\(\int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx\) [49]

Optimal result

Integrand size = 20, antiderivative size = 612 \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\frac {3 i b^3 d (c+d x)^2}{2 f^2}-\frac {3 i a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}+\frac {3 i a^2 b (c+d x)^4}{4 d}-\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {i b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d^2 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}-\frac {3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac {3 a b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f} \] Output:

3/4*I*b^3*d^3*polylog(4,-exp(2*I*(f*x+e)))/f^4-3/2*I*b^3*d*(d*x+c)^2*polyl 
og(2,-exp(2*I*(f*x+e)))/f^2+1/2*b^3*(d*x+c)^3/f+1/4*a^3*(d*x+c)^4/d+3/2*I* 
b^3*d^3*polylog(2,-exp(2*I*(f*x+e)))/f^4-3/4*a*b^2*(d*x+c)^4/d-3*I*a*b^2*( 
d*x+c)^3/f-3*b^3*d^2*(d*x+c)*ln(1+exp(2*I*(f*x+e)))/f^3+9*a*b^2*d*(d*x+c)^ 
2*ln(1+exp(2*I*(f*x+e)))/f^2-3*a^2*b*(d*x+c)^3*ln(1+exp(2*I*(f*x+e)))/f+b^ 
3*(d*x+c)^3*ln(1+exp(2*I*(f*x+e)))/f-1/4*I*b^3*(d*x+c)^4/d+9/2*I*a^2*b*d*( 
d*x+c)^2*polylog(2,-exp(2*I*(f*x+e)))/f^2+3/2*I*b^3*d*(d*x+c)^2/f^2-9/4*I* 
a^2*b*d^3*polylog(4,-exp(2*I*(f*x+e)))/f^4+9/2*a*b^2*d^3*polylog(3,-exp(2* 
I*(f*x+e)))/f^4-9/2*a^2*b*d^2*(d*x+c)*polylog(3,-exp(2*I*(f*x+e)))/f^3+3/2 
*b^3*d^2*(d*x+c)*polylog(3,-exp(2*I*(f*x+e)))/f^3+3/4*I*a^2*b*(d*x+c)^4/d- 
9*I*a*b^2*d^2*(d*x+c)*polylog(2,-exp(2*I*(f*x+e)))/f^3-3/2*b^3*d*(d*x+c)^2 
*tan(f*x+e)/f^2+3*a*b^2*(d*x+c)^3*tan(f*x+e)/f+1/2*b^3*(d*x+c)^3*tan(f*x+e 
)^2/f
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2594\) vs. \(2(612)=1224\).

Time = 7.32 (sec) , antiderivative size = 2594, normalized size of antiderivative = 4.24 \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\text {Result too large to show} \] Input:

Integrate[(c + d*x)^3*(a + b*Tan[e + f*x])^3,x]
 

Output:

