\(\int x^5 (a+b \csc (c+d x^2)) \, dx\) [1]

Optimal result

Integrand size = 16, antiderivative size = 129 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}+\frac {i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (c+d x^2\right )}\right )}{d^2}-\frac {b \operatorname {PolyLog}\left (3,-e^{i \left (c+d x^2\right )}\right )}{d^3}+\frac {b \operatorname {PolyLog}\left (3,e^{i \left (c+d x^2\right )}\right )}{d^3} \] Output:

1/6*a*x^6-b*x^4*arctanh(exp(I*(d*x^2+c)))/d+I*b*x^2*polylog(2,-exp(I*(d*x^ 
2+c)))/d^2-I*b*x^2*polylog(2,exp(I*(d*x^2+c)))/d^2-b*polylog(3,-exp(I*(d*x 
^2+c)))/d^3+b*polylog(3,exp(I*(d*x^2+c)))/d^3
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.23 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}-\frac {b \left (d^2 x^4 \text {arctanh}\left (\cos \left (c+d x^2\right )+i \sin \left (c+d x^2\right )\right )-i d x^2 \operatorname {PolyLog}\left (2,-\cos \left (c+d x^2\right )-i \sin \left (c+d x^2\right )\right )+i d x^2 \operatorname {PolyLog}\left (2,\cos \left (c+d x^2\right )+i \sin \left (c+d x^2\right )\right )+\operatorname {PolyLog}\left (3,-\cos \left (c+d x^2\right )-i \sin \left (c+d x^2\right )\right )-\operatorname {PolyLog}\left (3,\cos \left (c+d x^2\right )+i \sin \left (c+d x^2\right )\right )\right )}{d^3} \] Input:

Integrate[x^5*(a + b*Csc[c + d*x^2]),x]
 

Output:

(a*x^6)/6 - (b*(d^2*x^4*ArcTanh[Cos[c + d*x^2] + I*Sin[c + d*x^2]] - I*d*x 
^2*PolyLog[2, -Cos[c + d*x^2] - I*Sin[c + d*x^2]] + I*d*x^2*PolyLog[2, Cos 
[c + d*x^2] + I*Sin[c + d*x^2]] + PolyLog[3, -Cos[c + d*x^2] - I*Sin[c + d 
*x^2]] - PolyLog[3, Cos[c + d*x^2] + I*Sin[c + d*x^2]]))/d^3
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2010, 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.

\(\displaystyle \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \int \left (a x^5+b x^5 \csc \left (c+d x^2\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{i \left (c+d x^2\right )}\right )}{d}-\frac {b \operatorname {PolyLog}\left (3,-e^{i \left (d x^2+c\right )}\right )}{d^3}+\frac {b \operatorname {PolyLog}\left (3,e^{i \left (d x^2+c\right )}\right )}{d^3}+\frac {i b x^2 \operatorname {PolyLog}\left (2,-e^{i \left (d x^2+c\right )}\right )}{d^2}-\frac {i b x^2 \operatorname {PolyLog}\left (2,e^{i \left (d x^2+c\right )}\right )}{d^2}\)

Input:

Int[x^5*(a + b*Csc[c + d*x^2]),x]
 

Output:

(a*x^6)/6 - (b*x^4*ArcTanh[E^(I*(c + d*x^2))])/d + (I*b*x^2*PolyLog[2, -E^ 
(I*(c + d*x^2))])/d^2 - (I*b*x^2*PolyLog[2, E^(I*(c + d*x^2))])/d^2 - (b*P 
olyLog[3, -E^(I*(c + d*x^2))])/d^3 + (b*PolyLog[3, E^(I*(c + d*x^2))])/d^3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

Maple [F]

\[\int x^{5} \left (a +b \csc \left (d \,x^{2}+c \right )\right )d x\]

Input:

int(x^5*(a+b*csc(d*x^2+c)),x)
 

Output:

int(x^5*(a+b*csc(d*x^2+c)),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. 425 vs. \(2 (111) = 222\).

