Integrand size = 18, antiderivative size = 147 \[ \int (c+d x)^3 (a+b \cot (e+f x)) \, dx=\frac {a (c+d x)^4}{4 d}-\frac {i b (c+d x)^4}{4 d}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 i b d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4} \] Output:
1/4*a*(d*x+c)^4/d-1/4*I*b*(d*x+c)^4/d+b*(d*x+c)^3*ln(1-exp(2*I*(f*x+e)))/f -3/2*I*b*d*(d*x+c)^2*polylog(2,exp(2*I*(f*x+e)))/f^2+3/2*b*d^2*(d*x+c)*pol ylog(3,exp(2*I*(f*x+e)))/f^3+3/4*I*b*d^3*polylog(4,exp(2*I*(f*x+e)))/f^4
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(632\) vs. \(2(147)=294\).
Time = 3.19 (sec) , antiderivative size = 632, normalized size of antiderivative = 4.30 \[ \int (c+d x)^3 (a+b \cot (e+f x)) \, dx=\frac {4 a c^3 f^4 x+6 i b c^2 d f^3 \pi x+6 a c^2 d f^4 x^2+4 a c d^2 f^4 x^3+4 i b c d^2 f^4 x^3+a d^3 f^4 x^4+i b d^3 f^4 x^4-12 i b c^2 d f^3 x \arctan (\tan (e))+6 b c^2 d f^4 x^2 \cot (e)+6 b c^2 d f^2 \pi \log \left (1+e^{-2 i f x}\right )+12 b c d^2 f^3 x^2 \log \left (1-e^{-i (e+f x)}\right )+4 b d^3 f^3 x^3 \log \left (1-e^{-i (e+f x)}\right )+12 b c d^2 f^3 x^2 \log \left (1+e^{-i (e+f x)}\right )+4 b d^3 f^3 x^3 \log \left (1+e^{-i (e+f x)}\right )+12 b c^2 d f^3 x \log \left (1-e^{2 i (f x+\arctan (\tan (e)))}\right )+12 b c^2 d f^2 \arctan (\tan (e)) \log \left (1-e^{2 i (f x+\arctan (\tan (e)))}\right )-6 b c^2 d f^2 \pi \log (\cos (f x))+4 b c^3 f^3 \log (\sin (e+f x))-12 b c^2 d f^2 \arctan (\tan (e)) \log (\sin (f x+\arctan (\tan (e))))+12 i b d^2 f^2 x (2 c+d x) \operatorname {PolyLog}\left (2,-e^{-i (e+f x)}\right )+12 i b d^2 f^2 x (2 c+d x) \operatorname {PolyLog}\left (2,e^{-i (e+f x)}\right )-6 i b c^2 d f^2 \operatorname {PolyLog}\left (2,e^{2 i (f x+\arctan (\tan (e)))}\right )+24 b c d^2 f \operatorname {PolyLog}\left (3,-e^{-i (e+f x)}\right )+24 b d^3 f x \operatorname {PolyLog}\left (3,-e^{-i (e+f x)}\right )+24 b c d^2 f \operatorname {PolyLog}\left (3,e^{-i (e+f x)}\right )+24 b d^3 f x \operatorname {PolyLog}\left (3,e^{-i (e+f x)}\right )-24 i b d^3 \operatorname {PolyLog}\left (4,-e^{-i (e+f x)}\right )-24 i b d^3 \operatorname {PolyLog}\left (4,e^{-i (e+f x)}\right )-6 b c^2 d e^{i \arctan (\tan (e))} f^4 x^2 \cot (e) \sqrt {\sec ^2(e)}}{4 f^4} \] Input:
Integrate[(c + d*x)^3*(a + b*Cot[e + f*x]),x]
Output:
(4*a*c^3*f^4*x + (6*I)*b*c^2*d*f^3*Pi*x + 6*a*c^2*d*f^4*x^2 + 4*a*c*d^2*f^ 4*x^3 + (4*I)*b*c*d^2*f^4*x^3 + a*d^3*f^4*x^4 + I*b*d^3*f^4*x^4 - (12*I)*b *c^2*d*f^3*x*ArcTan[Tan[e]] + 6*b*c^2*d*f^4*x^2*Cot[e] + 6*b*c^2*d*f^2*Pi* Log[1 + E^((-2*I)*f*x)] + 12*b*c*d^2*f^3*x^2*Log[1 - E^((-I)*(e + f*x))] + 4*b*d^3*f^3*x^3*Log[1 - E^((-I)*(e + f*x))] + 12*b*c*d^2*f^3*x^2*Log[1 + E^((-I)*(e + f*x))] + 