\(\int (c+d x)^2 (a+b \cot (e+f x)) \, dx\) [38]

Optimal result

Integrand size = 18, antiderivative size = 112 \[ \int (c+d x)^2 (a+b \cot (e+f x)) \, dx=\frac {a (c+d x)^3}{3 d}-\frac {i b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {b d^2 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(406\) vs. \(2(112)=224\).

Time = 1.39 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.62 \[ \int (c+d x)^2 (a+b \cot (e+f x)) \, dx=\frac {3 a c^2 f^3 x+3 i b c d f^2 \pi x+3 a c d f^3 x^2+a d^2 f^3 x^3+i b d^2 f^3 x^3-6 i b c d f^2 x \arctan (\tan (e))+3 b c d f^3 x^2 \cot (e)+3 b c d f \pi \log \left (1+e^{-2 i f x}\right )+3 b d^2 f^2 x^2 \log \left (1-e^{-i (e+f x)}\right )+3 b d^2 f^2 x^2 \log \left (1+e^{-i (e+f x)}\right )+6 b c d f^2 x \log \left (1-e^{2 i (f x+\arctan (\tan (e)))}\right )+6 b c d f \arctan (\tan (e)) \log \left (1-e^{2 i (f x+\arctan (\tan (e)))}\right )-3 b c d f \pi \log (\cos (f x))+3 b c^2 f^2 \log (\sin (e+f x))-6 b c d f \arctan (\tan (e)) \log (\sin (f x+\arctan (\tan (e))))+6 i b d^2 f x \operatorname {PolyLog}\left (2,-e^{-i (e+f x)}\right )+6 i b d^2 f x \operatorname {PolyLog}\left (2,e^{-i (e+f x)}\right )-3 i b c d f \operatorname {PolyLog}\left (2,e^{2 i (f x+\arctan (\tan (e)))}\right )+6 b d^2 \operatorname {PolyLog}\left (3,-e^{-i (e+f x)}\right )+6 b d^2 \operatorname {PolyLog}\left (3,e^{-i (e+f x)}\right )-3 b c d e^{i \arctan (\tan (e))} f^3 x^2 \cot (e) \sqrt {\sec ^2(e)}}{3 f^3} \] Input:

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

Output:

(3*a*c^2*f^3*x + (3*I)*b*c*d*f^2*Pi*x + 3*a*c*d*f^3*x^2 + a*d^2*f^3*x^3 + 
I*b*d^2*f^3*x^3 - (6*I)*b*c*d*f^2*x*ArcTan[Tan[e]] + 3*b*c*d*f^3*x^2*Cot[e 
] + 3*b*c*d*f*Pi*Log[1 + E^((-2*I)*f*x)] + 3*b*d^2*f^2*x^2*Log[1 - E^((-I) 
*(e + f*x))] + 3*b*d^2*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] + 6*b*c*d*f^2*x 
*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + 6*b*c*d*f*ArcTan[Tan[e]]*Log[ 
1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] - 3*b*c*d*f*Pi*Log[Cos[f*x]] + 3*b*c 
^2*f^2*Log[Sin[e + f*x]] - 6*b*c*d*f*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[T 
an[e]]]] + (6*I)*b*d^2*f*x*PolyLog[2, -E^((-I)*(e + f*x))] + (6*I)*b*d^2*f 
*x*PolyLog[2, E^((-I)*(e + f*x))] - (3*I)*b*c*d*f*PolyLog[2, E^((2*I)*(f*x 
 + ArcTan[Tan[e]]))] + 6*b*d^2*PolyLog[3, -E^((-I)*(e + f*x))] + 6*b*d^2*P 
olyLog[3, E^((-I)*(e + f*x))] - 3*b*c*d*E^(I*ArcTan[Tan[e]])*f^3*x^2*Cot[e 
]*Sqrt[Sec[e]^2])/(3*f^3)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4205

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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. 534 vs. \(2 (98 ) = 196\).

