48/a^3/d*2^(1/2)*(cos(d*x+c)/sin(d*x+c))^(1/2)*(cos(d*x+c)+1)^2*(cos(d*x+c)-1)*(-15*I*sin(d*x+c)*cos(d*x+c)
*2^(1/2)-12*I*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)-16*cos(d*x+c)^7*2^(1/2)+18*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))*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)+16*cos(d*x+c)^6*2^(1/2)-16*I*cos(d*x+c)
^5*sin(d*x+c)*2^(1/2)+15*I*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)+16*I*cos(d*x+c)^6*sin(d*x+c)*2^(1/2)-4*2^(1/2)*cos(
d*x+c)^5-3*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-si
n(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))*si
n(d*x+c)+4*cos(d*x+c)^4*2^(1/2)-21*(-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2)*((cos(d*x+c)-1+sin(d*x+c))/si
n(d*x+c))^(1/2)*((cos(d*x+c)-1)/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))*sin(d*x+c)+18*((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))*sin(d*x+c)+3*(-(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))*sin(d*x+c)+12*I*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)-7*2^(1/2)*cos(d*x+c)^3+7*2^(1/2)*cos(d*x+c)^2)/sin(d*x+c
)^3/cos(d*x+c)

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

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 1.49048, size = 1485, normalized size = 5.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(((16*I*a^3*d*e^(2*I*d*x + 2*I*c) - 16*I*a^3*d)*
sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-1/64*I/(a^6*d^2)) + 2*I*e^(2*I*d*x + 2*I*c))
*e^(-2*I*d*x - 2*I*c)) - 12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(((-16*I*a^3*d*e^(2*I*d*x + 2
*I*c) + 16*I*a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(-1/64*I/(a^6*d^2)) + 2*I*
e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)) - 12*a^3*d*sqrt(9/16*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-1/4*(4*(
a^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(9/16*I/(a^
6*d^2)) + 3)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + 12*a^3*d*sqrt(9/16*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(1/4*(4*(a
^3*d*e^(2*I*d*x + 2*I*c) - a^3*d)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(9/16*I/(a^6
*d^2)) - 3)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*(-20*I
*e^(6*I*d*x + 6*I*c) + 14*I*e^(4*I*d*x + 4*I*c) + 5*I*e^(2*I*d*x + 2*I*c) + I))*e^(-6*I*d*x - 6*I*c)/(a^3*d)

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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(cot(d*x+c)**(1/2)/(a+I*a*tan(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\cot \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(cot(d*x + c))/(I*a*tan(d*x + c) + a)^3, x)