3.535 \(\int \frac{A+B \tan (c+d x)}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=367 \[ -\frac{7 (-4 B+i A)}{24 d \sqrt{\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}+\frac{5 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{\left (\frac{1}{32}+\frac{i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{1}{32}+\frac{i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+6 i) A-(29+i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^2}+\frac{-B+i A}{6 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^3} \]

[Out]

((1/16 + I/16)*((1 + 6*I)*A - (29 + I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + (((5 - 7*I
)*A + (28 + 30*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) + (5*(A + (6*I)*B))/(8*a^3*d*S
qrt[Cot[c + d*x]]) + (I*A - B)/(6*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^3) + (2*A + (5*I)*B)/(12*a*d*Sqr
t[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - (7*(I*A - 4*B))/(24*d*Sqrt[Cot[c + d*x]]*(I*a^3 + a^3*Cot[c + d*x]
)) + ((1/32 + I/32)*((6 + I)*A + (1 + 29*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^
3*d) - ((1/32 + I/32)*((6 + I)*A + (1 + 29*I)*B)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*
a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.925566, 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, 3596, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{7 (-4 B+i A)}{24 d \sqrt{\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}+\frac{5 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{\left (\frac{1}{32}+\frac{i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}-\frac{\left (\frac{1}{32}+\frac{i}{32}\right ) ((6+i) A+(1+29 i) B) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+6 i) A-(29+i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^2}+\frac{-B+i A}{6 d \sqrt{\cot (c+d x)} (a \cot (c+d x)+i a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3),x]

[Out]

((1/16 + I/16)*((1 + 6*I)*A - (29 + I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt[2]*a^3*d) + (((5 - 7*I
)*A + (28 + 30*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]])/(16*Sqrt[2]*a^3*d) + (5*(A + (6*I)*B))/(8*a^3*d*S
qrt[Cot[c + d*x]]) + (I*A - B)/(6*d*Sqrt[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^3) + (2*A + (5*I)*B)/(12*a*d*Sqr
t[Cot[c + d*x]]*(I*a + a*Cot[c + d*x])^2) - (7*(I*A - 4*B))/(24*d*Sqrt[Cot[c + d*x]]*(I*a^3 + a^3*Cot[c + d*x]
)) + ((1/32 + I/32)*((6 + I)*A + (1 + 29*I)*B)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(Sqrt[2]*a^
3*d) - ((1/32 + I/32)*((6 + I)*A + (1 + 29*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 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 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{A+B \tan (c+d x)}{\cot ^{\frac{7}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx &=\int \frac{B+A \cot (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))^3} \, dx\\ &=\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{\int \frac{-\frac{1}{2} a (A+13 i B)-\frac{7}{2} a (i A-B) \cot (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2}+\frac{\int \frac{a^2 (4 i A-31 B)-5 a^2 (2 A+5 i B) \cot (c+d x)}{\cot ^{\frac{3}{2}}(c+d x) (i a+a \cot (c+d x))} \, dx}{24 a^4}\\ &=\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac{7 (i A-4 B)}{24 d \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{15 a^3 (A+6 i B)+21 a^3 (i A-4 B) \cot (c+d x)}{\cot ^{\frac{3}{2}}(c+d x)} \, dx}{48 a^6}\\ &=\frac{5 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac{7 (i A-4 B)}{24 d \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{21 a^3 (i A-4 B)-15 a^3 (A+6 i B) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{48 a^6}\\ &=\frac{5 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac{7 (i A-4 B)}{24 d \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{-21 a^3 (i A-4 B)+15 a^3 (A+6 i B) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{24 a^6 d}\\ &=\frac{5 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac{7 (i A-4 B)}{24 d \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}-\frac{((5+7 i) A-(28-30 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{((5-7 i) A+(28+30 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 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac{7 (i A-4 B)}{24 d \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{((5+7 i) A-(28-30 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+7 i) A-(28-30 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{((5-7 i) A+(28+30 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{((5-7 i) A+(28+30 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{5 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac{7 (i A-4 B)}{24 d \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{((5+7 i) A-(28-30 i) B) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}-\frac{((5+7 i) A-(28-30 i) B) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{((5-7 i) A+(28+30 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{((5-7 i) A+(28+30 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{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{((5-7 i) A+(28+30 i) B) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{5 (A+6 i B)}{8 a^3 d \sqrt{\cot (c+d x)}}+\frac{i A-B}{6 d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3}+\frac{2 A+5 i B}{12 a d \sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2}-\frac{7 (i A-4 B)}{24 d \sqrt{\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{((5+7 i) A-(28-30 i) B) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}-\frac{((5+7 i) A-(28-30 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 = 4.33462, size = 280, normalized size = 0.76 \[ \frac{\sec ^2(c+d x) (\cos (d x)+i \sin (d x))^3 (A+B \tan (c+d x)) \left (\frac{2}{3} (\cos (3 d x)-i \sin (3 d x)) ((9 A+33 i B) \cos (c+d x)+21 (A+7 i B) \cos (3 (c+d x))+2 i \sin (c+d x) ((19 A+145 i B) \cos (2 (c+d x))+19 A+97 i B))-i (\cos (3 c)+i \sin (3 c)) \sqrt{\sin (2 (c+d x))} \csc (c+d x) \left (((7+5 i) A-(30-28 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))+(1-i) ((6+i) A+(1+29 i) B) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{32 d \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[c + d*x])/(Cot[c + d*x]^(7/2)*(a + I*a*Tan[c + d*x])^3),x]

