\(\int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx\) [50]

Optimal result

Integrand size = 20, antiderivative size = 427 \[ \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx=-\frac {3 i a b^2 (c+d x)^2}{f}+\frac {b^3 (c+d x)^2}{2 f}+\frac {a^3 (c+d x)^3}{3 d}+\frac {i a^2 b (c+d x)^3}{d}-\frac {a b^2 (c+d x)^3}{d}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}-\frac {3 i a b^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a^2 b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {i b^3 d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f} \] Output:

I*a^2*b*(d*x+c)^3/d+1/2*b^3*(d*x+c)^2/f+1/3*a^3*(d*x+c)^3/d-3*I*a*b^2*(d*x 
+c)^2/f-a*b^2*(d*x+c)^3/d-I*b^3*d*(d*x+c)*polylog(2,-exp(2*I*(f*x+e)))/f^2 
+6*a*b^2*d*(d*x+c)*ln(1+exp(2*I*(f*x+e)))/f^2-3*a^2*b*(d*x+c)^2*ln(1+exp(2 
*I*(f*x+e)))/f+b^3*(d*x+c)^2*ln(1+exp(2*I*(f*x+e)))/f-b^3*d^2*ln(cos(f*x+e 
))/f^3+3*I*a^2*b*d*(d*x+c)*polylog(2,-exp(2*I*(f*x+e)))/f^2-1/3*I*b^3*(d*x 
+c)^3/d-3*I*a*b^2*d^2*polylog(2,-exp(2*I*(f*x+e)))/f^3-3/2*a^2*b*d^2*polyl 
og(3,-exp(2*I*(f*x+e)))/f^3+1/2*b^3*d^2*polylog(3,-exp(2*I*(f*x+e)))/f^3-b 
^3*d*(d*x+c)*tan(f*x+e)/f^2+3*a*b^2*(d*x+c)^2*tan(f*x+e)/f+1/2*b^3*(d*x+c) 
^2*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. \(1846\) vs. \(2(427)=854\).

Time = 7.05 (sec) , antiderivative size = 1846, normalized size of antiderivative = 4.32 \[ \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx =\text {Too large to display} \] Input:

Integrate[(c + d*x)^2*(a + b*Tan[e + f*x])^3,x]
 

Output:

((-1/4*I)*a^2*b*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))*PolyLog[3, -E^((-2*I)*(e + f*x))])*Sec[e] 
)/(E^(I*e)*f^3) + ((I/12)*b^3*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))*PolyLog[3, -E^((-2*I)*(e + 
f*x))])*Sec[e])/(E^(I*e)*f^3) - (b^3*d^2*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x 
] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f^3*(Cos[e]^2 + Sin[e]^2)) + (6*a*b^2 
*c*d*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/ 
(f^2*(Cos[e]^2 + Sin[e]^2)) - (3*a^2*b*c^2*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f 
*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (b^3*c^2 
*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f*( 
Cos[e]^2 + Sin[e]^2)) + (3*a*b^2*d^2*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]] 
) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 
 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*L 
og[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog 
[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(f^3*S 
qrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a^2*b*c*d*Csc[e]*((f^2*x^2)/E^(I 
*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^ 
((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan...
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 427, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4205

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 (c+d x)^3}{3 d}+\frac {3 i a^2 b d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac {3 i a b^2 (c+d x)^2}{f}-\frac {a b^2 (c+d x)^3}{d}-\frac {3 i a b^2 d^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}-\frac {i b^3 d (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\frac {b^3 (c+d x)^2}{2 f}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {b^3 d^2 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}\)

Input:

Int[(c + d*x)^2*(a + b*Tan[e + f*x])^3,x]
 

Output:

((-3*I)*a*b^2*(c + d*x)^2)/f + (b^3*(c + d*x)^2)/(2*f) + (a^3*(c + d*x)^3) 
/(3*d) + (I*a^2*b*(c + d*x)^3)/d - (a*b^2*(c + d*x)^3)/d - ((I/3)*b^3*(c + 
 d*x)^3)/d + (6*a*b^2*d*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))])/f^2 - (3*a 
^2*b*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f + (b^3*(c + d*x)^2*Log[1 
+ E^((2*I)*(e + f*x))])/f - (b^3*d^2*Log[Cos[e + f*x]])/f^3 - ((3*I)*a*b^2 
*d^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((3*I)*a^2*b*d*(c + d*x)*Poly 
Log[2, -E^((2*I)*(e + f*x))])/f^2 - (I*b^3*d*(c + d*x)*PolyLog[2, -E^((2*I 
)*(e + f*x))])/f^2 - (3*a^2*b*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3 
) + (b^3*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) - (b^3*d*(c + d*x)* 
Tan[e + f*x])/f^2 + (3*a*b^2*(c + d*x)^2*Tan[e + f*x])/f + (b^3*(c + d*x)^ 
2*Tan[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. 1137 vs. \(2 (393 ) = 786\).

