\(\int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [266]

Optimal result

Integrand size = 31, antiderivative size = 323 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\left (\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x\right )-\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \cot (c+d x)}{d}-\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (20 a^2 A b-13 A b^3+5 a^3 B-27 a b^2 B\right ) \cot ^3(c+d x)}{15 d}+\frac {a^2 \left (5 a^2 A-8 A b^2-12 a b B\right ) \cot ^4(c+d x)}{20 d}-\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\sin (c+d x))}{d}-\frac {a (3 A b+2 a B) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d} \]

[Out]

-(4*A*a^3*b-4*A*a*b^3+B*a^4-6*B*a^2*b^2+B*b^4)*x-(4*A*a^3*b-4*A*a*b^3+B*a^4-6*B*a^2*b^2+B*b^4)*cot(d*x+c)/d-1/
2*(A*a^4-6*A*a^2*b^2+A*b^4-4*B*a^3*b+4*B*a*b^3)*cot(d*x+c)^2/d+1/15*a*(20*A*a^2*b-13*A*b^3+5*B*a^3-27*B*a*b^2)
*cot(d*x+c)^3/d+1/20*a^2*(5*A*a^2-8*A*b^2-12*B*a*b)*cot(d*x+c)^4/d-(A*a^4-6*A*a^2*b^2+A*b^4-4*B*a^3*b+4*B*a*b^
3)*ln(sin(d*x+c))/d-1/10*a*(3*A*b+2*B*a)*cot(d*x+c)^5*(a+b*tan(d*x+c))^2/d-1/6*a*A*cot(d*x+c)^6*(a+b*tan(d*x+c
))^3/d

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3686, 3726, 3716, 3709, 3610, 3612, 3556} \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^2 \left (5 a^2 A-12 a b B-8 A b^2\right ) \cot ^4(c+d x)}{20 d}+\frac {a \left (5 a^3 B+20 a^2 A b-27 a b^2 B-13 A b^3\right ) \cot ^3(c+d x)}{15 d}-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \cot (c+d x)}{d}-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \log (\sin (c+d x))}{d}-x \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right )-\frac {a (2 a B+3 A b) \cot ^5(c+d x) (a+b \tan (c+d x))^2}{10 d}-\frac {a A \cot ^6(c+d x) (a+b \tan (c+d x))^3}{6 d} \]

[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 3556

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

Rule 3610

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 3612

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 3686

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 3709

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 3716

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 3726

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*Ta
n[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]

Rubi steps \begin{align*} \text {integral}& = -\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 (\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\right )-\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 (\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\right )-\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] (verified)

Result contains complex when optimal does not.

Time = 1.38 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.93 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {-60 \left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \cot (c+d x)-30 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \cot ^2(c+d x)+20 a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \cot ^3(c+d x)+15 a^2 \left (a^2 A-6 A b^2-4 a b B\right ) \cot ^4(c+d x)-12 a^3 (4 A b+a B) \cot ^5(c+d x)-10 a^4 A \cot ^6(c+d x)+30 (a+i b)^4 (A+i B) \log (i-\tan (c+d x))-60 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\tan (c+d x))+30 (a-i b)^4 (A-i B) \log (i+\tan (c+d x))}{60 d} \]

[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)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.02

method	result	size
parallelrisch	\(\frac {\left (30 A \,a^{4}-180 A \,a^{2} b^{2}+30 A \,b^{4}-120 B \,a^{3} b +120 B a \,b^{3}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (-60 A \,a^{4}+360 A \,a^{2} b^{2}-60 A \,b^{4}+240 B \,a^{3} b -240 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-10 A \left (\cot ^{6}\left (d x +c \right )\right ) a^{4}+\left (-48 A \,a^{3} b -12 B \,a^{4}\right ) \left (\cot ^{5}\left (d x +c \right )\right )+15 a^{2} \left (\cot ^{4}\left (d x +c \right )\right ) \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )+\left (80 A \,a^{3} b -80 A a \,b^{3}+20 B \,a^{4}-120 B \,a^{2} b^{2}\right ) \left (\cot ^{3}\left (d x +c \right )\right )+\left (-30 A \,a^{4}+180 A \,a^{2} b^{2}-30 A \,b^{4}+120 B \,a^{3} b -120 B a \,b^{3}\right ) \left (\cot ^{2}\left (d x +c \right )\right )+\left (-240 A \,a^{3} b +240 A a \,b^{3}-60 B \,a^{4}+360 B \,a^{2} b^{2}-60 B \,b^{4}\right ) \cot \left (d x +c \right )-240 d \left (A \,a^{3} b -A a \,b^{3}+\frac {1}{4} B \,a^{4}-\frac {3}{2} B \,a^{2} b^{2}+\frac {1}{4} B \,b^{4}\right ) x}{60 d}\)	\(330\)
derivativedivides	\(\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}}{\tan \left (d x +c \right )}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}}{2 \tan \left (d x +c \right )^{2}}-\frac {A \,a^{4}}{6 \tan \left (d x +c \right )^{6}}-\frac {a^{3} \left (4 A b +B a \right )}{5 \tan \left (d x +c \right )^{5}}+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right )}{3 \tan \left (d x +c \right )^{3}}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{4 \tan \left (d x +c \right )^{4}}}{d}\)	\(331\)
default	\(\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}}{\tan \left (d x +c \right )}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}}{2 \tan \left (d x +c \right )^{2}}-\frac {A \,a^{4}}{6 \tan \left (d x +c \right )^{6}}-\frac {a^{3} \left (4 A b +B a \right )}{5 \tan \left (d x +c \right )^{5}}+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right )}{3 \tan \left (d x +c \right )^{3}}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{4 \tan \left (d x +c \right )^{4}}}{d}\)	\(331\)
norman	\(\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) x \left (\tan ^{6}\left (d x +c \right )\right )-\frac {A \,a^{4}}{6 d}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{d}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{2 d}+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{4 d}-\frac {a^{3} \left (4 A b +B a \right ) \tan \left (d x +c \right )}{5 d}}{\tan \left (d x +c \right )^{6}}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(352\)
risch	\(\text {Expression too large to display}\)	\(1137\)

