3.1.57 \(\int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx\) [57]

3.1.57.1 Optimal result

Integrand size = 20, antiderivative size = 839 \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}-\frac {2 b^2 (c+d x)^3}{(a-i b) (a+i b)^2 \left (i a+b-(i a-b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a+i b)^2 d}-\frac {b (c+d x)^4}{(a+i b)^2 (i a+b) d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {2 b (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a-i b) (a+i b)^2 f}-\frac {2 i b^2 (c+d x)^3 \log \left (1-\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a+i b)^2 (i a+b) f^2}-\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}-\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{(a-i b) (a+i b)^2 f^3}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 (a+i b)^2 (i a+b) f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,\frac {(a+i b) e^{2 i e+2 i f x}}{a-i b}\right )}{2 \left (a^2+b^2\right )^2 f^4} \]

output

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

3.1.57.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1733\) vs. \(2(839)=1678\).

Time = 11.55 (sec) , antiderivative size = 1733, normalized size of antiderivative = 2.07 \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx =\text {Too large to display} \]

input

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

output

(b*((4*c^2*(a*(-1 + E^((2*I)*e)) + I*b*(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*(a*(-1 + E^((2*I)*e)) + I*b*(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*(a*(-1 + E^((2*I)*e)) + I*b*(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*(a*(-1 + E^((2*I)*e)) + I*b*(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*(a*(-1 + E^((2*I)*e)) + I*b*(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*(a*(-1 + E^((2*I)*e)) + I*b*(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*(a*(-1 + E^((2*I)*e)) + I*b*(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*(a*(-1 + E^((2*I)*e)) + I*b*(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*PolyLog[ 
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)*( 
(-I)*a*(-1 + E^((2*I)*e)) + b*(1 + E^((2*I)*e)))*f) + (3*x^2*(-(a*c^2*d...
 

3.1.57.3 Rubi [A] (verified)

Time = 2.21 (sec) , antiderivative size = 839, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4217

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

\(\Big \downarrow \) 2009

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

input

Int[(c + d*x)^3/(a + b*Cot[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 
)*(a + I*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)/((a + I*b)^2*(I*a + b)*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)*(a + I*b 
)^2*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)])/((a 
+ I*b)^2*(I*a + b)*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)*(a + I*b)^2*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*(a + I*b)^2 
*(I*a + b)*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)
 

3.1.57.3.1 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 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]
 

3.1.57.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5930 vs. \(2 (754 ) = 1508\).

Time = 0.78 (sec) , antiderivative size = 5931, normalized size of antiderivative = 7.07

method	result	size
risch	\(\text {Expression too large to display}\)	\(5931\)

input

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

output

result too large to display
 

3.1.57.5 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. 3363 vs. \(2 (687) = 1374\).

Time = 0.38 (sec) , antiderivative size = 3363, normalized size of antiderivative = 4.01 \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx=\text {Too large to display} \]

input

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

output

1/4*((a^2*b - b^3)*d^3*f^4*x^4 - 4*a*b^2*c^3*f^3 - 4*(a*b^2*d^3*f^3 - (a^2 
*b - b^3)*c*d^2*f^4)*x^3 - 6*(2*a*b^2*c*d^2*f^3 - (a^2*b - b^3)*c^2*d*f^4) 
*x^2 - 4*(3*a*b^2*c^2*d*f^3 - (a^2*b - b^3)*c^3*f^4)*x + ((a^2*b - b^3)*d^ 
3*f^4*x^4 - 4*a*b^2*c^3*f^3 - 4*(a*b^2*d^3*f^3 - (a^2*b - b^3)*c*d^2*f^4)* 
x^3 - 6*(2*a*b^2*c*d^2*f^3 - (a^2*b - b^3)*c^2*d*f^4)*x^2 - 4*(3*a*b^2*c^2 
*d*f^3 - (a^2*b - b^3)*c^3*f^4)*x)*cos(2*f*x + 2*e) - 6*(-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 + (-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)*cos(2*f*x + 2*e) + (-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) 
*sin(2*f*x + 2*e))*dilog(-(a^2 + b^2 - (a^2 + 2*I*a*b - b^2)*cos(2*f*x + 2 
*e) + (-I*a^2 + 2*a*b + I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2) + 1) - 6*(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 + (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)*cos(2*f*x + 2*e) + (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)*sin(2*f*x + 2*e))*dilog(-(a^2 + b^2 - (a^2 - 2*I*a*b - b^2)*co 
s(2*f*x + 2*e) + (I*a^2 + 2*a*b - I*b^2)*sin(2*f*x + 2*e))/(a^2 + b^2) + 1 
) + 2*(2*a*b^2*d^3*e^3 - 2*a*b^2*c^3*f^3 + 3*b^3*d^3*e^2 + 3*(2*a*b^2*c^2* 
d*e + b^3*c^2*d)*f^2 - 6*(a*b^2*c*d^2*e^2 + b^3*c*d^2*e)*f + (2*a*b^2*d...
 

3.1.57.6 Sympy [F]

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

input

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

output

Integral((c + d*x)**3/(a + b*cot(e + f*x))**2, x)
 

3.1.57.7 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. 4641 vs. \(2 (687) = 1374\).

Time = 3.18 (sec) , antiderivative size = 4641, normalized size of antiderivative = 5.53 \[ \int \frac {(c+d x)^3}{(a+b \cot (e+f x))^2} \, dx=\text {Too large to display} \]

input

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

output

1/12*(36*(b^2/((a^4 + a^2*b^2)*f*tan(f*x + e) + (a^3*b + a*b^3)*f) + 2*a*b 
*log(a*tan(f*x + e) + b)/((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) - (a^2 - b^2)*(f*x + e)/((a^4 + 2*a^2 
*b^2 + b^4)*f))*c^2*d*e - 12*(2*a*b*log(a*tan(f*x + e) + b)/(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^2/(a^3*b + a*b^3 + (a^4 + a^2*b^ 
2)*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)*sin(2*f*x + 2*e))*arctan2(b*cos(2*f*x + 2*e) + a*sin(2*f*x + 2...
 

3.1.57.8 Giac [F]

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

input

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

output

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

3.1.57.9 Mupad [F(-1)]

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

input

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

output

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