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

Optimal. Leaf size=342 \[ \frac {a^{2/3} (3 B+2 i A) \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d}-\frac {\sqrt {3} a^{2/3} (B+i A) \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}-\frac {a^{2/3} (3 B+2 i A) \log (\tan (c+d x))}{6 d}+\frac {a^{2/3} (3 B+2 i A) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 a^{2/3} (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac {a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {a^{2/3} x (A-i B)}{2 \sqrt [3]{2}}-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d} \]

[Out]

1/4*a^(2/3)*(A-I*B)*x*2^(2/3)-1/4*a^(2/3)*(I*A+B)*ln(cos(d*x+c))*2^(2/3)/d-1/6*a^(2/3)*(2*I*A+3*B)*ln(tan(d*x+
c))/d+1/2*a^(2/3)*(2*I*A+3*B)*ln(a^(1/3)-(a+I*a*tan(d*x+c))^(1/3))/d-3/4*a^(2/3)*(I*A+B)*ln(2^(1/3)*a^(1/3)-(a
+I*a*tan(d*x+c))^(1/3))*2^(2/3)/d+1/3*a^(2/3)*(2*I*A+3*B)*arctan(1/3*(a^(1/3)+2*(a+I*a*tan(d*x+c))^(1/3))/a^(1
/3)*3^(1/2))/d*3^(1/2)-1/2*a^(2/3)*(I*A+B)*arctan(1/3*(a^(1/3)+2^(2/3)*(a+I*a*tan(d*x+c))^(1/3))/a^(1/3)*3^(1/
2))*3^(1/2)*2^(2/3)/d-A*cot(d*x+c)*(a+I*a*tan(d*x+c))^(2/3)/d

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Rubi [A]  time = 0.59, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3598, 3600, 3481, 55, 617, 204, 31, 3599} \[ \frac {a^{2/3} (3 B+2 i A) \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} d}-\frac {\sqrt {3} a^{2/3} (B+i A) \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{2} d}-\frac {a^{2/3} (3 B+2 i A) \log (\tan (c+d x))}{6 d}+\frac {a^{2/3} (3 B+2 i A) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 a^{2/3} (B+i A) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac {a^{2/3} (B+i A) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}+\frac {a^{2/3} x (A-i B)}{2 \sqrt [3]{2}}-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(2/3)*(A - I*B)*x)/(2*2^(1/3)) + (a^(2/3)*((2*I)*A + 3*B)*ArcTan[(a^(1/3) + 2*(a + I*a*Tan[c + d*x])^(1/3))
/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*d) - (Sqrt[3]*a^(2/3)*(I*A + B)*ArcTan[(a^(1/3) + 2^(2/3)*(a + I*a*Tan[c + d*x])
^(1/3))/(Sqrt[3]*a^(1/3))])/(2^(1/3)*d) - (a^(2/3)*(I*A + B)*Log[Cos[c + d*x]])/(2*2^(1/3)*d) - (a^(2/3)*((2*I
)*A + 3*B)*Log[Tan[c + d*x]])/(6*d) + (a^(2/3)*((2*I)*A + 3*B)*Log[a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2
*d) - (3*a^(2/3)*(I*A + B)*Log[2^(1/3)*a^(1/3) - (a + I*a*Tan[c + d*x])^(1/3)])/(2*2^(1/3)*d) - (A*Cot[c + d*x
]*(a + I*a*Tan[c + d*x])^(2/3))/d

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

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 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 3481

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

Rule 3598

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*d - B*c)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(f
*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n
+ 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3599

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

Rule 3600

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

Rubi steps

\begin {align*} \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx &=-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+\frac {\int \cot (c+d x) (a+i a \tan (c+d x))^{2/3} \left (\frac {1}{3} a (2 i A+3 B)-\frac {1}{3} a A \tan (c+d x)\right ) \, dx}{a}\\ &=-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+(-A+i B) \int (a+i a \tan (c+d x))^{2/3} \, dx+\frac {(2 i A+3 B) \int \cot (c+d x) (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{2/3} \, dx}{3 a}\\ &=-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+\frac {(a (i A+B)) \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac {(a (2 i A+3 B)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{a+i a x}} \, dx,x,\tan (c+d x)\right )}{3 d}\\ &=\frac {a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}-\frac {a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac {a^{2/3} (2 i A+3 B) \log (\tan (c+d x))}{6 d}-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+\frac {\left (3 a^{2/3} (i A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac {(3 a (i A+B)) \operatorname {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {\left (a^{2/3} (2 i A+3 B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}+\frac {(a (2 i A+3 B)) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}\\ &=\frac {a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}-\frac {a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac {a^{2/3} (2 i A+3 B) \log (\tan (c+d x))}{6 d}+\frac {a^{2/3} (2 i A+3 B) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}+\frac {\left (3 a^{2/3} (i A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{\sqrt [3]{2} d}-\frac {\left (a^{2/3} (2 i A+3 B)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{d}\\ &=\frac {a^{2/3} (A-i B) x}{2 \sqrt [3]{2}}+\frac {a^{2/3} (2 i A+3 B) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} d}-\frac {\sqrt {3} a^{2/3} (i A+B) \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{2} d}-\frac {a^{2/3} (i A+B) \log (\cos (c+d x))}{2 \sqrt [3]{2} d}-\frac {a^{2/3} (2 i A+3 B) \log (\tan (c+d x))}{6 d}+\frac {a^{2/3} (2 i A+3 B) \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 d}-\frac {3 a^{2/3} (i A+B) \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d}-\frac {A \cot (c+d x) (a+i a \tan (c+d x))^{2/3}}{d}\\ \end {align*}