[In]

int(cot(d*x+c)^7*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/60*((30*A*a^4-180*A*a^2*b^2+30*A*b^4-120*B*a^3*b+120*B*a*b^3)*ln(sec(d*x+c)^2)+(-60*A*a^4+360*A*a^2*b^2-60*A
*b^4+240*B*a^3*b-240*B*a*b^3)*ln(tan(d*x+c))-10*A*cot(d*x+c)^6*a^4+(-48*A*a^3*b-12*B*a^4)*cot(d*x+c)^5+15*a^2*
cot(d*x+c)^4*(A*a^2-6*A*b^2-4*B*a*b)+(80*A*a^3*b-80*A*a*b^3+20*B*a^4-120*B*a^2*b^2)*cot(d*x+c)^3+(-30*A*a^4+18
0*A*a^2*b^2-30*A*b^4+120*B*a^3*b-120*B*a*b^3)*cot(d*x+c)^2+(-240*A*a^3*b+240*A*a*b^3-60*B*a^4+360*B*a^2*b^2-60
*B*b^4)*cot(d*x+c)-240*d*(A*a^3*b-A*a*b^3+1/4*B*a^4-3/2*B*a^2*b^2+1/4*B*b^4)*x)/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.08 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{6} + 5 \, {\left (11 \, A a^{4} - 36 \, B a^{3} b - 54 \, A a^{2} b^{2} + 24 \, B a b^{3} + 6 \, A b^{4} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{6} + 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )^{5} + 10 \, A a^{4} + 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} - 20 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{60 \, d \tan \left (d x + c\right )^{6}} \]

[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)

Sympy [A] (verification not implemented)

