3.264 \(\int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=225 \[ \frac{a^2 \left (6 a^2 A-16 a b B-13 A b^2\right ) \cot ^2(c+d x)}{12 d}+\frac{a \left (24 a^2 A b+6 a^3 B-34 a b^2 B-19 A b^3\right ) \cot (c+d x)}{6 d}+\frac{\left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \log (\sin (c+d x))}{d}+x \left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right )-\frac{a (4 a B+7 A b) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d} \]

[Out]

(4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x + (a*(24*a^2*A*b - 19*A*b^3 + 6*a^3*B - 34*a*b^2*B)*Co
t[c + d*x])/(6*d) + (a^2*(6*a^2*A - 13*A*b^2 - 16*a*b*B)*Cot[c + d*x]^2)/(12*d) + ((a^4*A - 6*a^2*A*b^2 + A*b^
4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Sin[c + d*x]])/d - (a*(7*A*b + 4*a*B)*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(1
2*d) - (a*A*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3)/(4*d)

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Rubi [A]  time = 0.644813, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3605, 3645, 3635, 3628, 3531, 3475} \[ \frac{a^2 \left (6 a^2 A-16 a b B-13 A b^2\right ) \cot ^2(c+d x)}{12 d}+\frac{a \left (24 a^2 A b+6 a^3 B-34 a b^2 B-19 A b^3\right ) \cot (c+d x)}{6 d}+\frac{\left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \log (\sin (c+d x))}{d}+x \left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right )-\frac{a (4 a B+7 A b) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x + (a*(24*a^2*A*b - 19*A*b^3 + 6*a^3*B - 34*a*b^2*B)*Co
t[c + d*x])/(6*d) + (a^2*(6*a^2*A - 13*A*b^2 - 16*a*b*B)*Cot[c + d*x]^2)/(12*d) + ((a^4*A - 6*a^2*A*b^2 + A*b^
4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Sin[c + d*x]])/d - (a*(7*A*b + 4*a*B)*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(1
2*d) - (a*A*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3)/(4*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 3628

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

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*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] && NeQ[a*c + b*d, 0]

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 ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (a (7 A b+4 a B)-4 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-b (a A-4 b B) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\frac{1}{12} \int \cot ^3(c+d x) (a+b \tan (c+d x)) \left (-2 a \left (6 a^2 A-13 A b^2-16 a b B\right )-12 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-2 b \left (5 a A b+2 a^2 B-6 b^2 B\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}-\frac{a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\frac{1}{12} \int \cot ^2(c+d x) \left (-2 a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right )+12 \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)-2 b^2 \left (5 a A b+2 a^2 B-6 b^2 B\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right ) \cot (c+d x)}{6 d}+\frac{a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}-\frac{a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\frac{1}{12} \int \cot (c+d x) \left (12 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right )+12 \left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \tan (c+d x)\right ) \, dx\\ &=\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x+\frac{a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right ) \cot (c+d x)}{6 d}+\frac{a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}-\frac{a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \int \cot (c+d x) \, dx\\ &=\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x+\frac{a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right ) \cot (c+d x)}{6 d}+\frac{a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}+\frac{\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\sin (c+d x))}{d}-\frac{a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}\\ \end{align*}

Mathematica [C]  time = 0.919856, size = 211, normalized size = 0.94 \[ \frac{6 a^2 \left (a^2 A-4 a b B-6 A b^2\right ) \cot ^2(c+d x)+12 a \left (4 a^2 A b+a^3 B-6 a b^2 B-4 A b^3\right ) \cot (c+d x)+12 \left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \log (\tan (c+d x))-4 a^3 (a B+4 A b) \cot ^3(c+d x)-3 a^4 A \cot ^4(c+d x)-6 (a-i b)^4 (A-i B) \log (\tan (c+d x)+i)-6 (a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(12*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Cot[c + d*x] + 6*a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x]^
2 - 4*a^3*(4*A*b + a*B)*Cot[c + d*x]^3 - 3*a^4*A*Cot[c + d*x]^4 - 6*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]
] + 12*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Tan[c + d*x]] - 6*(a - I*b)^4*(A - I*B)*Log[I
+ Tan[c + d*x]])/(12*d)

