3.263 \(\int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=187 \[ \frac{a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}-\frac{a \left (4 a^2 A b+a^3 B-6 a b^2 B-4 A b^3\right ) \log (\sin (c+d x))}{d}+x \left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right )-\frac{a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}-\frac{b^4 B \log (\cos (c+d x))}{d} \]

[Out]

(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x + (a^2*(a^2*A - 3*A*b^2 - 3*a*b*B)*Cot[c + d*x])/d - (
b^4*B*Log[Cos[c + d*x]])/d - (a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Log[Sin[c + d*x]])/d - (a*(2*A*b + a
*B)*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2)/(2*d) - (a*A*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(3*d)

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Rubi [A]  time = 0.530705, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3605, 3645, 3635, 3624, 3475} \[ \frac{a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}-\frac{a \left (4 a^2 A b+a^3 B-6 a b^2 B-4 A b^3\right ) \log (\sin (c+d x))}{d}+x \left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right )-\frac{a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}-\frac{b^4 B \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x + (a^2*(a^2*A - 3*A*b^2 - 3*a*b*B)*Cot[c + d*x])/d - (
b^4*B*Log[Cos[c + d*x]])/d - (a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Log[Sin[c + d*x]])/d - (a*(2*A*b + a
*B)*Cot[c + d*x]^2*(a + b*Tan[c + d*x])^2)/(2*d) - (a*A*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(3*d)

Rule 3605

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

Rule 3645

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

Rule 3635

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

Rule 3624

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/tan[(e_.) + (f_.)*(x_)], x_Symbol
] :> Simp[B*x, x] + (Dist[A, Int[1/Tan[e + f*x], x], x] + Dist[C, Int[Tan[e + f*x], x], x]) /; FreeQ[{e, f, A,
B, C}, x] && NeQ[A, C]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}+\frac{1}{3} \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 \left (3 a (2 A b+a B)-3 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+3 b^2 B \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}+\frac{1}{6} \int \cot ^2(c+d x) (a+b \tan (c+d x)) \left (-6 a \left (a^2 A-3 A b^2-3 a b B\right )-6 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+6 b^3 B \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \left (a^2 A-3 A b^2-3 a b B\right ) \cot (c+d x)}{d}-\frac{a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}+\frac{1}{6} \int \cot (c+d x) \left (-6 a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right )+6 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \tan (c+d x)+6 b^4 B \tan ^2(c+d x)\right ) \, dx\\ &=\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x+\frac{a^2 \left (a^2 A-3 A b^2-3 a b B\right ) \cot (c+d x)}{d}-\frac{a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}+\left (b^4 B\right ) \int \tan (c+d x) \, dx-\left (a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right )\right ) \int \cot (c+d x) \, dx\\ &=\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x+\frac{a^2 \left (a^2 A-3 A b^2-3 a b B\right ) \cot (c+d x)}{d}-\frac{b^4 B \log (\cos (c+d x))}{d}-\frac{a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \log (\sin (c+d x))}{d}-\frac{a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac{a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\\ \end{align*}

Mathematica [C]  time = 1.09511, size = 167, normalized size = 0.89 \[ \frac{6 a^2 \left (a^2 A-4 a b B-6 A b^2\right ) \cot (c+d x)-6 a \left (4 a^2 A b+a^3 B-6 a b^2 B-4 A b^3\right ) \log (\tan (c+d x))-3 a^3 (a B+4 A b) \cot ^2(c+d x)-2 a^4 A \cot ^3(c+d x)+3 (a+i b)^4 (B-i A) \log (-\tan (c+d x)+i)+3 (a-i b)^4 (B+i A) \log (\tan (c+d x)+i)}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(6*a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x] - 3*a^3*(4*A*b + a*B)*Cot[c + d*x]^2 - 2*a^4*A*Cot[c + d*x]^3
+ 3*(a + I*b)^4*((-I)*A + B)*Log[I - Tan[c + d*x]] - 6*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Log[Tan[c +
d*x]] + 3*(a - I*b)^4*(I*A + B)*Log[I + Tan[c + d*x]])/(6*d)

