\(\int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx\) [59]

Optimal result

Integrand size = 20, antiderivative size = 848 \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a+i b) (i a+b)^2 \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f^3}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1724\) vs. \(2(848)=1696\).

Time = 9.24 (sec) , antiderivative size = 1724, normalized size of antiderivative = 2.03 \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(b*((-4*c^2*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*(3*b*d + 2*a 
*c*f)*x)/(a^2 + b^2) + (4*b*(c + d*x)^3)/(a - I*b) + (2*a*f*(c + d*x)^4)/( 
(a - I*b)*d) + (12*c*d*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*( 
b*d + a*c*f)*x*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(e + f*x)))])/((a + I 
*b)*(I*a + b)*f) + (6*d^2*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)) 
)*(b*d + 2*a*c*f)*x^2*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(e + f*x)))])/ 
((a + I*b)*(I*a + b)*f) + (4*a*d^3*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^( 
(2*I)*e)))*x^3*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(e + f*x)))])/((a + I 
*b)*(I*a + b)) + (2*c^2*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))* 
(3*b*d + 2*a*c*f)*Log[a + I*b + (a - I*b)*E^((2*I)*(e + f*x))])/((a + I*b) 
*(I*a + b)*f) + (6*c*d*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*( 
b*d + a*c*f)*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x)))])/((a^2 
 + b^2)*f^2) + (3*d^2*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*(b 
*d + 2*a*c*f)*(2*f*x*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x))) 
] - I*PolyLog[3, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2 
)*f^3) + (3*a*d^3*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*(2*f^2 
*x^2*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x)))] - (2*I)*f*x*Po 
lyLog[3, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x)))] - PolyLog[4, (-a - I* 
b)/((a - I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2)*f^3)))/(2*(a - I*b)*(a 
+ I*b)*(b - b*E^((2*I)*e) - I*a*(1 + E^((2*I)*e)))*f) + (3*x^2*(a*c^2*d...
 

Rubi [A] (verified)

Time = 2.28 (sec) , antiderivative size = 848, 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, 4217, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4217

\(\displaystyle \int \left (-\frac {4 b^2 (c+d x)^3}{(b+i a)^2 \left (i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}+i a \left (1+\frac {i b}{a}\right )\right )^2}+\frac {4 b (c+d x)^3}{(a-i b)^2 \left (i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}+i a \left (1+\frac {i b}{a}\right )\right )}+\frac {(c+d x)^3}{(a-i b)^2}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((-2*I)*b^2*(c + d*x)^3)/((a^2 + b^2)^2*f) + (2*b^2*(c + d*x)^3)/((a + I*b 
)*(I*a + b)^2*(I*a - b + (I*a + b)*E^((2*I)*e + (2*I)*f*x))*f) + (c + d*x) 
^4/(4*(a - I*b)^2*d) + (b*(c + d*x)^4)/((I*a - b)*(a - I*b)^2*d) - (b^2*(c 
 + d*x)^4)/((a^2 + b^2)^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + ((a - I*b)*E^( 
(2*I)*e + (2*I)*f*x))/(a + I*b)])/((a^2 + b^2)^2*f^2) + (2*b*(c + d*x)^3*L 
og[1 + ((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b)])/((a - I*b)^2*(a + I 
*b)*f) - ((2*I)*b^2*(c + d*x)^3*Log[1 + ((a - I*b)*E^((2*I)*e + (2*I)*f*x) 
)/(a + I*b)])/((a^2 + b^2)^2*f) - ((3*I)*b^2*d^2*(c + d*x)*PolyLog[2, -((( 
a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^3) + (3*b* 
d*(c + d*x)^2*PolyLog[2, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))] 
)/((I*a - b)*(a - I*b)^2*f^2) - (3*b^2*d*(c + d*x)^2*PolyLog[2, -(((a - I* 
b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^2) + (3*b^2*d^3* 
PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/(2*(a^2 + b^ 
2)^2*f^4) + (3*b*d^2*(c + d*x)*PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)* 
f*x))/(a + I*b))])/((a - I*b)^2*(a + I*b)*f^3) - ((3*I)*b^2*d^2*(c + d*x)* 
PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2) 
^2*f^3) - (3*b*d^3*PolyLog[4, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I 
*b))])/(2*(I*a - b)*(a - I*b)^2*f^4) + (3*b^2*d^3*PolyLog[4, -(((a - I*b)* 
E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/(2*(a^2 + b^2)^2*f^4)
 

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 4217

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + 
 b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5958 vs. \(2 (763 ) = 1526\).

Time = 0.94 (sec) , antiderivative size = 5959, normalized size of antiderivative = 7.03

method	result	size
risch	\(\text {Expression too large to display}\)	\(5959\)

Input:

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

Output:

result too large to display
 

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. 2537 vs. \(2 (695) = 1390\).

Time = 0.14 (sec) , antiderivative size = 2537, normalized size of antiderivative = 2.99 \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral((c + d*x)**3/(a + b*tan(e + f*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. 4631 vs. \(2 (695) = 1390\).