4*b*d^3*f^3*x^3*Log[1 + E^((-I)*(e + f*x))] + 12*b*c ^2*d*f^3*x*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + 12*b*c^2*d*f^2*ArcT an[Tan[e]]*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] - 6*b*c^2*d*f^2*Pi*Lo g[Cos[f*x]] + 4*b*c^3*f^3*Log[Sin[e + f*x]] - 12*b*c^2*d*f^2*ArcTan[Tan[e] ]*Log[Sin[f*x + ArcTan[Tan[e]]]] + (12*I)*b*d^2*f^2*x*(2*c + d*x)*PolyLog[ 2, -E^((-I)*(e + f*x))] + (12*I)*b*d^2*f^2*x*(2*c + d*x)*PolyLog[2, E^((-I )*(e + f*x))] - (6*I)*b*c^2*d*f^2*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e] ]))] + 24*b*c*d^2*f*PolyLog[3, -E^((-I)*(e + f*x))] + 24*b*d^3*f*x*PolyLog [3, -E^((-I)*(e + f*x))] + 24*b*c*d^2*f*PolyLog[3, E^((-I)*(e + f*x))] + 2 4*b*d^3*f*x*PolyLog[3, E^((-I)*(e + f*x))] - (24*I)*b*d^3*PolyLog[4, -E^(( -I)*(e + f*x))] - (24*I)*b*d^3*PolyLog[4, E^((-I)*(e + f*x))] - 6*b*c^2*d* E^(I*ArcTan[Tan[e]])*f^4*x^2*Cot[e]*Sqrt[Sec[e]^2])/(4*f^4)
Time = 0.51 (sec) , antiderivative size = 147, 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.
| \(\displaystyle \int (c+d x)^3 (a+b \cot (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^3 \left (a-b \tan \left (e+f x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4205 |
| \(\displaystyle \int \left (a (c+d x)^3+b (c+d x)^3 \cot (e+f x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a (c+d x)^4}{4 d}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 i b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i b (c+d x)^4}{4 d}+\frac {3 i b d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}\) |
Input:
Int[(c + d*x)^3*(a + b*Cot[e + f*x]),x]
Output:
(a*(c + d*x)^4)/(4*d) - ((I/4)*b*(c + d*x)^4)/d + (b*(c + d*x)^3*Log[1 - E ^((2*I)*(e + f*x))])/f - (((3*I)/2)*b*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (3*b*d^2*(c + d*x)*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^ 3) + (((3*I)/4)*b*d^3*PolyLog[4, E^((2*I)*(e + f*x))])/f^4
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (126 ) = 252\).
Time = 0.68 (sec) , antiderivative size = 876, normalized size of antiderivative = 5.96
Input:
int((d*x+c)^3*(a+b*cot(f*x+e)),x,method=_RETURNVERBOSE)
Output:
1/f*b*d^3*ln(1-exp(I*(f*x+e)))*x^3+1/f*b*d^3*ln(exp(I*(f*x+e))+1)*x^3+1/f* b*c^3*ln(exp(I*(f*x+e))-1)-2/f*b*c^3*ln(exp(I*(f*x+e)))+1/f*b*c^3*ln(exp(I *(f*x+e))+1)+d^2*a*c*x^3+3/2*d*a*c^2*x^2+a*c^3*x+I*b*c^3*x+1/4*I/d*b*c^4-1 /4*I*d^3*b*x^4-I*d^2*b*c*x^3+6/f^3*b*c*d^2*polylog(3,exp(I*(f*x+e)))+6/f^3 *b*c*d^2*polylog(3,-exp(I*(f*x+e)))+6/f^3*b*d^3*polylog(3,exp(I*(f*x+e)))* x+6/f^3*b*d^3*polylog(3,-exp(I*(f*x+e)))*x+1/f^4*b*d^3*ln(1-exp(I*(f*x+e)) )*e^3-1/f^4*b*e^3*d^3*ln(exp(I*(f*x+e))-1)+2/f^4*b*e^3*d^3*ln(exp(I*(f*x+e )))+6*I/f^4*b*d^3*polylog(4,exp(I*(f*x+e)))+6*I/f^4*b*d^3*polylog(4,-exp(I *(f*x+e)))-3/2*I/f^4*b*d^3*e^4-3/2*I*d*b*c^2*x^2-6*I/f*b*d*c^2*e*x-6*I/f^2 *b*c*d^2*polylog(2,exp(I*(f*x+e)))*x-6*I/f^2*b*c*d^2*polylog(2,-exp(I*(f*x +e)))*x+6*I/f^2*b*c*d^2*e^2*x-3/f^2*b*e*c^2*d*ln(exp(I*(f*x+e))-1)+6/f^2*b *e*c^2*d*ln(exp(I*(f*x+e)))+1/4*d^3*a*x^4+1/4/d*a*c^4+3/f^3*b*e^2*c*d^2*ln (exp(I*(f*x+e))-1)-6/f^3*b*e^2*c*d^2*ln(exp(I*(f*x+e)))+3/f*b*c*d^2*ln(1-e xp(I*(f*x+e)))*x^2+3/f*b*c*d^2*ln(exp(I*(f*x+e))+1)*x^2+3/f*b*d*c^2*ln(1-e xp(I*(f*x+e)))*x+3/f*b*d*c^2*ln(exp(I*(f*x+e))+1)*x-3/f^3*b*c*d^2*ln(1-exp (I*(f*x+e)))*e^2+3/f^2*b*d*c^2*ln(1-exp(I*(f*x+e)))*e+4*I/f^3*b*c*d^2*e^3- 3*I/f^2*b*d^3*polylog(2,exp(I*(f*x+e)))*x^2-3*I/f^2*b*d^3*polylog(2,-exp(I *(f*x+e)))*x^2-3*I/f^2*b*d*c^2*e^2-3*I/f^2*b*d*c^2*polylog(2,exp(I*(f*x+e) ))-3*I/f^2*b*d*c^2*polylog(2,-exp(I*(f*x+e)))-2*I/f^3*b*d^3*e^3*x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (122) = 244\).
Time = 0.10 (sec) , antiderivative size = 628, normalized size of antiderivative = 4.27 \[ \int (c+d x)^3 (a+b \cot (e+f x)) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^3*(a+b*cot(f*x+e)),x, algorithm="fricas")
Output:
1/8*(2*a*d^3*f^4*x^4 + 8*a*c*d^2*f^4*x^3 + 12*a*c^2*d*f^4*x^2 + 8*a*c^3*f^ 4*x + 3*I*b*d^3*polylog(4, cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) - 3*I*b* d^3*polylog(4, cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e)) - 6*(I*b*d^3*f^2*x^2 + 2*I*b*c*d^2*f^2*x + I*b*c^2*d*f^2)*dilog(cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) - 6*(-I*b*d^3*f^2*x^2 - 2*I*b*c*d^2*f^2*x - I*b*c^2*d*f^2)*dilog( cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e)) - 4*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*log(-1/2*cos(2*f*x + 2*e) + 1/2*I*sin(2*f*x + 2*e) + 1/2) - 4*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^ 3)*log(-1/2*cos(2*f*x + 2*e) - 1/2*I*sin(2*f*x + 2*e) + 1/2) + 4*(b*d^3*f^ 3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2)*log(-cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e) + 1) + 4*(b* d^3*f^3*x^3 + 3*b*c*d^2*f^3*x^2 + 3*b*c^2*d*f^3*x + b*d^3*e^3 - 3*b*c*d^2* e^2*f + 3*b*c^2*d*e*f^2)*log(-cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e) + 1) + 6*(b*d^3*f*x + b*c*d^2*f)*polylog(3, cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e )) + 6*(b*d^3*f*x + b*c*d^2*f)*polylog(3, cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e)))/f^4
\[ \int (c+d x)^3 (a+b \cot (e+f x)) \, dx=\int \left (a + b \cot {\left (e + f x \right )}\right ) \left (c + d x\right )^{3}\, dx \] Input:
integrate((d*x+c)**3*(a+b*cot(f*x+e)),x)
Output:
Integral((a + b*cot(e + f*x))*(c + d*x)**3, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (122) = 244\).