(((3*I)/4)*a*b^2*d^3*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E 
^((-2*I)*(e + f*x))]) + 6*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + 
 f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e + f*x))])*Sec[e 
])/(E^(I*e)*f^4) - (((3*I)/4)*a^2*b*c*d^2*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E 
^((2*I)*e))*Log[1 + E^((-2*I)*(e + f*x))]) + 6*(1 + E^((2*I)*e))*f*x*PolyL 
og[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2* 
I)*(e + f*x))])*Sec[e])/(E^(I*e)*f^3) + ((I/4)*b^3*c*d^2*(2*f^2*x^2*(2*f*x 
 - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((-2*I)*(e + f*x))]) + 6*(1 + E^((2*I 
)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*Poly 
Log[3, -E^((-2*I)*(e + f*x))])*Sec[e])/(E^(I*e)*f^3) - (((3*I)/8)*a^2*b*d^ 
3*E^(I*e)*((2*f^4*x^4)/E^((2*I)*e) - (4*I)*(1 + E^((-2*I)*e))*f^3*x^3*Log[ 
1 + E^((-2*I)*(e + f*x))] + 6*(1 + E^((-2*I)*e))*f^2*x^2*PolyLog[2, -E^((- 
2*I)*(e + f*x))] - (6*I)*(1 + E^((-2*I)*e))*f*x*PolyLog[3, -E^((-2*I)*(e + 
 f*x))] - 3*(1 + E^((-2*I)*e))*PolyLog[4, -E^((-2*I)*(e + f*x))])*Sec[e])/ 
f^4 + ((I/8)*b^3*d^3*E^(I*e)*((2*f^4*x^4)/E^((2*I)*e) - (4*I)*(1 + E^((-2* 
I)*e))*f^3*x^3*Log[1 + E^((-2*I)*(e + f*x))] + 6*(1 + E^((-2*I)*e))*f^2*x^ 
2*PolyLog[2, -E^((-2*I)*(e + f*x))] - (6*I)*(1 + E^((-2*I)*e))*f*x*PolyLog 
[3, -E^((-2*I)*(e + f*x))] - 3*(1 + E^((-2*I)*e))*PolyLog[4, -E^((-2*I)*(e 
 + f*x))])*Sec[e])/f^4 + ((b^3*c^3 + 3*b^3*c^2*d*x + 3*b^3*c*d^2*x^2 + b^3 
*d^3*x^3)*Sec[e + f*x]^2)/(2*f) - (3*b^3*c*d^2*Sec[e]*(Cos[e]*Log[Cos[e...
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4205, 2009}

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.

\(\displaystyle \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^3 (a+b \tan (e+f x))^3dx\)

\(\Big \downarrow \) 4205

\(\displaystyle \int \left (a^3 (c+d x)^3+3 a^2 b (c+d x)^3 \tan (e+f x)+3 a b^2 (c+d x)^3 \tan ^2(e+f x)+b^3 (c+d x)^3 \tan ^3(e+f x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 (c+d x)^4}{4 d}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 a^2 b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 i a^2 b (c+d x)^4}{4 d}-\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {3 a b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {3 i a b^2 (c+d x)^3}{f}-\frac {3 a b^2 (c+d x)^4}{4 d}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac {b^3 (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}+\frac {3 i b^3 d (c+d x)^2}{2 f^2}+\frac {b^3 (c+d x)^3}{2 f}-\frac {i b^3 (c+d x)^4}{4 d}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}\)

Input:

Int[(c + d*x)^3*(a + b*Tan[e + f*x])^3,x]
 

Output:

(((3*I)/2)*b^3*d*(c + d*x)^2)/f^2 - ((3*I)*a*b^2*(c + d*x)^3)/f + (b^3*(c 
+ d*x)^3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) + (((3*I)/4)*a^2*b*(c + d*x)^4)/ 
d - (3*a*b^2*(c + d*x)^4)/(4*d) - ((I/4)*b^3*(c + d*x)^4)/d - (3*b^3*d^2*( 
c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 
+ E^((2*I)*(e + f*x))])/f^2 - (3*a^2*b*(c + d*x)^3*Log[1 + E^((2*I)*(e + f 
*x))])/f + (b^3*(c + d*x)^3*Log[1 + E^((2*I)*(e + f*x))])/f + (((3*I)/2)*b 
^3*d^3*PolyLog[2, -E^((2*I)*(e + f*x))])/f^4 - ((9*I)*a*b^2*d^2*(c + d*x)* 
PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + (((9*I)/2)*a^2*b*d*(c + d*x)^2*Pol 
yLog[2, -E^((2*I)*(e + f*x))])/f^2 - (((3*I)/2)*b^3*d*(c + d*x)^2*PolyLog[ 
2, -E^((2*I)*(e + f*x))])/f^2 + (9*a*b^2*d^3*PolyLog[3, -E^((2*I)*(e + f*x 
))])/(2*f^4) - (9*a^2*b*d^2*(c + d*x)*PolyLog[3, -E^((2*I)*(e + f*x))])/(2 
*f^3) + (3*b^3*d^2*(c + d*x)*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) - ( 
((9*I)/4)*a^2*b*d^3*PolyLog[4, -E^((2*I)*(e + f*x))])/f^4 + (((3*I)/4)*b^3 
*d^3*PolyLog[4, -E^((2*I)*(e + f*x))])/f^4 - (3*b^3*d*(c + d*x)^2*Tan[e + 
f*x])/(2*f^2) + (3*a*b^2*(c + d*x)^3*Tan[e + f*x])/f + (b^3*(c + d*x)^3*Ta 
n[e + f*x]^2)/(2*f)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4205

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1929 vs. \(2 (544 ) = 1088\).

