3.266 \(\int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=323 \[ \frac{a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{20 d}+\frac{a \left (20 a^2 A b+5 a^3 B-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{15 d}-\frac{\left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac{\left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}-\frac{\left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \log (\sin (c+d x))}{d}-x \left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right )-\frac{a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d} \]

[Out]

-((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x) - ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b
^4*B)*Cot[c + d*x])/d - ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Cot[c + d*x]^2)/(2*d) + (a*(20*
a^2*A*b - 13*A*b^3 + 5*a^3*B - 27*a*b^2*B)*Cot[c + d*x]^3)/(15*d) + (a^2*(5*a^2*A - 8*A*b^2 - 12*a*b*B)*Cot[c
+ d*x]^4)/(20*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*(3*A*b + 2
*a*B)*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(10*d) - (a*A*Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3)/(6*d)

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Rubi [A]  time = 0.858379, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {3605, 3645, 3635, 3628, 3529, 3531, 3475} \[ \frac{a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{20 d}+\frac{a \left (20 a^2 A b+5 a^3 B-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{15 d}-\frac{\left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac{\left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}-\frac{\left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \log (\sin (c+d x))}{d}-x \left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right )-\frac{a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*(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) - ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b
^4*B)*Cot[c + d*x])/d - ((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Cot[c + d*x]^2)/(2*d) + (a*(20*
a^2*A*b - 13*A*b^3 + 5*a^3*B - 27*a*b^2*B)*Cot[c + d*x]^3)/(15*d) + (a^2*(5*a^2*A - 8*A*b^2 - 12*a*b*B)*Cot[c
+ d*x]^4)/(20*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*(3*A*b + 2
*a*B)*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(10*d) - (a*A*Cot[c + d*x]^6*(a + b*Tan[c + d*x])^3)/(6*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 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 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 ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}+\frac{1}{6} \int \cot ^6(c+d x) (a+b \tan (c+d x))^2 \left (3 a (3 A b+2 a B)-6 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-3 b (a A-2 b B) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{a (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}+\frac{1}{30} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (-6 a \left (5 a^2 A-8 A b^2-12 a b B\right )-30 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-6 b \left (7 a A b+3 a^2 B-5 b^2 B\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \left (5 a^2 A-8 A b^2-12 a b B\right ) \cot ^4(c+d x)}{20 d}-\frac{a (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}+\frac{1}{30} \int \cot ^4(c+d x) \left (-6 a \left (20 a^2 A b-13 A b^3+5 a^3 B-27 a b^2 B\right )+30 \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^2 \left (7 a A b+3 a^2 B-5 b^2 B\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (20 a^2 A b-13 A b^3+5 a^3 B-27 a b^2 B\right ) \cot ^3(c+d x)}{15 d}+\frac{a^2 \left (5 a^2 A-8 A b^2-12 a b B\right ) \cot ^4(c+d x)}{20 d}-\frac{a (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}+\frac{1}{30} \int \cot ^3(c+d x) \left (30 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right )+30 \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\\ &=-\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 ) \cot ^2(c+d x)}{2 d}+\frac{a \left (20 a^2 A b-13 A b^3+5 a^3 B-27 a b^2 B\right ) \cot ^3(c+d x)}{15 d}+\frac{a^2 \left (5 a^2 A-8 A b^2-12 a b B\right ) \cot ^4(c+d x)}{20 d}-\frac{a (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}+\frac{1}{30} \int \cot ^2(c+d x) \left (30 \left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right )-30 \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)\right ) \, dx\\ &=-\frac{\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \cot (c+d x)}{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 ) \cot ^2(c+d x)}{2 d}+\frac{a \left (20 a^2 A b-13 A b^3+5 a^3 B-27 a b^2 B\right ) \cot ^3(c+d x)}{15 d}+\frac{a^2 \left (5 a^2 A-8 A b^2-12 a b B\right ) \cot ^4(c+d x)}{20 d}-\frac{a (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}+\frac{1}{30} \int \cot (c+d x) \left (-30 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right )-30 \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{\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \cot (c+d x)}{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 ) \cot ^2(c+d x)}{2 d}+\frac{a \left (20 a^2 A b-13 A b^3+5 a^3 B-27 a b^2 B\right ) \cot ^3(c+d x)}{15 d}+\frac{a^2 \left (5 a^2 A-8 A b^2-12 a b B\right ) \cot ^4(c+d x)}{20 d}-\frac{a (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 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{\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \cot (c+d x)}{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 ) \cot ^2(c+d x)}{2 d}+\frac{a \left (20 a^2 A b-13 A b^3+5 a^3 B-27 a b^2 B\right ) \cot ^3(c+d x)}{15 d}+\frac{a^2 \left (5 a^2 A-8 A b^2-12 a b B\right ) \cot ^4(c+d x)}{20 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 (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac{a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d}\\ \end{align*}

