3.749 \(\int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx\)
Optimal. Leaf size=222 \[ -\frac{\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{8 \sqrt{2} a^3 d}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{8 \sqrt{2} a^3 d}+\frac{\sqrt{\cot (c+d x)}}{12 a d (a \cot (c+d x)+i a)^2}+\frac{i \sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3} \]
[Out]
-ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/(8*Sqrt[2]*a^3*d) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/(8*Sqrt[2]*
a^3*d) + ((I/6)*Sqrt[Cot[c + d*x]])/(d*(I*a + a*Cot[c + d*x])^3) + Sqrt[Cot[c + d*x]]/(12*a*d*(I*a + a*Cot[c +
d*x])^2) - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(16*Sqrt[2]*a^3*d) + Log[1 + Sqrt[2]*Sqrt[Cot[c
+ d*x]] + Cot[c + d*x]]/(16*Sqrt[2]*a^3*d)
________________________________________________________________________________________
Rubi [A] time = 0.306715, antiderivative size = 222, normalized size of antiderivative =
1., number of steps used = 15, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.462, Rules used = {3673, 3557, 3596, 21, 3476, 329, 211, 1165, 628, 1162, 617, 204}
\[ -\frac{\log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{\log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{8 \sqrt{2} a^3 d}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{8 \sqrt{2} a^3 d}+\frac{\sqrt{\cot (c+d x)}}{12 a d (a \cot (c+d x)+i a)^2}+\frac{i \sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3} \]
Antiderivative was successfully verified.
[In]
Int[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3),x]
[Out]
-ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/(8*Sqrt[2]*a^3*d) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]/(8*Sqrt[2]*
a^3*d) + ((I/6)*Sqrt[Cot[c + d*x]])/(d*(I*a + a*Cot[c + d*x])^3) + Sqrt[Cot[c + d*x]]/(12*a*d*(I*a + a*Cot[c +
d*x])^2) - Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(16*Sqrt[2]*a^3*d) + Log[1 + Sqrt[2]*Sqrt[Cot[c
+ d*x]] + Cot[c + d*x]]/(16*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 3557
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> -Sim
p[(b*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/(2*a*f*m), x] + Dist[1/(4*a^2*m), Int[((a + b*Tan[e + f*
x])^(m + 1)*Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x])/Sqrt[c + d*Tan[e + f*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] && IntegersQ[2
*m]
Rule 3596
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*A + b*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2
*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && !GtQ[n,
0]
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])
Rule 3476
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Rule 329
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 211
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))
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]
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])
Rubi steps
\begin{align*} \int \frac{1}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx &=\int \frac{\sqrt{\cot (c+d x)}}{(i a+a \cot (c+d x))^3} \, dx\\ &=\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\int \frac{a-5 i a \cot (c+d x)}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2} \, dx}{12 a^2}\\ &=\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{\int \frac{-6 i a^2-6 a^2 \cot (c+d x)}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))} \, dx}{48 a^4}\\ &=\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}-\frac{\int \frac{1}{\sqrt{\cot (c+d x)}} \, dx}{8 a^3}\\ &=\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\cot (c+d x)\right )}{8 a^3 d}\\ &=\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{4 a^3 d}\\ &=\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{8 a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{8 a^3 d}\\ &=\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}\\ &=\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}-\frac{\log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{16 \sqrt{2} a^3 d}+\frac{\log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{16 \sqrt{2} a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{8 \sqrt{2} a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{8 \sqrt{2} a^3 d}\\ &=-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{8 \sqrt{2} a^3 d}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{8 \sqrt{2} a^3 d}+\frac{i \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}-\frac{\log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{16 \sqrt{2} a^3 d}+\frac{\log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{16 \sqrt{2} a^3 d}\\ \end{align*}
Mathematica [A] time = 1.82886, size = 224, normalized size = 1.01 \[ \frac{(\cos (3 (c+d x))-i \sin (3 (c+d x))) \left (6 \tan ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right ) (\sin (3 (c+d x))-i \cos (3 (c+d x)))+\sqrt{i \tan (c+d x)} (\sin (c+d x)+\sin (3 (c+d x))+3 i \cos (c+d x)-3 i \cos (3 (c+d x)))+6 i \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right ) (\cos (3 (c+d x))+i \sin (3 (c+d x)))\right )}{48 a^3 d \sqrt{i \tan (c+d x)} \sqrt{\cot (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Cot[c + d*x]^(5/2)*(a + I*a*Tan[c + d*x])^3),x]
[Out]
((Cos[3*(c + d*x)] - I*Sin[3*(c + d*x)])*((6*I)*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x
)))]]*(Cos[3*(c + d*x)] + I*Sin[3*(c + d*x)]) + 6*ArcTan[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*
x)))]]*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)]) + ((3*I)*Cos[c + d*x] - (3*I)*Cos[3*(c + d*x)] + Sin[c + d*x
] + Sin[3*(c + d*x)])*Sqrt[I*Tan[c + d*x]]))/(48*a^3*d*Sqrt[Cot[c + d*x]]*Sqrt[I*Tan[c + d*x]])
________________________________________________________________________________________
Maple [C] time = 0.324, size = 2095, normalized size = 9.4 \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))^3,x)
[Out]
1/48/a^3/d*2^(1/2)*(cos(d*x+c)-1)*(-6*I*sin(d*x+c)*cos(d*x+c)*2^(1/2)-12*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)-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)^2*sin(d*x+c)+9*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)*Ellipti
cPi((-(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)-9*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)-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)-12*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)^3*((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)-12*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*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)-12*(-(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)^3+12*((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)^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*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)+
12*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)-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)
^3-12*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)-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)^2*sin(d*x+c)+6*I*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+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-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)+9*((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))/s
in(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)+3*((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)+9*(-(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)-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*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))^(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)^3+2*2^(1/2)*cos(d*x+c)^3-2*2^(1/2)*cos(d*x+c)^2)*cos(d*x+c)^2*(c
os(d*x+c)+1)^2/(4*I*sin(d*x+c)*cos(d*x+c)^2+4*cos(d*x+c)^3-I*sin(d*x+c)-3*cos(d*x+c))/(cos(d*x+c)/sin(d*x+c))^
(5/2)/sin(d*x+c)^5
________________________________________________________________________________________
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))^3,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
________________________________________________________________________________________
Fricas [B] time = 1.45904, size = 1451, normalized size = 6.54 \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))^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(2*(8*(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(1/64*I/(a^6*d^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(-2*(8*(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(1/64*I/(a^6*d^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(1/8*(8*(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(-1/64*I/(a^6*d^2)) + I)*e^(-2*
I*d*x - 2*I*c)/(a^3*d)) - 12*a^3*d*sqrt(-1/64*I/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-1/8*(8*(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(-1/64*I/(a^6*d^2)) - I)*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))*(2*e^(6*I*d*x + 6*I*c)
- 5*e^(4*I*d*x + 4*I*c) + 4*e^(2*I*d*x + 2*I*c) - 1))*e^(-6*I*d*x - 6*I*c)/(a^3*d)
________________________________________________________________________________________
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))**3,x)
[Out]
Exception raised: AttributeError
________________________________________________________________________________________
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 )}^{3} \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))^3,x, algorithm="giac")
[Out]
integrate(1/((I*a*tan(d*x + c) + a)^3*cot(d*x + c)^(5/2)), x)