Time = 1.00 (sec) , antiderivative size = 1930, normalized size of antiderivative = 3.15

method	result	size
risch	\(\text {Expression too large to display}\)	\(1930\)

Input:

int((d*x+c)^3*(a+b*tan(f*x+e))^3,x,method=_RETURNVERBOSE)
 

Output:

d^2*a^3*c*x^3+3/2*d*a^3*c^2*x^2+a^3*c^3*x-3/4*d^3*a*b^2*x^4-3*a*b^2*c^3*x- 
3/4/d*a*b^2*c^4+I*b^3*c^3*x+1/4*I/d*b^3*c^4+3/4*I*d^3*a^2*b*x^4-3/2*I*b^3* 
d*c^2*x^2+9/2*a*b^2*d^3*polylog(3,-exp(2*I*(f*x+e)))/f^4-I*d^2*b^3*c*x^3-3 
*d^2*a*b^2*c*x^3-9/2*d*a*b^2*c^2*x^2+b^2*(-6*I*b*c*d^2*x+18*I*a*c*d^2*f*x^ 
2+2*b*d^3*f*x^3*exp(2*I*(f*x+e))-3*I*b*c^2*d*exp(2*I*(f*x+e))-6*I*b*c*d^2* 
x*exp(2*I*(f*x+e))+6*I*a*c^3*f*exp(2*I*(f*x+e))+6*b*c*d^2*f*x^2*exp(2*I*(f 
*x+e))-3*I*b*d^3*x^2+18*I*a*c^2*d*f*x*exp(2*I*(f*x+e))-3*I*b*d^3*x^2*exp(2 
*I*(f*x+e))+6*b*c^2*d*f*x*exp(2*I*(f*x+e))-3*I*b*c^2*d+6*I*a*d^3*f*x^3*exp 
(2*I*(f*x+e))+18*I*a*c^2*d*f*x+2*b*c^3*f*exp(2*I*(f*x+e))+6*I*a*d^3*f*x^3+ 
18*I*a*c*d^2*f*x^2*exp(2*I*(f*x+e))+6*I*a*c^3*f)/f^2/(1+exp(2*I*(f*x+e)))^ 
2+1/4*d^3*a^3*x^4+1/4/d*a^3*c^4-2/f*b^3*c^3*ln(exp(I*(f*x+e)))+1/f*b^3*c^3 
*ln(1+exp(2*I*(f*x+e)))-1/4*I*b^3*d^3*x^4-36*I/f^2*b^2*d^2*c*a*e*x+1/f*b^3 
*d^3*ln(1+exp(2*I*(f*x+e)))*x^3-6/f^4*b^3*e*d^3*ln(exp(I*(f*x+e)))+2/f^4*b 
^3*e^3*d^3*ln(exp(I*(f*x+e)))+3/2/f^3*b^3*d^3*polylog(3,-exp(2*I*(f*x+e))) 
*x-3/f^3*b^3*d^3*ln(1+exp(2*I*(f*x+e)))*x+3/2/f^3*b^3*c*d^2*polylog(3,-exp 
(2*I*(f*x+e)))+6/f*b*a^2*c^3*ln(exp(I*(f*x+e)))-3/f*b*a^2*c^3*ln(1+exp(2*I 
*(f*x+e)))+6/f^3*b^3*c*d^2*ln(exp(I*(f*x+e)))-3/f^3*b^3*c*d^2*ln(1+exp(2*I 
*(f*x+e)))-3/2*I/f^4*b^3*d^3*e^4+3*I/f^2*b^3*d^3*x^2+3*I/f^4*b^3*d^3*e^2+1 
8*I/f*b*d*c^2*a^2*e*x-18*I/f^2*b*e^2*d^2*c*a^2*x+9*I/f^2*b*a^2*c*d^2*polyl 
og(2,-exp(2*I*(f*x+e)))*x-3*I*a^2*b*c^3*x-3/4*I/d*a^2*b*c^4+3/f*b^3*d*c...
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (530) = 1060\).

