\(\int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [263]

Optimal result

Integrand size = 31, antiderivative size = 187 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x+\frac {a^2 \left (a^2 A-3 A b^2-3 a b B\right ) \cot (c+d x)}{d}-\frac {b^4 B \log (\cos (c+d x))}{d}-\frac {a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \log (\sin (c+d x))}{d}-\frac {a (2 A b+a B) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.89 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 a^2 \left (a^2 A-6 A b^2-4 a b B\right ) \cot (c+d x)-3 a^3 (4 A b+a B) \cot ^2(c+d x)-2 a^4 A \cot ^3(c+d x)+3 (a+i b)^4 (-i A+B) \log (i-\tan (c+d x))-6 a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \log (\tan (c+d x))+3 (a-i b)^4 (i A+B) \log (i+\tan (c+d x))}{6 d} \] Input:

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

Output:

(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 - T 
an[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)
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.01, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4088, 27, 3042, 4128, 27, 3042, 4118, 3042, 4107, 3042, 25, 3956}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^4 (A+B \tan (c+d x))}{\tan (c+d x)^4}dx\)

\(\Big \downarrow \) 4088

\(\displaystyle \frac {1}{3} \int 3 \cot ^3(c+d x) (a+b \tan (c+d x))^2 \left (b^2 B \tan ^2(c+d x)-\left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (2 A b+a B)\right )dx-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 \left (b^2 B \tan ^2(c+d x)-\left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (2 A b+a B)\right )dx-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^2 \left (b^2 B \tan (c+d x)^2-\left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)+a (2 A b+a B)\right )}{\tan (c+d x)^3}dx-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 4128

\(\displaystyle \frac {1}{2} \int -2 \cot ^2(c+d x) (a+b \tan (c+d x)) \left (-B \tan ^2(c+d x) b^3+a \left (A a^2-3 b B a-3 A b^2\right )+\left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)\right )dx-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\int \cot ^2(c+d x) (a+b \tan (c+d x)) \left (-B \tan ^2(c+d x) b^3+a \left (A a^2-3 b B a-3 A b^2\right )+\left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)\right )dx-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \frac {(a+b \tan (c+d x)) \left (-B \tan (c+d x)^2 b^3+a \left (A a^2-3 b B a-3 A b^2\right )+\left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)\right )}{\tan (c+d x)^2}dx-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 4118

\(\displaystyle -\int \cot (c+d x) \left (-B \tan ^2(c+d x) b^4+a \left (B a^3+4 A b a^2-6 b^2 B a-4 A b^3\right )-\left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4\right ) \tan (c+d x)\right )dx+\frac {a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \frac {-B \tan (c+d x)^2 b^4+a \left (B a^3+4 A b a^2-6 b^2 B a-4 A b^3\right )-\left (A a^4-4 b B a^3-6 A b^2 a^2+4 b^3 B a+A b^4\right ) \tan (c+d x)}{\tan (c+d x)}dx+\frac {a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 4107

\(\displaystyle -a \left (a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right ) \int \cot (c+d x)dx+b^4 B \int \tan (c+d x)dx+\frac {a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}+x \left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right )-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -a \left (a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right ) \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+b^4 B \int \tan (c+d x)dx+\frac {a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}+x \left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right )-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 25

\(\displaystyle a \left (a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right ) \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+b^4 B \int \tan (c+d x)dx+\frac {a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}+x \left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right )-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {a^2 \left (a^2 A-3 a b B-3 A b^2\right ) \cot (c+d x)}{d}-\frac {a \left (a^3 B+4 a^2 A b-6 a b^2 B-4 A b^3\right ) \log (-\sin (c+d x))}{d}+x \left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right )-\frac {a (a B+2 A b) \cot ^2(c+d x) (a+b \tan (c+d x))^2}{2 d}-\frac {a A \cot ^3(c+d x) (a+b \tan (c+d x))^3}{3 d}-\frac {b^4 B \log (\cos (c+d x))}{d}\)

Input:

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

Output:

(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)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

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

rule 4088

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[(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] - Simp[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] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
 

rule 4107

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

rule 4118

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] + Simp[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 4128