Mathematica [C]  time = 1.26809, size = 299, normalized size = 0.93 \[ \frac{15 a^2 \left (a^2 A-4 a b B-6 A b^2\right ) \cot ^4(c+d x)+20 a \left (4 a^2 A b+a^3 B-6 a b^2 B-4 A b^3\right ) \cot ^3(c+d x)-30 \left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)-60 \left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right ) \cot (c+d x)-60 \left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \log (\tan (c+d x))-12 a^3 (a B+4 A b) \cot ^5(c+d x)-10 a^4 A \cot ^6(c+d x)+30 (a+i b)^4 (A+i B) \log (-\tan (c+d x)+i)+30 (a-i b)^4 (A-i B) \log (\tan (c+d x)+i)}{60 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(-60*(4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*Cot[c + d*x] - 30*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*
a^3*b*B + 4*a*b^3*B)*Cot[c + d*x]^2 + 20*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Cot[c + d*x]^3 + 15*a^2*(
a^2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x]^4 - 12*a^3*(4*A*b + a*B)*Cot[c + d*x]^5 - 10*a^4*A*Cot[c + d*x]^6 + 30
*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]] - 60*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Ta
n[c + d*x]] + 30*(a - I*b)^4*(A - I*B)*Log[I + Tan[c + d*x]])/(60*d)

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Maple [A]  time = 0.106, size = 532, normalized size = 1.7 \begin{align*} 4\,Aa{b}^{3}x+6\,B{a}^{2}{b}^{2}x-4\,Ax{a}^{3}b-{\frac{A{b}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{B\cot \left ( dx+c \right ){b}^{4}}{d}}+4\,{\frac{Aa{b}^{3}c}{d}}-{\frac{A{b}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{4\,Aa{b}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{Ba{b}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{3\,A{a}^{2}{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{2\,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 ) ^{6}}{6\,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}^{2}{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{4\,A{a}^{3}b \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{B{a}^{3}b \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{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}}-{\frac{B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^7*(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-1/2/d*A*b^4*cot(d*x+c)^2-1/d*B*cot(d*x+c)*b^4+4/d*A*a*b^3*c-1/d*A*b^4*ln
(sin(d*x+c))-4/3/d*A*a*b^3*cot(d*x+c)^3-2/d*B*a*b^3*cot(d*x+c)^2-3/2/d*A*a^2*b^2*cot(d*x+c)^4+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-1/6/d*A*a^4*cot(d*x+c)^6-1/2/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^2*b^2*cot(d*x+c)^3-4/5/d*A*a^3*b*cot(d*x+c)^5
-1/d*B*a^3*b*cot(d*x+c)^4+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)^3-1/5/d*B*a^4*cot(d*x+c)^5

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Maxima [A]  time = 1.50254, size = 450, normalized size = 1.39 \begin{align*} -\frac{60 \,{\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 )} - 30 \,{\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 ) + 60 \,{\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{60 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )^{5} + 10 \, A a^{4} + 30 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} - 20 \,{\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} - 15 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 12 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(60*(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) - 30*(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) + 60*(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)) + (60*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*tan(d*x + c)^5 + 10*A*a^4 + 30*(A*a^
4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*tan(d*x + c)^4 - 20*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*
b^3)*tan(d*x + c)^3 - 15*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2)*tan(d*x + c)^2 + 12*(B*a^4 + 4*A*a^3*b)*tan(d*x + c
))/tan(d*x + c)^6)/d