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Maple [A]  time = 0.089, size = 278, normalized size = 1.5 \begin{align*} A{b}^{4}x+{\frac{A{b}^{4}c}{d}}-{\frac{B{b}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{Aa{b}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+4\,Ba{b}^{3}x+4\,{\frac{Ba{b}^{3}c}{d}}-6\,A{a}^{2}{b}^{2}x-6\,{\frac{A\cot \left ( dx+c \right ){a}^{2}{b}^{2}}{d}}-6\,{\frac{A{a}^{2}{b}^{2}c}{d}}+6\,{\frac{B{a}^{2}{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-2\,{\frac{A{a}^{3}b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{A{a}^{3}b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-4\,B{a}^{3}bx-4\,{\frac{B\cot \left ( dx+c \right ){a}^{3}b}{d}}-4\,{\frac{B{a}^{3}bc}{d}}-{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{A\cot \left ( dx+c \right ){a}^{4}}{d}}+A{a}^{4}x+{\frac{A{a}^{4}c}{d}}-{\frac{B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{B{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.49889, size = 273, normalized size = 1.46 \begin{align*} \frac{6 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )}{\left (d x + c\right )} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{2 \, A a^{4} - 6 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.28901, size = 520, normalized size = 2.78 \begin{align*} -\frac{3 \, B b^{4} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 2 \, A a^{4} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b - 2 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 6 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 41.3728, size = 369, normalized size = 1.97 \begin{align*} \begin{cases} \tilde{\infty } A a^{4} x & \text{for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (A + B \tan{\left (c \right )}\right ) \left (a + b \tan{\left (c \right )}\right )^{4} \cot ^{4}{\left (c \right )} & \text{for}\: d = 0 \\A a^{4} x + \frac{A a^{4}}{d \tan{\left (c + d x \right )}} - \frac{A a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac{2 A a^{3} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac{4 A a^{3} b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{2 A a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - 6 A a^{2} b^{2} x - \frac{6 A a^{2} b^{2}}{d \tan{\left (c + d x \right )}} - \frac{2 A a b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{4 A a b^{3} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + A b^{4} x + \frac{B a^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{B a^{4} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{B a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - 4 B a^{3} b x - \frac{4 B a^{3} b}{d \tan{\left (c + d x \right )}} - \frac{3 B a^{2} b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{6 B a^{2} b^{2} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 4 B a b^{3} x + \frac{B b^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+b*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

Piecewise((zoo*A*a**4*x, (Eq(c, 0) | Eq(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x*(A + B*tan(c))*(a + b*tan(c)
)**4*cot(c)**4, Eq(d, 0)), (A*a**4*x + A*a**4/(d*tan(c + d*x)) - A*a**4/(3*d*tan(c + d*x)**3) + 2*A*a**3*b*log
(tan(c + d*x)**2 + 1)/d - 4*A*a**3*b*log(tan(c + d*x))/d - 2*A*a**3*b/(d*tan(c + d*x)**2) - 6*A*a**2*b**2*x -
6*A*a**2*b**2/(d*tan(c + d*x)) - 2*A*a*b**3*log(tan(c + d*x)**2 + 1)/d + 4*A*a*b**3*log(tan(c + d*x))/d + A*b*
*4*x + B*a**4*log(tan(c + d*x)**2 + 1)/(2*d) - B*a**4*log(tan(c + d*x))/d - B*a**4/(2*d*tan(c + d*x)**2) - 4*B
*a**3*b*x - 4*B*a**3*b/(d*tan(c + d*x)) - 3*B*a**2*b**2*log(tan(c + d*x)**2 + 1)/d + 6*B*a**2*b**2*log(tan(c +
d*x))/d + 4*B*a*b**3*x + B*b**4*log(tan(c + d*x)**2 + 1)/(2*d), True))

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Giac [A]  time = 2.80737, size = 379, normalized size = 2.03 \begin{align*} \frac{6 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )}{\left (d x + c\right )} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + \frac{11 \, B a^{4} \tan \left (d x + c\right )^{3} + 44 \, A a^{3} b \tan \left (d x + c\right )^{3} - 66 \, B a^{2} b^{2} \tan \left (d x + c\right )^{3} - 44 \, A a b^{3} \tan \left (d x + c\right )^{3} + 6 \, A a^{4} \tan \left (d x + c\right )^{2} - 24 \, B a^{3} b \tan \left (d x + c\right )^{2} - 36 \, A a^{2} b^{2} \tan \left (d x + c\right )^{2} - 3 \, B a^{4} \tan \left (d x + c\right ) - 12 \, A a^{3} b \tan \left (d x + c\right ) - 2 \, A a^{4}}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

1/6*(6*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*(d*x + c) + 3*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 -
4*A*a*b^3 + B*b^4)*log(tan(d*x + c)^2 + 1) - 6*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3)*log(abs(tan(d*x +
c))) + (11*B*a^4*tan(d*x + c)^3 + 44*A*a^3*b*tan(d*x + c)^3 - 66*B*a^2*b^2*tan(d*x + c)^3 - 44*A*a*b^3*tan(d*
x + c)^3 + 6*A*a^4*tan(d*x + c)^2 - 24*B*a^3*b*tan(d*x + c)^2 - 36*A*a^2*b^2*tan(d*x + c)^2 - 3*B*a^4*tan(d*x
+ c) - 12*A*a^3*b*tan(d*x + c) - 2*A*a^4)/tan(d*x + c)^3)/d