[Out]

(Sec[c + d*x]^2*(Cos[d*x] + I*Sin[d*x])^3*((2*(Cos[3*d*x] - I*Sin[3*d*x])*((9*A + (33*I)*B)*Cos[c + d*x] + 21*
(A + (7*I)*B)*Cos[3*(c + d*x)] + (2*I)*(19*A + (97*I)*B + (19*A + (145*I)*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/
3 - I*Csc[c + d*x]*(((7 + 5*I)*A - (30 - 28*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] + (1 - I)*((6 + I)*A + (
1 + 29*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*(Cos[3*c] + I*Sin[3*c])*Sqrt[Sin[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*Tan[c + d*
x])^3)

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Maple [C]  time = 0.635, size = 6350, normalized size = 17.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(d*x+c))/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^3,x)

[Out]

result too large to display

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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((A+B*tan(d*x+c))/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [B]  time = 1.767, size = 2064, normalized size = 5.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/96*(3*(a^3*d*e^(8*I*d*x + 8*I*c) + a^3*d*e^(6*I*d*x + 6*I*c))*sqrt((-I*A^2 - 2*A*B + I*B^2)/(a^6*d^2))*log(-
2*((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((-I*A^
2 - 2*A*B + I*B^2)/(a^6*d^2)) + (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*e^(8
*I*d*x + 8*I*c) + a^3*d*e^(6*I*d*x + 6*I*c))*sqrt((-I*A^2 - 2*A*B + I*B^2)/(a^6*d^2))*log(2*((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((-I*A^2 - 2*A*B + I*B^2)/(
a^6*d^2)) - (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*e^(8*I*d*x + 8*I*c) + a^
3*d*e^(6*I*d*x + 6*I*c))*sqrt((36*I*A^2 - 348*A*B - 841*I*B^2)/(a^6*d^2))*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((36*I*A^2 - 348*A*B - 841*I*B^2)/(a^
6*d^2)) + 6*A + 29*I*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) - 3*(a^3*d*e^(8*I*d*x + 8*I*c) + a^3*d*e^(6*I*d*x + 6*I*
c))*sqrt((36*I*A^2 - 348*A*B - 841*I*B^2)/(a^6*d^2))*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((36*I*A^2 - 348*A*B - 841*I*B^2)/(a^6*d^2)) - 6*A - 29*I
*B)*e^(-2*I*d*x - 2*I*c)/(a^3*d)) + 2*((-20*I*A + 146*B)*e^(8*I*d*x + 8*I*c) + (6*I*A - 105*B)*e^(6*I*d*x + 6*
I*c) + (19*I*A - 49*B)*e^(4*I*d*x + 4*I*c) + (-6*I*A + 9*B)*e^(2*I*d*x + 2*I*c) + I*A - B)*sqrt((I*e^(2*I*d*x
+ 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1)))/(a^3*d*e^(8*I*d*x + 8*I*c) + a^3*d*e^(6*I*d*x + 6*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((A+B*tan(d*x+c))/cot(d*x+c)**(7/2)/(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{B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/cot(d*x+c)^(7/2)/(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)/((I*a*tan(d*x + c) + a)^3*cot(d*x + c)^(7/2)), x)