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Mathematica [F]  time = 6.95, size = 0, normalized size = 0.00 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^{2/3} (A+B \tan (c+d x)) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.67, size = 1096, normalized size = 3.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*2^(2/3)*(-2*I*A*e^(2*I*d*x + 2*I*c) - 2*I*A)*(a/(e^(2*I*d*x + 2*I*c) + 1))^(2/3)*e^(4/3*I*d*x + 4/3*I*c
) + 6*(1/2)^(1/3)*(d*e^(2*I*d*x + 2*I*c) - d)*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(1/3)*log((2^(1/3)
*(A^2 - 2*I*A*B - B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) + 2*(1/2)^(2/3)*d^2*((I*A
^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(2/3))/((A^2 - 2*I*A*B - B^2)*a)) + 3*(1/2)^(1/3)*((I*sqrt(3)*d - d)*
e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(1/3)*log((2^(1/3)*(A^2 -
2*I*A*B - B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - (1/2)^(2/3)*(I*sqrt(3)*d^2 + d
^2)*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(2/3))/((A^2 - 2*I*A*B - B^2)*a)) + 3*(1/2)^(1/3)*((-I*sqrt(
3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d)*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(1/3)*log((2^(1
/3)*(A^2 - 2*I*A*B - B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - (1/2)^(2/3)*(-I*sqrt
(3)*d^2 + d^2)*((I*A^3 + 3*A^2*B - 3*I*A*B^2 - B^3)*a^2/d^3)^(2/3))/((A^2 - 2*I*A*B - B^2)*a)) + ((I*sqrt(3)*d
- d)*e^(2*I*d*x + 2*I*c) - I*sqrt(3)*d + d)*((-8*I*A^3 - 36*A^2*B + 54*I*A*B^2 + 27*B^3)*a^2/d^3)^(1/3)*log(1
/9*(9*2^(1/3)*(8*A^2 - 24*I*A*B - 18*B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*c) - (9*I
*sqrt(3)*d^2 + 9*d^2)*((-8*I*A^3 - 36*A^2*B + 54*I*A*B^2 + 27*B^3)*a^2/d^3)^(2/3))/((8*A^2 - 24*I*A*B - 18*B^2
)*a)) + ((-I*sqrt(3)*d - d)*e^(2*I*d*x + 2*I*c) + I*sqrt(3)*d + d)*((-8*I*A^3 - 36*A^2*B + 54*I*A*B^2 + 27*B^3
)*a^2/d^3)^(1/3)*log(1/9*(9*2^(1/3)*(8*A^2 - 24*I*A*B - 18*B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I
*d*x + 2/3*I*c) - (-9*I*sqrt(3)*d^2 + 9*d^2)*((-8*I*A^3 - 36*A^2*B + 54*I*A*B^2 + 27*B^3)*a^2/d^3)^(2/3))/((8*
A^2 - 24*I*A*B - 18*B^2)*a)) + 2*(d*e^(2*I*d*x + 2*I*c) - d)*((-8*I*A^3 - 36*A^2*B + 54*I*A*B^2 + 27*B^3)*a^2/
d^3)^(1/3)*log((2^(1/3)*(4*A^2 - 12*I*A*B - 9*B^2)*a*(a/(e^(2*I*d*x + 2*I*c) + 1))^(1/3)*e^(2/3*I*d*x + 2/3*I*
c) + d^2*((-8*I*A^3 - 36*A^2*B + 54*I*A*B^2 + 27*B^3)*a^2/d^3)^(2/3))/((4*A^2 - 12*I*A*B - 9*B^2)*a)))/(d*e^(2
*I*d*x + 2*I*c) - d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \cot \left (d x + c\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.14, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +B \tan \left (d x +c \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 298, normalized size = 0.87 \[ -\frac {i \, {\left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (A - i \, B\right )} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {3 \cdot 2^{\frac {2}{3}} {\left (A - i \, B\right )} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} + \frac {6 \cdot 2^{\frac {2}{3}} {\left (A - i \, B\right )} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {4 \, \sqrt {3} {\left (2 \, A - 3 i \, B\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} + \frac {2 \, {\left (2 \, A - 3 i \, B\right )} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}} - \frac {4 \, {\left (2 \, A - 3 i \, B\right )} \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} - \frac {12 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} A}{a \tan \left (d x + c\right )}\right )} a}{12 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*I*(6*sqrt(3)*2^(2/3)*(A - I*B)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(I*a*tan(d*x + c) + a)^(1
/3))/a^(1/3))/a^(1/3) - 3*2^(2/3)*(A - I*B)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(I*a*tan(d*x + c) + a)^(1/3)*a^(1/3)
+ (I*a*tan(d*x + c) + a)^(2/3))/a^(1/3) + 6*2^(2/3)*(A - I*B)*log(-2^(1/3)*a^(1/3) + (I*a*tan(d*x + c) + a)^(
1/3))/a^(1/3) - 4*sqrt(3)*(2*A - 3*I*B)*arctan(1/3*sqrt(3)*(2*(I*a*tan(d*x + c) + a)^(1/3) + a^(1/3))/a^(1/3))
/a^(1/3) + 2*(2*A - 3*I*B)*log((I*a*tan(d*x + c) + a)^(2/3) + (I*a*tan(d*x + c) + a)^(1/3)*a^(1/3) + a^(2/3))/
a^(1/3) - 4*(2*A - 3*I*B)*log((I*a*tan(d*x + c) + a)^(1/3) - a^(1/3))/a^(1/3) - 12*I*(I*a*tan(d*x + c) + a)^(2
/3)*A/(a*tan(d*x + c)))*a/d