Time = 0.63 (sec) , antiderivative size = 535, normalized size of antiderivative = 4.78

method	result	size
risch	\(\frac {d^{2} a \,x^{3}}{3}+\frac {a \,c^{3}}{3 d}+\frac {4 i b \,d^{2} e^{3}}{3 f^{3}}-\frac {2 b \,d^{2} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {i b \,c^{3}}{3 d}-\frac {2 b \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}+i b \,c^{2} x -\frac {i d^{2} b \,x^{3}}{3}+d a c \,x^{2}+a \,c^{2} x -i d b c \,x^{2}+\frac {2 b \,d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {2 b \,d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {b \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f}+\frac {b \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{f}+\frac {b \,d^{2} \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{f}+\frac {b \,d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x^{2}}{f}+\frac {b \,e^{2} d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f^{3}}-\frac {b \,d^{2} \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) e^{2}}{f^{3}}-\frac {2 i b d c \,e^{2}}{f^{2}}+\frac {2 i b \,d^{2} e^{2} x}{f^{2}}+\frac {4 b c d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {4 i b d c e x}{f}-\frac {2 i b d c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 i b d c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {2 b d c \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) e}{f^{2}}-\frac {2 b e c d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{f^{2}}+\frac {2 b d c \ln \left (1-{\mathrm e}^{i \left (f x +e \right )}\right ) x}{f}+\frac {2 b d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{f}-\frac {2 i b \,d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f^{2}}-\frac {2 i b \,d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (f x +e \right )}\right ) x}{f^{2}}\)	\(535\)

Input:

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

Output:

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

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. 405 vs. \(2 (95) = 190\).

Time = 0.09 (sec) , antiderivative size = 405, normalized size of antiderivative = 3.62 \[ \int (c+d x)^2 (a+b \cot (e+f x)) \, dx=\frac {4 \, a d^{2} f^{3} x^{3} + 12 \, a c d f^{3} x^{2} + 12 \, a c^{2} f^{3} x + 3 \, b d^{2} {\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + 3 \, b d^{2} {\rm polylog}\left (3, \cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) - 6 \, {\left (i \, b d^{2} f x + i \, b c d f\right )} {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right )\right ) - 6 \, {\left (-i \, b d^{2} f x - i \, b c d f\right )} {\rm Li}_2\left (\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right )\right ) + 6 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) + \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) + 6 \, {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, f x + 2 \, e\right ) - \frac {1}{2} i \, \sin \left (2 \, f x + 2 \, e\right ) + \frac {1}{2}\right ) + 6 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) + i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right ) + 6 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x - b d^{2} e^{2} + 2 \, b c d e f\right )} \log \left (-\cos \left (2 \, f x + 2 \, e\right ) - i \, \sin \left (2 \, f x + 2 \, e\right ) + 1\right )}{12 \, f^{3}} \] Input:

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

Output:

1/12*(4*a*d^2*f^3*x^3 + 12*a*c*d*f^3*x^2 + 12*a*c^2*f^3*x + 3*b*d^2*polylo 
g(3, cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) + 3*b*d^2*polylog(3, cos(2*f*x 
 + 2*e) - I*sin(2*f*x + 2*e)) - 6*(I*b*d^2*f*x + I*b*c*d*f)*dilog(cos(2*f* 
x + 2*e) + I*sin(2*f*x + 2*e)) - 6*(-I*b*d^2*f*x - I*b*c*d*f)*dilog(cos(2* 
f*x + 2*e) - I*sin(2*f*x + 2*e)) + 6*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2) 
*log(-1/2*cos(2*f*x + 2*e) + 1/2*I*sin(2*f*x + 2*e) + 1/2) + 6*(b*d^2*e^2 
- 2*b*c*d*e*f + b*c^2*f^2)*log(-1/2*cos(2*f*x + 2*e) - 1/2*I*sin(2*f*x + 2 
*e) + 1/2) + 6*(b*d^2*f^2*x^2 + 2*b*c*d*f^2*x - b*d^2*e^2 + 2*b*c*d*e*f)*l 
og(-cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e) + 1) + 6*(b*d^2*f^2*x^2 + 2*b*c* 
d*f^2*x - b*d^2*e^2 + 2*b*c*d*e*f)*log(-cos(2*f*x + 2*e) - I*sin(2*f*x + 2 
*e) + 1))/f^3
 

Sympy [F]

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

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

Output:

Integral((a + b*cot(e + f*x))*(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. 527 vs. \(2 (95) = 190\).

