\(\int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx\) [86]

Optimal result

Integrand size = 14, antiderivative size = 197 \[ \int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx=x \cos (x)-i x \log \left (1-i \left (-1+\sqrt {2}\right ) e^{i x}\right )+i x \log \left (1+i \left (-1+\sqrt {2}\right ) e^{i x}\right )+i x \log \left (1-i \left (1+\sqrt {2}\right ) e^{i x}\right )-i x \log \left (1+i \left (1+\sqrt {2}\right ) e^{i x}\right )+\operatorname {PolyLog}\left (2,-i \left (-1+\sqrt {2}\right ) e^{i x}\right )-\operatorname {PolyLog}\left (2,i \left (-1+\sqrt {2}\right ) e^{i x}\right )-\operatorname {PolyLog}\left (2,-i \left (1+\sqrt {2}\right ) e^{i x}\right )+\operatorname {PolyLog}\left (2,i \left (1+\sqrt {2}\right ) e^{i x}\right )-\sin (x) \] Output:

x*cos(x)-I*x*ln(1-I*(2^(1/2)-1)*exp(I*x))+I*x*ln(1+I*(2^(1/2)-1)*exp(I*x)) 
+I*x*ln(1-I*(1+2^(1/2))*exp(I*x))-I*x*ln(1+I*(1+2^(1/2))*exp(I*x))+polylog 
(2,-I*(2^(1/2)-1)*exp(I*x))-polylog(2,I*(2^(1/2)-1)*exp(I*x))-polylog(2,-I 
*(1+2^(1/2))*exp(I*x))+polylog(2,I*(1+2^(1/2))*exp(I*x))-sin(x)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1037\) vs. \(2(197)=394\).

Time = 0.76 (sec) , antiderivative size = 1037, normalized size of antiderivative = 5.26 \[ \int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx =\text {Too large to display} \] Input:

Integrate[(x*Sin[x]^3)/(1 + Cos[x]^2),x]
 

Output:

2*ArcCos[-Sqrt[2]]*ArcTan[(-1 + Sqrt[2])*Cot[(Pi + 2*x)/4]] - 2*ArcCos[Sqr 
t[2]]*ArcTan[(1 + Sqrt[2])*Cot[(Pi + 2*x)/4]] - Pi*ArcTan[1 - Sqrt[2]*Tan[ 
x/2]] - Pi*ArcTan[1 + Sqrt[2]*Tan[x/2]] - (Pi - 2*x)*ArcTan[(-1 + Sqrt[2]) 
*Tan[(Pi + 2*x)/4]] + (Pi - 2*x)*ArcTan[(1 + Sqrt[2])*Tan[(Pi + 2*x)/4]] + 
 x*Cos[x] + I*(ArcCos[Sqrt[2]] - 2*ArcTan[(1 + Sqrt[2])*Cot[(Pi + 2*x)/4]] 
)*Log[(Sqrt[2]*(1 - I*Cot[(Pi + 2*x)/4]))/(-1 + Sqrt[2] + I*Cot[(Pi + 2*x) 
/4])] - I*(ArcCos[-Sqrt[2]] + 2*ArcTan[(-1 + Sqrt[2])*Cot[(Pi + 2*x)/4]])* 
Log[(Sqrt[2]*(1 + I*Cot[(Pi + 2*x)/4]))/(1 + Sqrt[2] + I*Cot[(Pi + 2*x)/4] 
)] + I*(ArcCos[Sqrt[2]] + 2*ArcTan[(1 + Sqrt[2])*Cot[(Pi + 2*x)/4]])*Log[( 
(-I)*(-2 + Sqrt[2])*(-I + Cot[(Pi + 2*x)/4]))/(-1 + Sqrt[2] + I*Cot[(Pi + 
2*x)/4])] - I*(ArcCos[-Sqrt[2]] - 2*ArcTan[(-1 + Sqrt[2])*Cot[(Pi + 2*x)/4 
]])*Log[(I*(2 + Sqrt[2])*(I + Cot[(Pi + 2*x)/4]))/(1 + Sqrt[2] + I*Cot[(Pi 
 + 2*x)/4])] - I*(ArcCos[Sqrt[2]] + 2*ArcTan[(1 + Sqrt[2])*Cot[(Pi + 2*x)/ 
4]] + 2*ArcTan[(-1 + Sqrt[2])*Tan[(Pi + 2*x)/4]])*Log[(1/2 + I/2)/(E^((I/2 
)*x)*Sqrt[Sqrt[2] - Sin[x]])] - I*(ArcCos[Sqrt[2]] - 2*ArcTan[(1 + Sqrt[2] 
)*Cot[(Pi + 2*x)/4]] - 2*ArcTan[(-1 + Sqrt[2])*Tan[(Pi + 2*x)/4]])*Log[((1 
/2 - I/2)*E^((I/2)*x))/Sqrt[Sqrt[2] - Sin[x]]] + I*(ArcCos[-Sqrt[2]] + 2*A 
rcTan[(-1 + Sqrt[2])*Cot[(Pi + 2*x)/4]] + 2*ArcTan[(1 + Sqrt[2])*Tan[(Pi + 
 2*x)/4]])*Log[(-1/2 + I/2)/(E^((I/2)*x)*Sqrt[Sqrt[2] + Sin[x]])] + I*(Arc 
Cos[-Sqrt[2]] - 2*ArcTan[(-1 + Sqrt[2])*Cot[(Pi + 2*x)/4]] - 2*ArcTan[(...
 