Time = 0.11 (sec) , antiderivative size = 1177, normalized size of antiderivative = 1.92 \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3*(a+b*tan(f*x+e))^3,x, algorithm="fricas")
 

Output:

1/8*(2*(a^3 - 3*a*b^2)*d^3*f^4*x^4 + 3*I*(3*a^2*b - b^3)*d^3*polylog(4, (t 
an(f*x + e)^2 + 2*I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 3*I*(3*a^2*b 
 - b^3)*d^3*polylog(4, (tan(f*x + e)^2 - 2*I*tan(f*x + e) - 1)/(tan(f*x + 
e)^2 + 1)) + 4*(b^3*d^3*f^3 + 2*(a^3 - 3*a*b^2)*c*d^2*f^4)*x^3 + 12*(b^3*c 
*d^2*f^3 + (a^3 - 3*a*b^2)*c^2*d*f^4)*x^2 + 4*(b^3*d^3*f^3*x^3 + 3*b^3*c*d 
^2*f^3*x^2 + 3*b^3*c^2*d*f^3*x + b^3*c^3*f^3)*tan(f*x + e)^2 + 4*(3*b^3*c^ 
2*d*f^3 + 2*(a^3 - 3*a*b^2)*c^3*f^4)*x - 6*(I*(3*a^2*b - b^3)*d^3*f^2*x^2 
- 6*I*a*b^2*c*d^2*f + I*b^3*d^3 + I*(3*a^2*b - b^3)*c^2*d*f^2 - 2*I*(3*a*b 
^2*d^3*f - (3*a^2*b - b^3)*c*d^2*f^2)*x)*dilog(2*(I*tan(f*x + e) - 1)/(tan 
(f*x + e)^2 + 1) + 1) - 6*(-I*(3*a^2*b - b^3)*d^3*f^2*x^2 + 6*I*a*b^2*c*d^ 
2*f - I*b^3*d^3 - I*(3*a^2*b - b^3)*c^2*d*f^2 + 2*I*(3*a*b^2*d^3*f - (3*a^ 
2*b - b^3)*c*d^2*f^2)*x)*dilog(2*(-I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1 
) + 1) - 4*((3*a^2*b - b^3)*d^3*f^3*x^3 - 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2* 
f + (3*a^2*b - b^3)*c^3*f^3 - 3*(3*a*b^2*d^3*f^2 - (3*a^2*b - b^3)*c*d^2*f 
^3)*x^2 - 3*(6*a*b^2*c*d^2*f^2 - b^3*d^3*f - (3*a^2*b - b^3)*c^2*d*f^3)*x) 
*log(-2*(I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 4*((3*a^2*b - b^3)*d^ 
3*f^3*x^3 - 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b - b^3)*c^3*f^3 - 
3*(3*a*b^2*d^3*f^2 - (3*a^2*b - b^3)*c*d^2*f^3)*x^2 - 3*(6*a*b^2*c*d^2*f^2 
 - b^3*d^3*f - (3*a^2*b - b^3)*c^2*d*f^3)*x)*log(-2*(-I*tan(f*x + e) - 1)/ 
(tan(f*x + e)^2 + 1)) + 6*(3*a*b^2*d^3 - (3*a^2*b - b^3)*d^3*f*x - (3*a...
 

Sympy [F]

\[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{3}\, dx \] Input:

integrate((d*x+c)**3*(a+b*tan(f*x+e))**3,x)
 

Output:

Integral((a + b*tan(e + f*x))**3*(c + d*x)**3, x)
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6861 vs. \(2 (530) = 1060\).

