3.530 \(\int \frac{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx\)
Optimal. Leaf size=367 \[ \frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3 \cot (c+d x)+i a^3\right )}-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}-\frac{\left (\frac{1}{32}-\frac{i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{1}{32}-\frac{i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{1}{16}-\frac{i}{16}\right ) ((1+29 i) A-(6+i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{((30+28 i) A-(7-5 i) B) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (a \cot (c+d x)+i a)^2} \]
[Out]
((-1/16 + I/16)*((1 + 29*I)*A - (6 + I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + (((30 + 2
8*I)*A - (7 - 5*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) - (5*(6*A + I*B)*Sqrt[Cot[c +
d*x]])/(8*a^3*d) + ((A + I*B)*Cot[c + d*x]^(7/2))/(6*d*(I*a + a*Cot[c + d*x])^3) + ((5*A + (2*I)*B)*Cot[c + d
*x]^(5/2))/(12*a*d*(I*a + a*Cot[c + d*x])^2) + (7*(4*A + I*B)*Cot[c + d*x]^(3/2))/(24*d*(I*a^3 + a^3*Cot[c + d
*x])) - ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]
*a^3*d) + ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[
2]*a^3*d)
________________________________________________________________________________________
Rubi [A] time = 0.918336, antiderivative size = 367, normalized size of antiderivative = 1.,
number of steps used = 15, number of rules used = 10, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used
= {3581, 3595, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (a^3 \cot (c+d x)+i a^3\right )}-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}-\frac{\left (\frac{1}{32}-\frac{i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{1}{32}-\frac{i}{32}\right ) ((29+i) A+(1+6 i) B) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{1}{16}-\frac{i}{16}\right ) ((1+29 i) A-(6+i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{((30+28 i) A-(7-5 i) B) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (a \cot (c+d x)+i a)^2} \]
Antiderivative was successfully verified.
[In]
Int[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]
[Out]
((-1/16 + I/16)*((1 + 29*I)*A - (6 + I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + (((30 + 2
8*I)*A - (7 - 5*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) - (5*(6*A + I*B)*Sqrt[Cot[c +
d*x]])/(8*a^3*d) + ((A + I*B)*Cot[c + d*x]^(7/2))/(6*d*(I*a + a*Cot[c + d*x])^3) + ((5*A + (2*I)*B)*Cot[c + d
*x]^(5/2))/(12*a*d*(I*a + a*Cot[c + d*x])^2) + (7*(4*A + I*B)*Cot[c + d*x]^(3/2))/(24*d*(I*a^3 + a^3*Cot[c + d
*x])) - ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]
*a^3*d) + ((1/32 - I/32)*((29 + I)*A + (1 + 6*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[
2]*a^3*d)
Rule 3581
Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.)
+ (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d
+ c*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && !IntegerQ[p] && IntegerQ[m] && IntegerQ
[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 3528
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 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{\cot ^{\frac{3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx &=\int \frac{\cot ^{\frac{7}{2}}(c+d x) (B+A \cot (c+d x))}{(i a+a \cot (c+d x))^3} \, dx\\ &=\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\int \frac{\cot ^{\frac{5}{2}}(c+d x) \left (-\frac{7}{2} a (i A-B)+\frac{1}{2} a (13 A+i B) \cot (c+d x)\right )}{(i a+a \cot (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac{\int \frac{\cot ^{\frac{3}{2}}(c+d x) \left (-5 a^2 (5 i A-2 B)+a^2 (31 A+4 i B) \cot (c+d x)\right )}{i a+a \cot (c+d x)} \, dx}{24 a^4}\\ &=\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\int \sqrt{\cot (c+d x)} \left (-21 a^3 (4 i A-B)+15 a^3 (6 A+i B) \cot (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{-15 a^3 (6 A+i B)-21 a^3 (4 i A-B) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{48 a^6}\\ &=-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{15 a^3 (6 A+i B)+21 a^3 (4 i A-B) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{24 a^6 d}\\ &=-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{((30+28 i) A-(7-5 i) B) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 a^3 d}+\frac{((30-28 i) A+(7+5 i) B) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 a^3 d}\\ &=-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{((30+28 i) A-(7-5 i) B) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{32 a^3 d}+\frac{((30+28 i) A-(7-5 i) B) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{32 a^3 d}-\frac{((30-28 i) A+(7+5 i) B) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{32 \sqrt{2} a^3 d}-\frac{((30-28 i) A+(7+5 i) B) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{32 \sqrt{2} a^3 d}\\ &=-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac{((30-28 i) A+(7+5 i) B) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{((30-28 i) A+(7+5 i) B) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{((30+28 i) A-(7-5 i) B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}-\frac{((30+28 i) A-(7-5 i) B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}\\ &=-\frac{((30+28 i) A-(7-5 i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{((30+28 i) A-(7-5 i) B) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}-\frac{5 (6 A+i B) \sqrt{\cot (c+d x)}}{8 a^3 d}+\frac{(A+i B) \cot ^{\frac{7}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{(5 A+2 i B) \cot ^{\frac{5}{2}}(c+d x)}{12 a d (i a+a \cot (c+d x))^2}+\frac{7 (4 A+i B) \cot ^{\frac{3}{2}}(c+d x)}{24 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac{((30-28 i) A+(7+5 i) B) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{((30-28 i) A+(7+5 i) B) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}\\ \end{align*}