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Fricas [A]  time = 1.80574, size = 813, normalized size = 2.52 \begin{align*} -\frac{30 \,{\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 )^{6} + 5 \,{\left (11 \, A a^{4} - 36 \, B a^{3} b - 54 \, A a^{2} b^{2} + 24 \, B a b^{3} + 6 \, A b^{4} + 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 )} d x\right )} \tan \left (d x + c\right )^{6} + 60 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )^{5} + 10 \, A a^{4} + 30 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} - 20 \,{\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} - 15 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 12 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{60 \, d \tan \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/60*(30*(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)^6 + 5*(11*A*a^4 - 36*B*a^3*b - 54*A*a^2*b^2 + 24*B*a*b^3 + 6*A*b^4 + 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)*tan(d*x + c)^6 + 60*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*tan(d*x
+ c)^5 + 10*A*a^4 + 30*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*tan(d*x + c)^4 - 20*(B*a^4 + 4*A
*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3)*tan(d*x + c)^3 - 15*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2)*tan(d*x + c)^2 + 12*(B
*a^4 + 4*A*a^3*b)*tan(d*x + c))/(d*tan(d*x + c)^6)

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

[Out]

Timed out

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Giac [B]  time = 3.02937, size = 1273, normalized size = 3.94 \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)^7*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/1920*(5*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 12*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^3*b*tan(1/2*d*x + 1/2*c)^5
- 60*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 120*B*a^3*b*tan(1/2*d*x + 1/2*c)^4 + 180*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^4
+ 140*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 560*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 480*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3
- 320*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 435*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 1440*B*a^3*b*tan(1/2*d*x + 1/2*c)^2
- 2160*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 960*B*a*b^3*tan(1/2*d*x + 1/2*c)^2 + 240*A*b^4*tan(1/2*d*x + 1/2*c)
^2 - 1320*B*a^4*tan(1/2*d*x + 1/2*c) - 5280*A*a^3*b*tan(1/2*d*x + 1/2*c) + 7200*B*a^2*b^2*tan(1/2*d*x + 1/2*c)
+ 4800*A*a*b^3*tan(1/2*d*x + 1/2*c) - 960*B*b^4*tan(1/2*d*x + 1/2*c) + 1920*(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) - 1920*(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) + 1920*(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))) - (4
704*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 18816*B*a^3*b*tan(1/2*d*x + 1/2*c)^6 - 28224*A*a^2*b^2*tan(1/2*d*x + 1/2*c)
^6 + 18816*B*a*b^3*tan(1/2*d*x + 1/2*c)^6 + 4704*A*b^4*tan(1/2*d*x + 1/2*c)^6 - 1320*B*a^4*tan(1/2*d*x + 1/2*c
)^5 - 5280*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 7200*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 4800*A*a*b^3*tan(1/2*d*x +
1/2*c)^5 - 960*B*b^4*tan(1/2*d*x + 1/2*c)^5 - 435*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 1440*B*a^3*b*tan(1/2*d*x + 1
/2*c)^4 + 2160*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 - 960*B*a*b^3*tan(1/2*d*x + 1/2*c)^4 - 240*A*b^4*tan(1/2*d*x +
1/2*c)^4 + 140*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 560*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 480*B*a^2*b^2*tan(1/2*d*x
+ 1/2*c)^3 - 320*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 60*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 120*B*a^3*b*tan(1/2*d*x +
1/2*c)^2 - 180*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 12*B*a^4*tan(1/2*d*x + 1/2*c) - 48*A*a^3*b*tan(1/2*d*x + 1/2
*c) - 5*A*a^4)/tan(1/2*d*x + 1/2*c)^6)/d