Rubi [F]

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.

\(\displaystyle \int \frac {x \sin ^3(x)}{\cos ^2(x)+1} \, dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \frac {x \sin ^3(x)}{\cos ^2(x)+1}dx\)

Input:

Int[(x*Sin[x]^3)/(1 + Cos[x]^2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 927 vs. \(2 (153 ) = 306\).

Time = 0.26 (sec) , antiderivative size = 928, normalized size of antiderivative = 4.71

method	result	size
risch	\(\frac {\sqrt {2}\, x \ln \left (\frac {-i \sqrt {2}+i-{\mathrm e}^{i x}}{-i \sqrt {2}+i}\right )}{-4 i \sqrt {2}+4 i}-\frac {i \sqrt {2}\, \ln \left (\frac {-i \sqrt {2}-i-{\mathrm e}^{i x}}{-i \sqrt {2}-i}\right ) x}{4}+\frac {i \sqrt {2}\, \ln \left (\frac {i \sqrt {2}-i-{\mathrm e}^{i x}}{i \sqrt {2}-i}\right ) x}{4}-\frac {i \sqrt {2}\, \ln \left (\frac {-i \sqrt {2}+i-{\mathrm e}^{i x}}{-i \sqrt {2}+i}\right ) x}{4}-\frac {i \sqrt {2}\, \operatorname {dilog}\left (\frac {i \sqrt {2}-i-{\mathrm e}^{i x}}{i \sqrt {2}-i}\right )}{4 \left (i \sqrt {2}-i\right )}-\frac {i \sqrt {2}\, \operatorname {dilog}\left (\frac {-i \sqrt {2}+i-{\mathrm e}^{i x}}{-i \sqrt {2}+i}\right )}{4 \left (-i \sqrt {2}+i\right )}-\frac {\sqrt {2}\, x \ln \left (\frac {i \sqrt {2}+i-{\mathrm e}^{i x}}{i \sqrt {2}+i}\right )}{4 \left (i \sqrt {2}+i\right )}-\frac {\sqrt {2}\, x \ln \left (\frac {-i \sqrt {2}-i-{\mathrm e}^{i x}}{-i \sqrt {2}-i}\right )}{4 \left (-i \sqrt {2}-i\right )}+\frac {i \sqrt {2}\, \ln \left (\frac {i \sqrt {2}+i-{\mathrm e}^{i x}}{i \sqrt {2}+i}\right ) x}{4}+\frac {\operatorname {dilog}\left (\frac {i \sqrt {2}+i-{\mathrm e}^{i x}}{i \sqrt {2}+i}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-i \sqrt {2}-i-{\mathrm e}^{i x}}{-i \sqrt {2}-i}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {i \sqrt {2}-i-{\mathrm e}^{i x}}{i \sqrt {2}-i}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-i \sqrt {2}+i-{\mathrm e}^{i x}}{-i \sqrt {2}+i}\right )}{2}+\frac {\sqrt {2}\, x \ln \left (\frac {i \sqrt {2}-i-{\mathrm e}^{i x}}{i \sqrt {2}-i}\right )}{4 i \sqrt {2}-4 i}+\frac {i \sqrt {2}\, \operatorname {dilog}\left (\frac {i \sqrt {2}+i-{\mathrm e}^{i x}}{i \sqrt {2}+i}\right )}{4 i \sqrt {2}+4 i}+\frac {i \sqrt {2}\, \operatorname {dilog}\left (\frac {-i \sqrt {2}-i-{\mathrm e}^{i x}}{-i \sqrt {2}-i}\right )}{-4 i \sqrt {2}-4 i}+\frac {\left (x +i\right ) {\mathrm e}^{i x}}{2}-\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {-i \sqrt {2}+i-{\mathrm e}^{i x}}{-i \sqrt {2}+i}\right )}{4}+\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {i \sqrt {2}+i-{\mathrm e}^{i x}}{i \sqrt {2}+i}\right )}{4}+\frac {\left (x -i\right ) {\mathrm e}^{-i x}}{2}-\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {-i \sqrt {2}-i-{\mathrm e}^{i x}}{-i \sqrt {2}-i}\right )}{4}+\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {i \sqrt {2}-i-{\mathrm e}^{i x}}{i \sqrt {2}-i}\right )}{4}+\frac {i x \ln \left (\frac {i \sqrt {2}+i-{\mathrm e}^{i x}}{i \sqrt {2}+i}\right )}{2}-\frac {i x \ln \left (\frac {-i \sqrt {2}-i-{\mathrm e}^{i x}}{-i \sqrt {2}-i}\right )}{2}-\frac {i x \ln \left (\frac {i \sqrt {2}-i-{\mathrm e}^{i x}}{i \sqrt {2}-i}\right )}{2}+\frac {i x \ln \left (\frac {-i \sqrt {2}+i-{\mathrm e}^{i x}}{-i \sqrt {2}+i}\right )}{2}\)	\(928\)