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mupad [B]  time = 8.07, size = 5825, normalized size = 17.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((18*d^3*(A^3*a^9*19i - A*B^2*a^9*27i + 45*A^2*B*a^9) - (1458*a^7*d^6*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2
*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A
^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1
/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3) - 9*d*(
a + a*tan(c + d*x)*1i)^(1/3)*(135*B^2*a^8*d^3 - 75*A^2*a^8*d^3 + A*B*a^8*d^3*198i))*(-((d^3*((((594*A^3*a^12 +
1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^1
4 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1
i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1
/3))*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^1
0*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5
*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^1
2*486i)*1i)/(5832*a^10*d^3))^(2/3) + 9*d*(a + a*tan(c + d*x)*1i)^(1/3)*(A^5*a^10*16i + 27*B^5*a^10 + A*B^4*a^1
0*126i + 92*A^4*B*a^10 - 231*A^2*B^3*a^10 - A^3*B^2*a^10*208i))*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i
)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^
14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i -
594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3) + log((18*d^3*(A
^3*a^9*19i - A*B^2*a^9*27i + 45*A^2*B*a^9) - (1458*a^7*d^6*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3
+ (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 1
0044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^
3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3) - 9*d*(a + a*tan(c + d
*x)*1i)^(1/3)*(135*B^2*a^8*d^3 - 75*A^2*a^8*d^3 + A*B*a^8*d^3*198i))*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12
)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^
4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1
i + 594*A^3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3))*(((d^3*((((
594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14
- 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A
^3*B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832
*a^10*d^3))^(2/3) + 9*d*(a + a*tan(c + d*x)*1i)^(1/3)*(A^5*a^10*16i + 27*B^5*a^10 + A*B^4*a^10*126i + 92*A^4*B
*a^10 - 231*A^2*B^3*a^10 - A^3*B^2*a^10*208i))*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*
a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2
*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^3*a^12 - B^3
*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3) + log(((3^(1/2)*1i)/2 - 1/2)^2*(18
*d^3*(A^3*a^9*19i - A*B^2*a^9*27i + 45*A^2*B*a^9) + ((3^(1/2)*1i)/2 - 1/2)*(9*d*(a + a*tan(c + d*x)*1i)^(1/3)*
(135*B^2*a^8*d^3 - 75*A^2*a^8*d^3 + A*B*a^8*d^3*198i) - 1458*a^7*d^6*((3^(1/2)*1i)/2 - 1/2)^2*(-((d^3*((((594*
A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 14
58*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B
^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^1
0*d^3))^(2/3))*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 -
11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14
+ 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 +
A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3))*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*
a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2
*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3
*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3) + 9*d*(a + a*tan(c + d*x)*1i)^(1/3
)*(A^5*a^10*16i + 27*B^5*a^10 + A*B^4*a^10*126i + 92*A^4*B*a^10 - 231*A^2*B^3*a^10 - A^3*B^2*a^10*208i))*((3^(
1/2)*1i)/2 - 1/2)*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^
2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^
14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^1
2 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3) + log(((3^(1/2)*1i)/2 - 1/2)^2*(18*d^3*(A^3*a^9*19i - A*B^2*a^
9*27i + 45*A^2*B*a^9) + ((3^(1/2)*1i)/2 - 1/2)*(9*d*(a + a*tan(c + d*x)*1i)^(1/3)*(135*B^2*a^8*d^3 - 75*A^2*a^
8*d^3 + A*B*a^8*d^3*198i) - 1458*a^7*d^6*((3^(1/2)*1i)/2 - 1/2)^2*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1
i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a