Time = 0.10 (sec) , antiderivative size = 425, normalized size of antiderivative = 3.29 \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\frac {2 \, a d^{3} x^{6} - 3 \, b d^{2} x^{4} \log \left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) - 3 \, b d^{2} x^{4} \log \left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) - 6 i \, b d x^{2} {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + 6 i \, b d x^{2} {\rm Li}_2\left (\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) - 6 i \, b d x^{2} {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + 6 i \, b d x^{2} {\rm Li}_2\left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) + 3 \, b c^{2} \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) + 3 \, b c^{2} \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) - \frac {1}{2} i \, \sin \left (d x^{2} + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (b d^{2} x^{4} - b c^{2}\right )} \log \left (-\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right ) + 1\right ) + 3 \, {\left (b d^{2} x^{4} - b c^{2}\right )} \log \left (-\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right ) + 1\right ) + 6 \, b {\rm polylog}\left (3, \cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) + 6 \, b {\rm polylog}\left (3, \cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right ) - 6 \, b {\rm polylog}\left (3, -\cos \left (d x^{2} + c\right ) + i \, \sin \left (d x^{2} + c\right )\right ) - 6 \, b {\rm polylog}\left (3, -\cos \left (d x^{2} + c\right ) - i \, \sin \left (d x^{2} + c\right )\right )}{12 \, d^{3}} \] Input:

integrate(x^5*(a+b*csc(d*x^2+c)),x, algorithm="fricas")
 

Output:

1/12*(2*a*d^3*x^6 - 3*b*d^2*x^4*log(cos(d*x^2 + c) + I*sin(d*x^2 + c) + 1) 
 - 3*b*d^2*x^4*log(cos(d*x^2 + c) - I*sin(d*x^2 + c) + 1) - 6*I*b*d*x^2*di 
log(cos(d*x^2 + c) + I*sin(d*x^2 + c)) + 6*I*b*d*x^2*dilog(cos(d*x^2 + c) 
- I*sin(d*x^2 + c)) - 6*I*b*d*x^2*dilog(-cos(d*x^2 + c) + I*sin(d*x^2 + c) 
) + 6*I*b*d*x^2*dilog(-cos(d*x^2 + c) - I*sin(d*x^2 + c)) + 3*b*c^2*log(-1 
/2*cos(d*x^2 + c) + 1/2*I*sin(d*x^2 + c) + 1/2) + 3*b*c^2*log(-1/2*cos(d*x 
^2 + c) - 1/2*I*sin(d*x^2 + c) + 1/2) + 3*(b*d^2*x^4 - b*c^2)*log(-cos(d*x 
^2 + c) + I*sin(d*x^2 + c) + 1) + 3*(b*d^2*x^4 - b*c^2)*log(-cos(d*x^2 + c 
) - I*sin(d*x^2 + c) + 1) + 6*b*polylog(3, cos(d*x^2 + c) + I*sin(d*x^2 + 
c)) + 6*b*polylog(3, cos(d*x^2 + c) - I*sin(d*x^2 + c)) - 6*b*polylog(3, - 
cos(d*x^2 + c) + I*sin(d*x^2 + c)) - 6*b*polylog(3, -cos(d*x^2 + c) - I*si 
n(d*x^2 + c)))/d^3
 

Sympy [F]

\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^{5} \left (a + b \csc {\left (c + d x^{2} \right )}\right )\, dx \] Input:

integrate(x**5*(a+b*csc(d*x**2+c)),x)
 

Output:

Integral(x**5*(a + b*csc(c + d*x**2)), x)
 

Maxima [F]

\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \] Input:

integrate(x^5*(a+b*csc(d*x^2+c)),x, algorithm="maxima")
 

Output:

1/6*a*x^6 + b*(integrate(x^5*sin(d*x^2 + c)/(cos(d*x^2 + c)^2 + sin(d*x^2 
+ c)^2 + 2*cos(d*x^2 + c) + 1), x) + integrate(x^5*sin(d*x^2 + c)/(cos(d*x 
^2 + c)^2 + sin(d*x^2 + c)^2 - 2*cos(d*x^2 + c) + 1), x))
 

Giac [F]

\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int { {\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \] Input:

integrate(x^5*(a+b*csc(d*x^2+c)),x, algorithm="giac")
 

Output:

integrate((b*csc(d*x^2 + c) + a)*x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\int x^5\,\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right ) \,d x \] Input:

int(x^5*(a + b/sin(c + d*x^2)),x)
 

Output:

int(x^5*(a + b/sin(c + d*x^2)), x)
 

Reduce [F]

\[ \int x^5 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx=\left (\int \csc \left (d \,x^{2}+c \right ) x^{5}d x \right ) b +\frac {a \,x^{6}}{6} \] Input:

int(x^5*(a+b*csc(d*x^2+c)),x)
 

Output:

(6*int(csc(c + d*x**2)*x**5,x)*b + a*x**6)/6