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

3.1.49.1 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

-1/4*I*b^3*(d*x+c)^4/d+3/4*I*b^3*d^3*polylog(4,-exp(2*I*(f*x+e)))/f^4+1/2* 
b^3*(d*x+c)^3/f+1/4*a^3*(d*x+c)^4/d+3/4*I*a^2*b*(d*x+c)^4/d-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+3/2*I*b^3*d^3*poly 
log(2,-exp(2*I*(f*x+e)))/f^4-9/4*I*a^2*b*d^3*polylog(4,-exp(2*I*(f*x+e)))/ 
f^4+9/2*I*a^2*b*d*(d*x+c)^2*polylog(2,-exp(2*I*(f*x+e)))/f^2-9*I*a*b^2*d^2 
*(d*x+c)*polylog(2,-exp(2*I*(f*x+e)))/f^3+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^3-3/2*I*b^3*d*(d*x+c)^2*pol 
ylog(2,-exp(2*I*(f*x+e)))/f^2+3/2*I*b^3*d*(d*x+c)^2/f^2-3/2*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
 

3.1.49.2 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 = 7.57 (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...
 

3.1.49.3 Rubi [A] (verified)

Time = 1.28 (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)
 

3.1.49.3.1 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]
 

3.1.49.4 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 = 1.21 (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

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

3.1.49.5 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 = 0.27 (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...
 

3.1.49.6 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)
 

3.1.49.7 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 = 10.88 (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 + ...
 

3.1.49.8 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)
 

3.1.49.9 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)