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Maple [A]  time = 0.094, size = 347, normalized size = 1.5 \begin{align*} -4\,Aa{b}^{3}x-6\,B{a}^{2}{b}^{2}x+4\,Ax{a}^{3}b-4\,{\frac{Aa{b}^{3}c}{d}}+{\frac{A{b}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+B{a}^{4}x+B{b}^{4}x+{\frac{A{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{B\cot \left ( dx+c \right ){a}^{4}}{d}}+{\frac{B{a}^{4}c}{d}}-6\,{\frac{A{a}^{2}{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{A\cot \left ( dx+c \right ){a}^{3}b}{d}}-4\,{\frac{B{a}^{3}b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{B{b}^{4}c}{d}}-6\,{\frac{B{a}^{2}{b}^{2}c}{d}}+4\,{\frac{A{a}^{3}bc}{d}}-2\,{\frac{B{a}^{3}b \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{A\cot \left ( dx+c \right ) a{b}^{3}}{d}}+4\,{\frac{Ba{b}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{A{a}^{2}{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-6\,{\frac{B\cot \left ( dx+c \right ){a}^{2}{b}^{2}}{d}}-{\frac{4\,A{a}^{3}b \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.5022, size = 332, normalized size = 1.48 \begin{align*} \frac{12 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )}{\left (d x + c\right )} - 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 )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{3 \, A a^{4} - 12 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\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} + 4 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(12*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*(d*x + c) - 6*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2
+ 4*B*a*b^3 + A*b^4)*log(tan(d*x + c)^2 + 1) + 12*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(ta
n(d*x + c)) - (3*A*a^4 - 12*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3)*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 + 4*(B*a^4 + 4*A*a^3*b)*tan(d*x + c))/tan(d*x + c)^4)/d

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Fricas [A]  time = 1.92654, size = 571, normalized size = 2.54 \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 )} \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 )^{4} - 3 \, A a^{4} + 3 \,{\left (3 \, A a^{4} - 8 \, B a^{3} b - 12 \, A a^{2} b^{2} + 4 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{4} + 12 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\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} - 4 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{12 \, d \tan \left (d x + c\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(6*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x
+ c)^4 - 3*A*a^4 + 3*(3*A*a^4 - 8*B*a^3*b - 12*A*a^2*b^2 + 4*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B
*b^4)*d*x)*tan(d*x + c)^4 + 12*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3)*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 - 4*(B*a^4 + 4*A*a^3*b)*tan(d*x + c))/(d*tan(d*x + c)^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**5*(a+b*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 2.90367, size = 788, normalized size = 3.5 \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)^5*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/192*(3*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 8*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 32*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 -
36*A*a^4*tan(1/2*d*x + 1/2*c)^2 + 96*B*a^3*b*tan(1/2*d*x + 1/2*c)^2 + 144*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 1
20*B*a^4*tan(1/2*d*x + 1/2*c) + 480*A*a^3*b*tan(1/2*d*x + 1/2*c) - 576*B*a^2*b^2*tan(1/2*d*x + 1/2*c) - 384*A*
a*b^3*tan(1/2*d*x + 1/2*c) - 192*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*(d*x + c) + 192*(A*a^4
- 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 192*(A*a^4 - 4*B*a^3*b - 6*A*
a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(abs(tan(1/2*d*x + 1/2*c))) + (400*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 1600*B*a^3*b
*tan(1/2*d*x + 1/2*c)^4 - 2400*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 1600*B*a*b^3*tan(1/2*d*x + 1/2*c)^4 + 400*A*
b^4*tan(1/2*d*x + 1/2*c)^4 - 120*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 480*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 576*B*a^2
*b^2*tan(1/2*d*x + 1/2*c)^3 + 384*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 36*A*a^4*tan(1/2*d*x + 1/2*c)^2 + 96*B*a^3*
b*tan(1/2*d*x + 1/2*c)^2 + 144*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 8*B*a^4*tan(1/2*d*x + 1/2*c) + 32*A*a^3*b*ta
n(1/2*d*x + 1/2*c) + 3*A*a^4)/tan(1/2*d*x + 1/2*c)^4)/d