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*d^2 + c*(c*C - B*d))*(a + b*Tan[e + 
 f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim 
p[1/(d*(n + 1)*(c^2 + d^2))   Int[(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]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}}{3 \tan \left (d x +c \right )^{3}}-\frac {a^{3} \left (4 A b +B a \right )}{2 \tan \left (d x +c \right )^{2}}-a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{\tan \left (d x +c \right )}}{d}\)	\(195\)
default	\(\frac {\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}}{3 \tan \left (d x +c \right )^{3}}-\frac {a^{3} \left (4 A b +B a \right )}{2 \tan \left (d x +c \right )^{2}}-a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{\tan \left (d x +c \right )}}{d}\)	\(195\)
parallelrisch	\(\frac {3 \left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+6 \left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-2 A \cot \left (d x +c \right )^{3} a^{4}+3 \left (-4 A \,a^{3} b -B \,a^{4}\right ) \cot \left (d x +c \right )^{2}+6 a^{2} \cot \left (d x +c \right ) \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )+6 d x \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right )}{6 d}\)	\(195\)
norman	\(\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) x \tan \left (d x +c \right )^{3}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right ) \tan \left (d x +c \right )^{2}}{d}-\frac {A \,a^{4}}{3 d}-\frac {a^{3} \left (4 A b +B a \right ) \tan \left (d x +c \right )}{2 d}}{\tan \left (d x +c \right )^{3}}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}-\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}\)	\(208\)
risch	\(\frac {8 i A \,a^{3} b c}{d}+i B \,a^{4} x -6 i B \,a^{2} b^{2} x -\frac {8 i A a \,b^{3} c}{d}-4 i A a \,b^{3} x -\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,b^{4}}{d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A b}{d}+A \,a^{4} x -6 A \,a^{2} b^{2} x -4 B \,a^{3} b x +4 B a \,b^{3} x +\frac {4 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A \,b^{3}}{d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B \,b^{2}}{d}+\frac {2 i B \,a^{4} c}{d}-\frac {2 i a^{2} \left (-6 A \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+18 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12 B a b \,{\mathrm e}^{4 i \left (d x +c \right )}+12 i A a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-24 B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-12 i A a b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 A \,a^{2}+18 A \,b^{2}+12 B a b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+4 i A \,a^{3} b x +\frac {2 i B \,b^{4} c}{d}-\frac {12 i B \,a^{2} b^{2} c}{d}+i B \,b^{4} x +A \,b^{4} x\)	\(445\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.19 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {3 \, B b^{4} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 2 \, A a^{4} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b - 2 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \] Input:

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

Output:

-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*ta 
n(d*x + c)^3)
 

Sympy [A] (verification not implemented)

Time = 2.62 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.97 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a^{4} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{4}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a^{4} x & \text {for}\: c = - d x \\A a^{4} x + \frac {A a^{4}}{d \tan {\left (c + d x \right )}} - \frac {A a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {2 A a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {4 A a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {2 A a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - 6 A a^{2} b^{2} x - \frac {6 A a^{2} b^{2}}{d \tan {\left (c + d x \right )}} - \frac {2 A a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 A a b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + A b^{4} x + \frac {B a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - 4 B a^{3} b x - \frac {4 B a^{3} b}{d \tan {\left (c + d x \right )}} - \frac {3 B a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {6 B a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 4 B a b^{3} x + \frac {B b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((zoo*A*a**4*x, Eq(c, 0) & Eq(d, 0)), (x*(A + B*tan(c))*(a + b*ta 
n(c))**4*cot(c)**4, Eq(d, 0)), (zoo*A*a**4*x, Eq(c, -d*x)), (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))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.08 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {2 \, A a^{4} - 6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{3}}}{6 \, d} \] Input:

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

Output:

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
 

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.12 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{d} + \frac {{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} - \frac {{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} - \frac {2 \, A a^{4} - 6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \] Input:

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

Output:

(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)/d + 1/2*(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 
)/d - (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))) 
/d - 1/6*(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))/(d*tan(d*x + c)^3)
 

Mupad [B] (verification not implemented)

Time = 3.62 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^4+4\,A\,a^3\,b-6\,B\,a^2\,b^2-4\,A\,a\,b^3\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{2}+2\,A\,b\,a^3\right )+\frac {A\,a^4}{3}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-A\,a^4+4\,B\,a^3\,b+6\,A\,a^2\,b^2\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d} \] Input:

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

Output:

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

Reduce [F]

\[ \int \cot ^4(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\int \cot \left (d x +c \right )^{4} \left (a +\tan \left (d x +c \right ) b \right )^{4} \left (A +B \tan \left (d x +c \right )\right )d x \] Input:

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

Output:

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