Time = 0.12 (sec) , antiderivative size = 527, normalized size of antiderivative = 4.71 \[ \int (c+d x)^2 (a+b \cot (e+f x)) \, dx=\frac {6 \, {\left (f x + e\right )} a c^{2} + \frac {2 \, {\left (f x + e\right )}^{3} a d^{2}}{f^{2}} - \frac {6 \, {\left (f x + e\right )}^{2} a d^{2} e}{f^{2}} + \frac {6 \, {\left (f x + e\right )} a d^{2} e^{2}}{f^{2}} + \frac {6 \, {\left (f x + e\right )}^{2} a c d}{f} - \frac {12 \, {\left (f x + e\right )} a c d e}{f} + 6 \, b c^{2} \log \left (\sin \left (f x + e\right )\right ) + \frac {6 \, b d^{2} e^{2} \log \left (\sin \left (f x + e\right )\right )}{f^{2}} - \frac {12 \, b c d e \log \left (\sin \left (f x + e\right )\right )}{f} + \frac {-2 i \, {\left (f x + e\right )}^{3} b d^{2} + 12 \, b d^{2} {\rm Li}_{3}(-e^{\left (i \, f x + i \, e\right )}) + 12 \, b d^{2} {\rm Li}_{3}(e^{\left (i \, f x + i \, e\right )}) - 6 \, {\left (-i \, b d^{2} e + i \, b c d f\right )} {\left (f x + e\right )}^{2} - 6 \, {\left (-i \, {\left (f x + e\right )}^{2} b d^{2} + 2 \, {\left (i \, b d^{2} e - i \, b c d f\right )} {\left (f x + e\right )}\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) - 6 \, {\left (i \, {\left (f x + e\right )}^{2} b d^{2} + 2 \, {\left (-i \, b d^{2} e + i \, b c d f\right )} {\left (f x + e\right )}\right )} \arctan \left (\sin \left (f x + e\right ), -\cos \left (f x + e\right ) + 1\right ) - 12 \, {\left (i \, {\left (f x + e\right )} b d^{2} - i \, b d^{2} e + i \, b c d f\right )} {\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) - 12 \, {\left (i \, {\left (f x + e\right )} b d^{2} - i \, b d^{2} e + i \, b c d f\right )} {\rm Li}_2\left (e^{\left (i \, f x + i \, e\right )}\right ) + 3 \, {\left ({\left (f x + e\right )}^{2} b d^{2} - 2 \, {\left (b d^{2} e - b c d f\right )} {\left (f x + e\right )}\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) + 3 \, {\left ({\left (f x + e\right )}^{2} b d^{2} - 2 \, {\left (b d^{2} e - b c d f\right )} {\left (f x + e\right )}\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1\right )}{f^{2}}}{6 \, f} \] Input:

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

Output:

1/6*(6*(f*x + e)*a*c^2 + 2*(f*x + e)^3*a*d^2/f^2 - 6*(f*x + e)^2*a*d^2*e/f 
^2 + 6*(f*x + e)*a*d^2*e^2/f^2 + 6*(f*x + e)^2*a*c*d/f - 12*(f*x + e)*a*c* 
d*e/f + 6*b*c^2*log(sin(f*x + e)) + 6*b*d^2*e^2*log(sin(f*x + e))/f^2 - 12 
*b*c*d*e*log(sin(f*x + e))/f + (-2*I*(f*x + e)^3*b*d^2 + 12*b*d^2*polylog( 
3, -e^(I*f*x + I*e)) + 12*b*d^2*polylog(3, e^(I*f*x + I*e)) - 6*(-I*b*d^2* 
e + I*b*c*d*f)*(f*x + e)^2 - 6*(-I*(f*x + e)^2*b*d^2 + 2*(I*b*d^2*e - I*b* 
c*d*f)*(f*x + e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) - 6*(I*(f*x + e) 
^2*b*d^2 + 2*(-I*b*d^2*e + I*b*c*d*f)*(f*x + e))*arctan2(sin(f*x + e), -co 
s(f*x + e) + 1) - 12*(I*(f*x + e)*b*d^2 - I*b*d^2*e + I*b*c*d*f)*dilog(-e^ 
(I*f*x + I*e)) - 12*(I*(f*x + e)*b*d^2 - I*b*d^2*e + I*b*c*d*f)*dilog(e^(I 
*f*x + I*e)) + 3*((f*x + e)^2*b*d^2 - 2*(b*d^2*e - b*c*d*f)*(f*x + e))*log 
(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) + 3*((f*x + e)^2*b* 
d^2 - 2*(b*d^2*e - b*c*d*f)*(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 
 - 2*cos(f*x + e) + 1))/f^2)/f
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*int(cot(e + f*x)*x**2,x)*b*d**2*f + 6*int(cot(e + f*x)*x,x)*b*c*d*f - 3 
*log(tan((e + f*x)/2)**2 + 1)*b*c**2 + 3*log(tan((e + f*x)/2))*b*c**2 + 3* 
a*c**2*f*x + 3*a*c*d*f*x**2 + a*d**2*f*x**3)/(3*f)