Time = 0.16 (sec) , antiderivative size = 978, normalized size of antiderivative = 6.65 \[ \int (c+d x)^3 (a+b \cot (e+f x)) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^3*(a+b*cot(f*x+e)),x, algorithm="maxima")
Output:
1/4*(4*(f*x + e)*a*c^3 + (f*x + e)^4*a*d^3/f^3 - 4*(f*x + e)^3*a*d^3*e/f^3 + 6*(f*x + e)^2*a*d^3*e^2/f^3 - 4*(f*x + e)*a*d^3*e^3/f^3 + 4*(f*x + e)^3 *a*c*d^2/f^2 - 12*(f*x + e)^2*a*c*d^2*e/f^2 + 12*(f*x + e)*a*c*d^2*e^2/f^2 + 6*(f*x + e)^2*a*c^2*d/f - 12*(f*x + e)*a*c^2*d*e/f + 4*b*c^3*log(sin(f* x + e)) - 4*b*d^3*e^3*log(sin(f*x + e))/f^3 + 12*b*c*d^2*e^2*log(sin(f*x + e))/f^2 - 12*b*c^2*d*e*log(sin(f*x + e))/f + (-I*(f*x + e)^4*b*d^3 + 24*I *b*d^3*polylog(4, -e^(I*f*x + I*e)) + 24*I*b*d^3*polylog(4, e^(I*f*x + I*e )) - 4*(-I*b*d^3*e + I*b*c*d^2*f)*(f*x + e)^3 - 6*(I*b*d^3*e^2 - 2*I*b*c*d ^2*e*f + I*b*c^2*d*f^2)*(f*x + e)^2 - 4*(-I*(f*x + e)^3*b*d^3 + 3*(I*b*d^3 *e - I*b*c*d^2*f)*(f*x + e)^2 + 3*(-I*b*d^3*e^2 + 2*I*b*c*d^2*e*f - I*b*c^ 2*d*f^2)*(f*x + e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) - 4*(I*(f*x + e)^3*b*d^3 + 3*(-I*b*d^3*e + I*b*c*d^2*f)*(f*x + e)^2 + 3*(I*b*d^3*e^2 - 2 *I*b*c*d^2*e*f + I*b*c^2*d*f^2)*(f*x + e))*arctan2(sin(f*x + e), -cos(f*x + e) + 1) - 12*(I*(f*x + e)^2*b*d^3 + I*b*d^3*e^2 - 2*I*b*c*d^2*e*f + I*b* c^2*d*f^2 + 2*(-I*b*d^3*e + I*b*c*d^2*f)*(f*x + e))*dilog(-e^(I*f*x + I*e) ) - 12*(I*(f*x + e)^2*b*d^3 + I*b*d^3*e^2 - 2*I*b*c*d^2*e*f + I*b*c^2*d*f^ 2 + 2*(-I*b*d^3*e + I*b*c*d^2*f)*(f*x + e))*dilog(e^(I*f*x + I*e)) + 2*((f *x + e)^3*b*d^3 - 3*(b*d^3*e - b*c*d^2*f)*(f*x + e)^2 + 3*(b*d^3*e^2 - 2*b *c*d^2*e*f + b*c^2*d*f^2)*(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) + 2*((f*x + e)^3*b*d^3 - 3*(b*d^3*e - b*c*d^2*f)*(...
\[ \int (c+d x)^3 (a+b \cot (e+f x)) \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \cot \left (f x + e\right ) + a\right )} \,d x } \] Input:
integrate((d*x+c)^3*(a+b*cot(f*x+e)),x, algorithm="giac")
Output:
integrate((d*x + c)^3*(b*cot(f*x + e) + a), x)
Timed out. \[ \int (c+d x)^3 (a+b \cot (e+f x)) \, dx=\int \left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3 \,d x \] Input:
int((a + b*cot(e + f*x))*(c + d*x)^3,x)
Output:
int((a + b*cot(e + f*x))*(c + d*x)^3, x)
\[ \int (c+d x)^3 (a+b \cot (e+f x)) \, dx=\frac {4 \left (\int \cot \left (f x +e \right ) x^{3}d x \right ) b \,d^{3} f +12 \left (\int \cot \left (f x +e \right ) x^{2}d x \right ) b c \,d^{2} f +12 \left (\int \cot \left (f x +e \right ) x d x \right ) b \,c^{2} d f -4 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right ) b \,c^{3}+4 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) b \,c^{3}+4 a \,c^{3} f x +6 a \,c^{2} d f \,x^{2}+4 a c \,d^{2} f \,x^{3}+a \,d^{3} f \,x^{4}}{4 f} \] Input:
int((d*x+c)^3*(a+b*cot(f*x+e)),x)
Output:
(4*int(cot(e + f*x)*x**3,x)*b*d**3*f + 12*int(cot(e + f*x)*x**2,x)*b*c*d** 2*f + 12*int(cot(e + f*x)*x,x)*b*c**2*d*f - 4*log(tan((e + f*x)/2)**2 + 1) *b*c**3 + 4*log(tan((e + f*x)/2))*b*c**3 + 4*a*c**3*f*x + 6*a*c**2*d*f*x** 2 + 4*a*c*d**2*f*x**3 + a*d**3*f*x**4)/(4*f)