Time = 8.17 (sec) , antiderivative size = 6861, normalized size of antiderivative = 11.21 \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3*(a+b*tan(f*x+e))^3,x, algorithm="maxima")
 

Output:

1/4*(4*(f*x + e)*a^3*c^3 + (f*x + e)^4*a^3*d^3/f^3 - 4*(f*x + e)^3*a^3*d^3 
*e/f^3 + 6*(f*x + e)^2*a^3*d^3*e^2/f^3 - 4*(f*x + e)*a^3*d^3*e^3/f^3 + 4*( 
f*x + e)^3*a^3*c*d^2/f^2 - 12*(f*x + e)^2*a^3*c*d^2*e/f^2 + 12*(f*x + e)*a 
^3*c*d^2*e^2/f^2 + 6*(f*x + e)^2*a^3*c^2*d/f - 12*(f*x + e)*a^3*c^2*d*e/f 
+ 12*a^2*b*c^3*log(sec(f*x + e)) - 12*a^2*b*d^3*e^3*log(sec(f*x + e))/f^3 
+ 36*a^2*b*c*d^2*e^2*log(sec(f*x + e))/f^2 - 36*a^2*b*c^2*d*e*log(sec(f*x 
+ e))/f - 4*(72*a*b^2*d^3*e^3 - 72*a*b^2*c^3*f^3 - 3*(3*a^2*b + 3*I*a*b^2 
- b^3)*(f*x + e)^4*d^3 + 36*b^3*d^3*e^2 + 12*((3*a^2*b + 3*I*a*b^2 - b^3)* 
d^3*e - (3*a^2*b + 3*I*a*b^2 - b^3)*c*d^2*f)*(f*x + e)^3 - 18*((3*a^2*b + 
3*I*a*b^2 - b^3)*d^3*e^2 - 2*(3*a^2*b + 3*I*a*b^2 - b^3)*c*d^2*e*f + (3*a^ 
2*b + 3*I*a*b^2 - b^3)*c^2*d*f^2)*(f*x + e)^2 + 36*(6*a*b^2*c^2*d*e + b^3* 
c^2*d)*f^2 + 12*((3*I*a*b^2 - b^3)*d^3*e^3 + 3*(-3*I*a*b^2 + b^3)*c*d^2*e^ 
2*f + 3*(3*I*a*b^2 - b^3)*c^2*d*e*f^2 + (-3*I*a*b^2 + b^3)*c^3*f^3)*(f*x + 
 e) - 72*(3*a*b^2*c*d^2*e^2 + b^3*c*d^2*e)*f + 4*(3*b^3*d^3*e^3 - 3*b^3*c^ 
3*f^3 - 27*a*b^2*d^3*e^2 + 4*(3*a^2*b - b^3)*(f*x + e)^3*d^3 - 9*b^3*d^3*e 
 - 9*(3*a*b^2*d^3 + (3*a^2*b - b^3)*d^3*e - (3*a^2*b - b^3)*c*d^2*f)*(f*x 
+ e)^2 + 9*(b^3*c^2*d*e - 3*a*b^2*c^2*d)*f^2 + 9*(6*a*b^2*d^3*e + b^3*d^3 
+ (3*a^2*b - b^3)*d^3*e^2 + (3*a^2*b - b^3)*c^2*d*f^2 - 2*(3*a*b^2*c*d^2 + 
 (3*a^2*b - b^3)*c*d^2*e)*f)*(f*x + e) - 9*(b^3*c*d^2*e^2 - 6*a*b^2*c*d^2* 
e - b^3*c*d^2)*f + (3*b^3*d^3*e^3 - 3*b^3*c^3*f^3 - 27*a*b^2*d^3*e^2 + ...
 