Input:

int(x*sin(x)^3/(cos(x)^2+1),x,method=_RETURNVERBOSE)
 

Output:

1/4*I*2^(1/2)*ln((I*2^(1/2)+I-exp(I*x))/(I*2^(1/2)+I))*x-1/4*I*2^(1/2)*ln( 
(-I*2^(1/2)-I-exp(I*x))/(-I*2^(1/2)-I))*x+1/4*I*2^(1/2)*ln((I*2^(1/2)-I-ex 
p(I*x))/(I*2^(1/2)-I))*x-1/4*I*2^(1/2)*ln((-I*2^(1/2)+I-exp(I*x))/(-I*2^(1 
/2)+I))*x+1/4*I/(I*2^(1/2)+I)*2^(1/2)*dilog((I*2^(1/2)+I-exp(I*x))/(I*2^(1 
/2)+I))+1/4*I/(-I*2^(1/2)-I)*2^(1/2)*dilog((-I*2^(1/2)-I-exp(I*x))/(-I*2^( 
1/2)-I))-1/4*I/(I*2^(1/2)-I)*2^(1/2)*dilog((I*2^(1/2)-I-exp(I*x))/(I*2^(1/ 
2)-I))-1/4*I/(-I*2^(1/2)+I)*2^(1/2)*dilog((-I*2^(1/2)+I-exp(I*x))/(-I*2^(1 
/2)+I))+1/4/(-I*2^(1/2)+I)*2^(1/2)*x*ln((-I*2^(1/2)+I-exp(I*x))/(-I*2^(1/2 
)+I))-1/4/(I*2^(1/2)+I)*2^(1/2)*x*ln((I*2^(1/2)+I-exp(I*x))/(I*2^(1/2)+I)) 
-1/4/(-I*2^(1/2)-I)*2^(1/2)*x*ln((-I*2^(1/2)-I-exp(I*x))/(-I*2^(1/2)-I))+1 
/4/(I*2^(1/2)-I)*2^(1/2)*x*ln((I*2^(1/2)-I-exp(I*x))/(I*2^(1/2)-I))+1/2*(x 
+I)*exp(I*x)+1/2*dilog((I*2^(1/2)+I-exp(I*x))/(I*2^(1/2)+I))-1/2*dilog((-I 
*2^(1/2)-I-exp(I*x))/(-I*2^(1/2)-I))-1/2*dilog((I*2^(1/2)-I-exp(I*x))/(I*2 
^(1/2)-I))+1/2*dilog((-I*2^(1/2)+I-exp(I*x))/(-I*2^(1/2)+I))-1/4*2^(1/2)*d 
ilog((-I*2^(1/2)+I-exp(I*x))/(-I*2^(1/2)+I))+1/4*2^(1/2)*dilog((I*2^(1/2)+ 
I-exp(I*x))/(I*2^(1/2)+I))+1/2*(x-I)*exp(-I*x)-1/4*2^(1/2)*dilog((-I*2^(1/ 
2)-I-exp(I*x))/(-I*2^(1/2)-I))+1/4*2^(1/2)*dilog((I*2^(1/2)-I-exp(I*x))/(I 
*2^(1/2)-I))+1/2*I*x*ln((I*2^(1/2)+I-exp(I*x))/(I*2^(1/2)+I))-1/2*I*x*ln(( 
-I*2^(1/2)-I-exp(I*x))/(-I*2^(1/2)-I))-1/2*I*x*ln((I*2^(1/2)-I-exp(I*x))/( 
I*2^(1/2)-I))+1/2*I*x*ln((-I*2^(1/2)+I-exp(I*x))/(-I*2^(1/2)+I))
 

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. 393 vs. \(2 (139) = 278\).