Mathematica [A] time = 3.57323, size = 284, normalized size = 0.77 \[ \frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left (\frac{2}{3} \cot (c+d x) (\cos (3 d x)-i \sin (3 d x)) ((49 A+19 i B) \cos (c+d x)-(145 A+19 i B) \cos (3 (c+d x))+6 \sin (c+d x) (7 (B-7 i A) \cos (2 (c+d x))-19 i A+2 B))+(-\sin (3 c)+i \cos (3 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left (((28-30 i) A+(5+7 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((29+i) A+(1+6 i) B) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cot[c + d*x]^(3/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^3,x]
[Out]
(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((2*Cot[c + d*x]*(Cos[3*d*x] - I*Sin[3*d*x])*((49*A + (19*I)*B)*Cos[
c + d*x] - (145*A + (19*I)*B)*Cos[3*(c + d*x)] + 6*((-19*I)*A + 2*B + 7*((-7*I)*A + B)*Cos[2*(c + d*x)])*Sin[c
+ d*x]))/3 + Csc[c + d*x]*(((28 - 30*I)*A + (5 + 7*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] - (1 + I)*((29 +
I)*A + (1 + 6*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(I*Cos[3*c] - Sin[3*c])*Sqrt[S
in[2*(c + d*x)]])*(A + B*Tan[c + d*x]))/(32*d*Sqrt[Cot[c + d*x]]*(A*Cos[c + d*x] + B*Sin[c + d*x])*(a + I*a*Ta
n[c + d*x])^3)
________________________________________________________________________________________
Maple [C] time = 0.741, size = 2577, normalized size = 7. \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)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x)
[Out]
1/48/a^3/d*2^(1/2)*sin(d*x+c)*(cos(d*x+c)/sin(d*x+c))^(3/2)*(-21*B*cos(d*x+c)*((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)*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)-1-sin(d*x+c))/sin(d*x+c))^(1/2)+87*A*cos(d*x+c)*(-(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))+18*A*cos(d*x+c)^3*2^(1/2)-90*A*cos(d*x+c)*2^(1/2)-16*I*A*
cos(d*x+c)^6*sin(d*x+c)*2^(1/2)-16*I*A*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)-28*I*A*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)+
18*B*cos(d*x+c)*(-(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)*((c
os(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))
+16*A*cos(d*x+c)^7*2^(1/2)+3*I*A*(-(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))-87*I*A*(-(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))+84*I*A*(-(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)*EllipticF((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))-3*I*B*Ell
ipticPi((-(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)*((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)+18*I*B*(-(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))+8*A*cos(d*x+c)^5*2^(1/2)+7
*B*cos(d*x+c)^2*sin(d*x+c)*2^(1/2)+87*A*(-(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))+18*B*(-(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))+3*A*((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))+3*B*cos(d*x+c)*((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))+3*A*cos(d*x+c)*((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))-21*B*((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)*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)-1-sin(d*x+c))/sin(d*x+c))^(1/2)+3*B*((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))-4*I*B*cos(d*x+c)^5*2^(1/2)+1
6*I*B*cos(d*x+c)^7*2^(1/2)+4*B*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)+18*I*B*cos(d*x+c)*(-(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)*((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))-87*I*A*cos(d*x+c)*(-(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))+84*I*A*cos(d*x+c)*(-(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)*Elliptic
F((-(cos(d*x+c)-1-sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))-3*I*B*cos(d*x+c)*(-(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))+16*B*cos(d*x+c)^6*sin(d*x+c)*2^(1/2)+3*I*B*cos(
d*x+c)^3*2^(1/2)-15*I*B*cos(d*x+c)*2^(1/2)+3*I*A*cos(d*x+c)*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
________________________________________________________________________________________
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)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")
[Out]
Exception raised: RuntimeError
________________________________________________________________________________________
Fricas [B] time = 1.67205, size = 1854, normalized size = 5.05 \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)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")
[Out]
1/96*(3*a^3*d*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(1/8*((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((I*A^2 + 2*A*B - I*B^2)/(a
^6*d^2)) - 16*(A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - 3*a^3*d*sqrt((I*A^2 + 2*A*B - I
*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(1/8*((-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((I*A^2 + 2*A*B - I*B^2)/(a^6*d^2)) - 16*(A - I*B)*e^(2*I*d*x +
2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) + 3*a^3*d*sqrt((-841*I*A^2 + 348*A*B + 36*I*B^2)/(a^6*d^2))*e^(6*I*d*
x + 6*I*c)*log(1/8*((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((-841*I*A^2 + 348*A*B + 36*I*B^2)/(a^6*d^2)) + 29*I*A - 6*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) - 3*a^3*
d*sqrt((-841*I*A^2 + 348*A*B + 36*I*B^2)/(a^6*d^2))*e^(6*I*d*x + 6*I*c)*log(-1/8*((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((-841*I*A^2 + 348*A*B + 36*I*B^2)/(a^
6*d^2)) - 29*I*A + 6*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) - 2*(2*(73*A + 10*I*B)*e^(6*I*d*x + 6*I*c) - (41*A + 14*
I*B)*e^(4*I*d*x + 4*I*c) - (8*A + 5*I*B)*e^(2*I*d*x + 2*I*c) - A - I*B)*sqrt((I*e^(2*I*d*x + 2*I*c) + I)/(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(cot(d*x+c)**(3/2)*(A+B*tan(d*x+c))/(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{{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac{3}{2}}}{{\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)^(3/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")
[Out]
integrate((B*tan(d*x + c) + A)*cot(d*x + c)^(3/2)/(I*a*tan(d*x + c) + a)^3, x)