Time = 2.83 (sec) , antiderivative size = 4631, normalized size of antiderivative = 5.46 \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/12*(36*c^2*d*e*(2*a*b*log(b*tan(f*x + e) + a)/((a^4 + 2*a^2*b^2 + b^4)* 
f) - a*b*log(tan(f*x + e)^2 + 1)/((a^4 + 2*a^2*b^2 + b^4)*f) - b/((a^2*b + 
 b^3)*f*tan(f*x + e) + (a^3 + a*b^2)*f) + (a^2 - b^2)*(f*x + e)/((a^4 + 2* 
a^2*b^2 + b^4)*f)) - 12*(2*a*b*log(b*tan(f*x + e) + a)/(a^4 + 2*a^2*b^2 + 
b^4) - a*b*log(tan(f*x + e)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*( 
f*x + e)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*tan(f*x 
+ e)))*c^3 - (3*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(f*x + e)^4*d^3 - 24*(I*a* 
b^2 - b^3)*d^3*e^3 - 72*(-I*a*b^2 + b^3)*c*d^2*e^2*f - 12*((a^3 - I*a^2*b 
+ a*b^2 - I*b^3)*d^3*e - (a^3 - I*a^2*b + a*b^2 - I*b^3)*c*d^2*f)*(f*x + e 
)^3 + 18*((a^3 - I*a^2*b + a*b^2 - I*b^3)*d^3*e^2 - 2*(a^3 - I*a^2*b + a*b 
^2 - I*b^3)*c*d^2*e*f + (a^3 - I*a^2*b + a*b^2 - I*b^3)*c^2*d*f^2)*(f*x + 
e)^2 - 12*((a^3 - I*a^2*b + a*b^2 - I*b^3)*d^3*e^3 - 3*(a^3 - I*a^2*b + a* 
b^2 - I*b^3)*c*d^2*e^2*f)*(f*x + e) - 12*(2*(I*a^2*b - a*b^2)*d^3*e^3 + 3* 
(-I*a*b^2 + b^3)*d^3*e^2 + 3*(-I*a*b^2 + b^3)*c^2*d*f^2 + 6*((-I*a^2*b + a 
*b^2)*c*d^2*e^2 + (I*a*b^2 - b^3)*c*d^2*e)*f + (2*(I*a^2*b + a*b^2)*d^3*e^ 
3 + 3*(-I*a*b^2 - b^3)*d^3*e^2 + 3*(-I*a*b^2 - b^3)*c^2*d*f^2 + 6*((-I*a^2 
*b - a*b^2)*c*d^2*e^2 + (I*a*b^2 + b^3)*c*d^2*e)*f)*cos(2*f*x + 2*e) - (2* 
(a^2*b - I*a*b^2)*d^3*e^3 - 3*(a*b^2 - I*b^3)*d^3*e^2 - 3*(a*b^2 - I*b^3)* 
c^2*d*f^2 - 6*((a^2*b - I*a*b^2)*c*d^2*e^2 - (a*b^2 - I*b^3)*c*d^2*e)*f)*s 
in(2*f*x + 2*e))*arctan2(-b*cos(2*f*x + 2*e) + a*sin(2*f*x + 2*e) + b, ...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(int(x**3/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*tan(e + f* 
x)*a**5*b*d**3*f + 2*int(x**3/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + 
 a**2),x)*tan(e + f*x)*a**3*b**3*d**3*f + int(x**3/(tan(e + f*x)**2*b**2 + 
 2*tan(e + f*x)*a*b + a**2),x)*tan(e + f*x)*a*b**5*d**3*f + int(x**3/(tan( 
e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*a**6*d**3*f + 2*int(x**3/ 
(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*a**4*b**2*d**3*f + i 
nt(x**3/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*a**2*b**4*d* 
*3*f + 3*int(x**2/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*ta 
n(e + f*x)*a**5*b*c*d**2*f + 6*int(x**2/(tan(e + f*x)**2*b**2 + 2*tan(e + 
f*x)*a*b + a**2),x)*tan(e + f*x)*a**3*b**3*c*d**2*f + 3*int(x**2/(tan(e + 
f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*tan(e + f*x)*a*b**5*c*d**2*f 
+ 3*int(x**2/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*a**6*c* 
d**2*f + 6*int(x**2/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)* 
a**4*b**2*c*d**2*f + 3*int(x**2/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b 
 + a**2),x)*a**2*b**4*c*d**2*f + 3*int(x/(tan(e + f*x)**2*b**2 + 2*tan(e + 
 f*x)*a*b + a**2),x)*tan(e + f*x)*a**5*b*c**2*d*f + 6*int(x/(tan(e + f*x)* 
*2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*tan(e + f*x)*a**3*b**3*c**2*d*f + 
3*int(x/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a**2),x)*tan(e + f*x) 
*a*b**5*c**2*d*f + 3*int(x/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a*b + a* 
*2),x)*a**6*c**2*d*f + 6*int(x/(tan(e + f*x)**2*b**2 + 2*tan(e + f*x)*a...