Time = 0.15 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.99 \[ \int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx =\text {Too large to display} \] Input:

integrate(x*sin(x)^3/(1+cos(x)^2),x, algorithm="fricas")
 

Output:

x*cos(x) - 1/2*I*x*log((I*sqrt(2) + I)*cos(x) + (sqrt(2) + 1)*sin(x) + 1) 
- 1/2*I*x*log((I*sqrt(2) + I)*cos(x) - (sqrt(2) + 1)*sin(x) + 1) + 1/2*I*x 
*log((I*sqrt(2) - I)*cos(x) + (sqrt(2) - 1)*sin(x) + 1) + 1/2*I*x*log((I*s 
qrt(2) - I)*cos(x) - (sqrt(2) - 1)*sin(x) + 1) - 1/2*I*x*log((-I*sqrt(2) + 
 I)*cos(x) + (sqrt(2) - 1)*sin(x) + 1) - 1/2*I*x*log((-I*sqrt(2) + I)*cos( 
x) - (sqrt(2) - 1)*sin(x) + 1) + 1/2*I*x*log((-I*sqrt(2) - I)*cos(x) + (sq 
rt(2) + 1)*sin(x) + 1) + 1/2*I*x*log((-I*sqrt(2) - I)*cos(x) - (sqrt(2) + 
1)*sin(x) + 1) - 1/2*dilog(-(I*sqrt(2) + I)*cos(x) + (sqrt(2) + 1)*sin(x)) 
 + 1/2*dilog(-(I*sqrt(2) + I)*cos(x) - (sqrt(2) + 1)*sin(x)) + 1/2*dilog(- 
(I*sqrt(2) - I)*cos(x) + (sqrt(2) - 1)*sin(x)) - 1/2*dilog(-(I*sqrt(2) - I 
)*cos(x) - (sqrt(2) - 1)*sin(x)) + 1/2*dilog(-(-I*sqrt(2) + I)*cos(x) + (s 
qrt(2) - 1)*sin(x)) - 1/2*dilog(-(-I*sqrt(2) + I)*cos(x) - (sqrt(2) - 1)*s 
in(x)) - 1/2*dilog(-(-I*sqrt(2) - I)*cos(x) + (sqrt(2) + 1)*sin(x)) + 1/2* 
dilog(-(-I*sqrt(2) - I)*cos(x) - (sqrt(2) + 1)*sin(x)) - sin(x)
 

Sympy [F]

\[ \int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx=\int \frac {x \sin ^{3}{\left (x \right )}}{\cos ^{2}{\left (x \right )} + 1}\, dx \] Input:

integrate(x*sin(x)**3/(1+cos(x)**2),x)
 

Output:

Integral(x*sin(x)**3/(cos(x)**2 + 1), x)
 

Maxima [F]

\[ \int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx=\int { \frac {x \sin \left (x\right )^{3}}{\cos \left (x\right )^{2} + 1} \,d x } \] Input:

integrate(x*sin(x)^3/(1+cos(x)^2),x, algorithm="maxima")
 

Output:

x*cos(x) - integrate(4*(6*x*cos(3*x)*sin(2*x) - 6*x*cos(x)*sin(2*x) + 6*x* 
cos(2*x)*sin(x) - (x*sin(3*x) - x*sin(x))*cos(4*x) + (x*cos(3*x) - x*cos(x 
))*sin(4*x) - (6*x*cos(2*x) + x)*sin(3*x) + x*sin(x))/(2*(6*cos(2*x) + 1)* 
cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin(4*x)*sin(2*x) 
+ 36*sin(2*x)^2 + 12*cos(2*x) + 1), x) - sin(x)
 

Giac [F]

\[ \int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx=\int { \frac {x \sin \left (x\right )^{3}}{\cos \left (x\right )^{2} + 1} \,d x } \] Input:

integrate(x*sin(x)^3/(1+cos(x)^2),x, algorithm="giac")
 

Output:

integrate(x*sin(x)^3/(cos(x)^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx=\int \frac {x\,{\sin \left (x\right )}^3}{{\cos \left (x\right )}^2+1} \,d x \] Input:

int((x*sin(x)^3)/(cos(x)^2 + 1),x)
 

Output:

int((x*sin(x)^3)/(cos(x)^2 + 1), x)
 

Reduce [F]

\[ \int \frac {x \sin ^3(x)}{1+\cos ^2(x)} \, dx=\int \frac {\sin \left (x \right )^{3} x}{\cos \left (x \right )^{2}+1}d x \] Input:

int(x*sin(x)^3/(1+cos(x)^2),x)
 

Output:

int((sin(x)**3*x)/(cos(x)**2 + 1),x)