Time = 12.58 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.99 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a^{4} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{7}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a^{4} x & \text {for}\: c = - d x \\\frac {A a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} + \frac {A a^{4}}{4 d \tan ^{4}{\left (c + d x \right )}} - \frac {A a^{4}}{6 d \tan ^{6}{\left (c + d x \right )}} - 4 A a^{3} b x - \frac {4 A a^{3} b}{d \tan {\left (c + d x \right )}} + \frac {4 A a^{3} b}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {4 A a^{3} b}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {3 A a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {6 A a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {3 A a^{2} b^{2}}{d \tan ^{2}{\left (c + d x \right )}} - \frac {3 A a^{2} b^{2}}{2 d \tan ^{4}{\left (c + d x \right )}} + 4 A a b^{3} x + \frac {4 A a b^{3}}{d \tan {\left (c + d x \right )}} - \frac {4 A a b^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {A b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {A b^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {A b^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - B a^{4} x - \frac {B a^{4}}{d \tan {\left (c + d x \right )}} + \frac {B a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {B a^{4}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {2 B a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 B a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {2 B a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - \frac {B a^{3} b}{d \tan ^{4}{\left (c + d x \right )}} + 6 B a^{2} b^{2} x + \frac {6 B a^{2} b^{2}}{d \tan {\left (c + d x \right )}} - \frac {2 B a^{2} b^{2}}{d \tan ^{3}{\left (c + d x \right )}} + \frac {2 B a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {4 B a b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {2 B a b^{3}}{d \tan ^{2}{\left (c + d x \right )}} - B b^{4} x - \frac {B b^{4}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate(cot(d*x+c)**7*(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(d, 0)), (x*(A + B*tan(c))*(a + b*tan(c))**4*cot(c)**7, Eq(d, 0)), (zoo*
A*a**4*x, Eq(c, -d*x)), (A*a**4*log(tan(c + d*x)**2 + 1)/(2*d) - A*a**4*log(tan(c + d*x))/d - A*a**4/(2*d*tan(
c + d*x)**2) + A*a**4/(4*d*tan(c + d*x)**4) - A*a**4/(6*d*tan(c + d*x)**6) - 4*A*a**3*b*x - 4*A*a**3*b/(d*tan(
c + d*x)) + 4*A*a**3*b/(3*d*tan(c + d*x)**3) - 4*A*a**3*b/(5*d*tan(c + d*x)**5) - 3*A*a**2*b**2*log(tan(c + d*
x)**2 + 1)/d + 6*A*a**2*b**2*log(tan(c + d*x))/d + 3*A*a**2*b**2/(d*tan(c + d*x)**2) - 3*A*a**2*b**2/(2*d*tan(
c + d*x)**4) + 4*A*a*b**3*x + 4*A*a*b**3/(d*tan(c + d*x)) - 4*A*a*b**3/(3*d*tan(c + d*x)**3) + A*b**4*log(tan(
c + d*x)**2 + 1)/(2*d) - A*b**4*log(tan(c + d*x))/d - A*b**4/(2*d*tan(c + d*x)**2) - B*a**4*x - B*a**4/(d*tan(
c + d*x)) + B*a**4/(3*d*tan(c + d*x)**3) - B*a**4/(5*d*tan(c + d*x)**5) - 2*B*a**3*b*log(tan(c + d*x)**2 + 1)/
d + 4*B*a**3*b*log(tan(c + d*x))/d + 2*B*a**3*b/(d*tan(c + d*x)**2) - B*a**3*b/(d*tan(c + d*x)**4) + 6*B*a**2*
b**2*x + 6*B*a**2*b**2/(d*tan(c + d*x)) - 2*B*a**2*b**2/(d*tan(c + d*x)**3) + 2*B*a*b**3*log(tan(c + d*x)**2 +
1)/d - 4*B*a*b**3*log(tan(c + d*x))/d - 2*B*a*b**3/(d*tan(c + d*x)**2) - B*b**4*x - B*b**4/(d*tan(c + d*x)),
True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.03 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )} - 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )^{5} + 10 \, A a^{4} + 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} - 20 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6}}}{60 \, d} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 943 vs. \(2 (313) = 626\).

Time = 1.85 (sec) , antiderivative size = 943, normalized size of antiderivative = 2.92 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[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

Mupad [B] (verification not implemented)

Time = 8.11 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.95 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {{\mathrm {cot}\left (c+d\,x\right )}^6\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{5}+\frac {4\,A\,b\,a^3}{5}\right )+\frac {A\,a^4}{6}-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {B\,a^4}{3}+\frac {4\,A\,a^3\,b}{3}-2\,B\,a^2\,b^2-\frac {4\,A\,a\,b^3}{3}\right )+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {A\,a^4}{4}+B\,a^3\,b+\frac {3\,A\,a^2\,b^2}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {A\,a^4}{2}-2\,B\,a^3\,b-3\,A\,a^2\,b^2+2\,B\,a\,b^3+\frac {A\,b^4}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (B\,a^4+4\,A\,a^3\,b-6\,B\,a^2\,b^2-4\,A\,a\,b^3+B\,b^4\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,a^4-4\,B\,a^3\,b-6\,A\,a^2\,b^2+4\,B\,a\,b^3+A\,b^4\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4}{2\,d} \]

[In]

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

[Out]

(log(tan(c + d*x) + 1i)*(A - B*1i)*(a*1i + b)^4)/(2*d) - (log(tan(c + d*x))*(A*a^4 + A*b^4 - 6*A*a^2*b^2 + 4*B
*a*b^3 - 4*B*a^3*b))/d - (cot(c + d*x)^6*(tan(c + d*x)*((B*a^4)/5 + (4*A*a^3*b)/5) + (A*a^4)/6 - tan(c + d*x)^
3*((B*a^4)/3 - 2*B*a^2*b^2 - (4*A*a*b^3)/3 + (4*A*a^3*b)/3) + tan(c + d*x)^2*((3*A*a^2*b^2)/2 - (A*a^4)/4 + B*
a^3*b) + tan(c + d*x)^4*((A*a^4)/2 + (A*b^4)/2 - 3*A*a^2*b^2 + 2*B*a*b^3 - 2*B*a^3*b) + tan(c + d*x)^5*(B*a^4
+ B*b^4 - 6*B*a^2*b^2 - 4*A*a*b^3 + 4*A*a^3*b)))/d + (log(tan(c + d*x) - 1i)*(A + B*1i)*(a*1i - b)^4)/(2*d)