Giac [F]

\[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \tan \left (f x + e\right ) + a\right )}^{3} \,d x } \] Input:

integrate((d*x+c)^3*(a+b*tan(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^3*(b*tan(f*x + e) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^3 \,d x \] Input:

int((a + b*tan(e + f*x))^3*(c + d*x)^3,x)
 

Output:

int((a + b*tan(e + f*x))^3*(c + d*x)^3, x)
 

Reduce [F]

\[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\frac {4 \left (\int \tan \left (f x +e \right )^{3} x^{3}d x \right ) b^{3} d^{3} f^{2}+12 \left (\int \tan \left (f x +e \right )^{3} x^{2}d x \right ) b^{3} c \,d^{2} f^{2}+12 \left (\int \tan \left (f x +e \right )^{3} x d x \right ) b^{3} c^{2} d \,f^{2}+12 \left (\int \tan \left (f x +e \right ) x^{3}d x \right ) a^{2} b \,d^{3} f^{2}+36 \left (\int \tan \left (f x +e \right ) x^{2}d x \right ) a^{2} b c \,d^{2} f^{2}-36 \left (\int \tan \left (f x +e \right ) x^{2}d x \right ) a \,b^{2} d^{3} f +36 \left (\int \tan \left (f x +e \right ) x d x \right ) a^{2} b \,c^{2} d \,f^{2}-72 \left (\int \tan \left (f x +e \right ) x d x \right ) a \,b^{2} c \,d^{2} f +6 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) a^{2} b \,c^{3} f -18 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) a \,b^{2} c^{2} d -2 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) b^{3} c^{3} f +2 \tan \left (f x +e \right )^{2} b^{3} c^{3} f +12 \tan \left (f x +e \right ) a \,b^{2} c^{3} f +36 \tan \left (f x +e \right ) a \,b^{2} c^{2} d f x +36 \tan \left (f x +e \right ) a \,b^{2} c \,d^{2} f \,x^{2}+12 \tan \left (f x +e \right ) a \,b^{2} d^{3} f \,x^{3}+4 a^{3} c^{3} f^{2} x +6 a^{3} c^{2} d \,f^{2} x^{2}+4 a^{3} c \,d^{2} f^{2} x^{3}+a^{3} d^{3} f^{2} x^{4}-12 a \,b^{2} c^{3} f^{2} x -18 a \,b^{2} c^{2} d \,f^{2} x^{2}-12 a \,b^{2} c \,d^{2} f^{2} x^{3}-3 a \,b^{2} d^{3} f^{2} x^{4}}{4 f^{2}} \] Input:

int((d*x+c)^3*(a+b*tan(f*x+e))^3,x)
 

Output:

(4*int(tan(e + f*x)**3*x**3,x)*b**3*d**3*f**2 + 12*int(tan(e + f*x)**3*x** 
2,x)*b**3*c*d**2*f**2 + 12*int(tan(e + f*x)**3*x,x)*b**3*c**2*d*f**2 + 12* 
int(tan(e + f*x)*x**3,x)*a**2*b*d**3*f**2 + 36*int(tan(e + f*x)*x**2,x)*a* 
*2*b*c*d**2*f**2 - 36*int(tan(e + f*x)*x**2,x)*a*b**2*d**3*f + 36*int(tan( 
e + f*x)*x,x)*a**2*b*c**2*d*f**2 - 72*int(tan(e + f*x)*x,x)*a*b**2*c*d**2* 
f + 6*log(tan(e + f*x)**2 + 1)*a**2*b*c**3*f - 18*log(tan(e + f*x)**2 + 1) 
*a*b**2*c**2*d - 2*log(tan(e + f*x)**2 + 1)*b**3*c**3*f + 2*tan(e + f*x)** 
2*b**3*c**3*f + 12*tan(e + f*x)*a*b**2*c**3*f + 36*tan(e + f*x)*a*b**2*c** 
2*d*f*x + 36*tan(e + f*x)*a*b**2*c*d**2*f*x**2 + 12*tan(e + f*x)*a*b**2*d* 
*3*f*x**3 + 4*a**3*c**3*f**2*x + 6*a**3*c**2*d*f**2*x**2 + 4*a**3*c*d**2*f 
**2*x**3 + a**3*d**3*f**2*x**4 - 12*a*b**2*c**3*f**2*x - 18*a*b**2*c**2*d* 
f**2*x**2 - 12*a*b**2*c*d**2*f**2*x**3 - 3*a*b**2*d**3*f**2*x**4)/(4*f**2)