Time = 0.86 (sec) , antiderivative size = 1138, normalized size of antiderivative = 2.67

method	result	size
risch	\(\text {Expression too large to display}\)	\(1138\)

Input:

int((d*x+c)^2*(a+b*tan(f*x+e))^3,x,method=_RETURNVERBOSE)
 

Output:

12*I/f*b*a^2*c*d*e*x-3/2*a^2*b*d^2*polylog(3,-exp(2*I*(f*x+e)))/f^3+2*b^2* 
(3*I*a*d^2*f*x^2*exp(2*I*(f*x+e))+6*I*a*c*d*f*x*exp(2*I*(f*x+e))+b*d^2*f*x 
^2*exp(2*I*(f*x+e))+3*I*a*c^2*f*exp(2*I*(f*x+e))+3*I*a*d^2*f*x^2-I*b*d^2*x 
*exp(2*I*(f*x+e))+2*b*c*d*f*x*exp(2*I*(f*x+e))+6*I*a*c*d*f*x-I*b*c*d*exp(2 
*I*(f*x+e))+b*c^2*f*exp(2*I*(f*x+e))+3*I*a*c^2*f-I*b*d^2*x-I*b*c*d)/f^2/(1 
+exp(2*I*(f*x+e)))^2+d*a^3*c*x^2+a^3*c^2*x-d^2*a*b^2*x^3-3*a*b^2*c^2*x-1/d 
*a*b^2*c^3+I*b^3*c^2*x+1/3*I/d*b^3*c^3+1/3*d^2*a^3*x^3+1/3/d*a^3*c^3+6/f*b 
*a^2*c^2*ln(exp(I*(f*x+e)))-3/f*b*a^2*c^2*ln(1+exp(2*I*(f*x+e)))-2/f^3*b^3 
*e^2*d^2*ln(exp(I*(f*x+e)))+1/f*b^3*d^2*ln(1+exp(2*I*(f*x+e)))*x^2+4/3*I/f 
^3*b^3*d^2*e^3-2/f*b^3*c^2*ln(exp(I*(f*x+e)))+1/f*b^3*c^2*ln(1+exp(2*I*(f* 
x+e)))+2/f^3*b^3*d^2*ln(exp(I*(f*x+e)))-1/f^3*b^3*d^2*ln(1+exp(2*I*(f*x+e) 
))-1/3*I*b^3*d^2*x^3-I*d*b^3*c*x^2-3*d*a*b^2*c*x^2+I*d^2*a^2*b*x^3-3*I*a^2 
*b*c^2*x-I/d*a^2*b*c^3-12/f^2*b*e*d*a^2*c*ln(exp(I*(f*x+e)))-6/f*b*a^2*c*d 
*ln(1+exp(2*I*(f*x+e)))*x+1/2*b^3*d^2*polylog(3,-exp(2*I*(f*x+e)))/f^3-4*I 
/f*b^3*c*d*e*x-12*I/f^2*b^2*a*d^2*e*x+3*I/f^2*b*a^2*c*d*polylog(2,-exp(2*I 
*(f*x+e)))+3*I/f^2*b*a^2*d^2*polylog(2,-exp(2*I*(f*x+e)))*x+6*I/f^2*b*a^2* 
c*d*e^2-6*I/f^2*b*a^2*d^2*e^2*x-3*I*a*b^2*d^2*polylog(2,-exp(2*I*(f*x+e))) 
/f^3+3*I*d*a^2*b*c*x^2+6/f^2*b^2*a*d^2*ln(1+exp(2*I*(f*x+e)))*x-2*I/f^2*b^ 
3*c*d*e^2-6*I/f*b^2*a*d^2*x^2+2*I/f^2*b^3*d^2*e^2*x-6*I/f^3*b^2*a*d^2*e^2- 
4*I/f^3*b*a^2*d^2*e^3-I/f^2*b^3*c*d*polylog(2,-exp(2*I*(f*x+e)))-I/f^2*...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.61 \[ \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^2*(a+b*tan(f*x+e))^3,x, algorithm="fricas")
 

Output:

1/12*(4*(a^3 - 3*a*b^2)*d^2*f^3*x^3 - 3*(3*a^2*b - b^3)*d^2*polylog(3, (ta 
n(f*x + e)^2 + 2*I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 3*(3*a^2*b - 
b^3)*d^2*polylog(3, (tan(f*x + e)^2 - 2*I*tan(f*x + e) - 1)/(tan(f*x + e)^ 
2 + 1)) + 6*(b^3*d^2*f^2 + 2*(a^3 - 3*a*b^2)*c*d*f^3)*x^2 + 6*(b^3*d^2*f^2 
*x^2 + 2*b^3*c*d*f^2*x + b^3*c^2*f^2)*tan(f*x + e)^2 + 12*(b^3*c*d*f^2 + ( 
a^3 - 3*a*b^2)*c^2*f^3)*x - 6*(-3*I*a*b^2*d^2 + I*(3*a^2*b - b^3)*d^2*f*x 
+ I*(3*a^2*b - b^3)*c*d*f)*dilog(2*(I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 
1) + 1) - 6*(3*I*a*b^2*d^2 - I*(3*a^2*b - b^3)*d^2*f*x - I*(3*a^2*b - b^3) 
*c*d*f)*dilog(2*(-I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1) + 1) - 6*((3*a^ 
2*b - b^3)*d^2*f^2*x^2 - 6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b - b^3)*c^2*f^2 
 - 2*(3*a*b^2*d^2*f - (3*a^2*b - b^3)*c*d*f^2)*x)*log(-2*(I*tan(f*x + e) - 
 1)/(tan(f*x + e)^2 + 1)) - 6*((3*a^2*b - b^3)*d^2*f^2*x^2 - 6*a*b^2*c*d*f 
 + b^3*d^2 + (3*a^2*b - b^3)*c^2*f^2 - 2*(3*a*b^2*d^2*f - (3*a^2*b - b^3)* 
c*d*f^2)*x)*log(-2*(-I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) + 12*(3*a*b 
^2*d^2*f^2*x^2 + 3*a*b^2*c^2*f^2 - b^3*c*d*f + (6*a*b^2*c*d*f^2 - b^3*d^2* 
f)*x)*tan(f*x + e))/f^3
 

Sympy [F]

\[ \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{2}\, dx \] Input:

integrate((d*x+c)**2*(a+b*tan(f*x+e))**3,x)
 

Output:

Integral((a + b*tan(e + f*x))**3*(c + d*x)**2, 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. 3407 vs. \(2 (384) = 768\).

Time = 1.59 (sec) , antiderivative size = 3407, normalized size of antiderivative = 7.98 \[ \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^2*(a+b*tan(f*x+e))^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx=\int { {\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{3} \,d x } \] Input:

integrate((d*x+c)^2*(a+b*tan(f*x+e))^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^2*(b*tan(f*x + e) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \] Input:

int((a + b*tan(e + f*x))^3*(c + d*x)^2,x)
 

Output:

int((a + b*tan(e + f*x))^3*(c + d*x)^2, x)
 

Reduce [F]

\[ \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx=\frac {6 \left (\int \tan \left (f x +e \right )^{3} x^{2}d x \right ) b^{3} d^{2} f^{2}+12 \left (\int \tan \left (f x +e \right )^{3} x d x \right ) b^{3} c d \,f^{2}+18 \left (\int \tan \left (f x +e \right ) x^{2}d x \right ) a^{2} b \,d^{2} f^{2}+36 \left (\int \tan \left (f x +e \right ) x d x \right ) a^{2} b c d \,f^{2}-36 \left (\int \tan \left (f x +e \right ) x d x \right ) a \,b^{2} d^{2} f +9 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) a^{2} b \,c^{2} f -18 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) a \,b^{2} c d -3 \,\mathrm {log}\left (\tan \left (f x +e \right )^{2}+1\right ) b^{3} c^{2} f +3 \tan \left (f x +e \right )^{2} b^{3} c^{2} f +18 \tan \left (f x +e \right ) a \,b^{2} c^{2} f +36 \tan \left (f x +e \right ) a \,b^{2} c d f x +18 \tan \left (f x +e \right ) a \,b^{2} d^{2} f \,x^{2}+6 a^{3} c^{2} f^{2} x +6 a^{3} c d \,f^{2} x^{2}+2 a^{3} d^{2} f^{2} x^{3}-18 a \,b^{2} c^{2} f^{2} x -18 a \,b^{2} c d \,f^{2} x^{2}-6 a \,b^{2} d^{2} f^{2} x^{3}}{6 f^{2}} \] Input:

int((d*x+c)^2*(a+b*tan(f*x+e))^3,x)
 

Output:

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