^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i +
594*A^3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3))*(((d^3*((((594
*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1
458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*
B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a^
10*d^3))^(1/3))*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 -
11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14
+ 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 -
A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3) + 9*d*(a + a*tan(c + d*x)*1i)^(1/3)*(A^5*a^10*16i + 27*B^5*a^10 +
A*B^4*a^10*126i + 92*A^4*B*a^10 - 231*A^2*B^3*a^10 - A^3*B^2*a^10*208i))*((3^(1/2)*1i)/2 - 1/2)*(((d^3*((((59
4*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 -
1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3
*B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a
^10*d^3))^(1/3) - log(((3^(1/2)*1i)/2 + 1/2)^2*(18*d^3*(A^3*a^9*19i - A*B^2*a^9*27i + 45*A^2*B*a^9) - ((3^(1/2
)*1i)/2 + 1/2)*(9*d*(a + a*tan(c + d*x)*1i)^(1/3)*(135*B^2*a^8*d^3 - 75*A^2*a^8*d^3 + A*B*a^8*d^3*198i) - 1458
*a^7*d^6*((3^(1/2)*1i)/2 + 1/2)^2*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A
^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 -
((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i
- 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3))*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1
i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a
^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i -
594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3))*(-((d^3*((((59
4*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 -
1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3
*B^3*a^14)*1i)/d^6))^(1/2)*1i - 594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a
^10*d^3))^(2/3) + 9*d*(a + a*tan(c + d*x)*1i)^(1/3)*(A^5*a^10*16i + 27*B^5*a^10 + A*B^4*a^10*126i + 92*A^4*B*a
^10 - 231*A^2*B^3*a^10 - A^3*B^2*a^10*208i))*((3^(1/2)*1i)/2 + 1/2)*(-((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12
)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^
4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1
i - 594*A^3*a^12 + B^3*a^12*1458i - 1458*A*B^2*a^12 + A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3) - log(((3^(1
/2)*1i)/2 + 1/2)^2*(18*d^3*(A^3*a^9*19i - A*B^2*a^9*27i + 45*A^2*B*a^9) - ((3^(1/2)*1i)/2 + 1/2)*(9*d*(a + a*t
an(c + d*x)*1i)^(1/3)*(135*B^2*a^8*d^3 - 75*A^2*a^8*d^3 + A*B*a^8*d^3*198i) - 1458*a^7*d^6*((3^(1/2)*1i)/2 + 1
/2)^2*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^1
0*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5
*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^1
2*486i)*1i)/(5832*a^10*d^3))^(2/3))*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486*
A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6
- ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^3*a^12 - B^3*a^12*1458i
+ 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3))*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1
i)/d^3 + (1458*B^3*a^12 + 486*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a
^14 - 10044*A^4*B^2*a^14)/d^6 - ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i +
594*A^3*a^12 - B^3*a^12*1458i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(2/3) + 9*d*(a + a*ta
n(c + d*x)*1i)^(1/3)*(A^5*a^10*16i + 27*B^5*a^10 + A*B^4*a^10*126i + 92*A^4*B*a^10 - 231*A^2*B^3*a^10 - A^3*B^
2*a^10*208i))*((3^(1/2)*1i)/2 + 1/2)*(((d^3*((((594*A^3*a^12 + 1458*A*B^2*a^12)*1i)/d^3 + (1458*B^3*a^12 + 486
*A^2*B*a^12)/d^3)^2 - 11664*a^10*((432*A^6*a^14 - 1458*B^6*a^14 + 15066*A^2*B^4*a^14 - 10044*A^4*B^2*a^14)/d^6
- ((7290*A*B^5*a^14 + 3240*A^5*B*a^14 - 16470*A^3*B^3*a^14)*1i)/d^6))^(1/2)*1i + 594*A^3*a^12 - B^3*a^12*1458
i + 1458*A*B^2*a^12 - A^2*B*a^12*486i)*1i)/(5832*a^10*d^3))^(1/3) - (A*(a + a*tan(c + d*x)*1i)^(2/3))/(d*tan(c
+ d*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {2}{3}} \left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((I*a*(tan(c + d*x) - I))**(2/3)*(A + B*tan(c + d*x))*cot(c + d*x)**2, x)

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