Integrand size = 16, antiderivative size = 104 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b \operatorname {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {b \operatorname {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3} \] Output:
1/6*a*x^6-b*x^4*arctanh(exp(d*x^2+c))/d-b*x^2*polylog(2,-exp(d*x^2+c))/d^2 +b*x^2*polylog(2,exp(d*x^2+c))/d^2+b*polylog(3,-exp(d*x^2+c))/d^3-b*polylo g(3,exp(d*x^2+c))/d^3
Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.28 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}+\frac {b x^4 \log \left (1-e^{c+d x^2}\right )}{2 d}-\frac {b x^4 \log \left (1+e^{c+d x^2}\right )}{2 d}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b \operatorname {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {b \operatorname {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3} \] Input:
Integrate[x^5*(a + b*Csch[c + d*x^2]),x]
Output:
(a*x^6)/6 + (b*x^4*Log[1 - E^(c + d*x^2)])/(2*d) - (b*x^4*Log[1 + E^(c + d *x^2)])/(2*d) - (b*x^2*PolyLog[2, -E^(c + d*x^2)])/d^2 + (b*x^2*PolyLog[2, E^(c + d*x^2)])/d^2 + (b*PolyLog[3, -E^(c + d*x^2)])/d^3 - (b*PolyLog[3, E^(c + d*x^2)])/d^3
Time = 0.34 (sec) , antiderivative size = 104, 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.
\(\displaystyle \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a x^5+b x^5 \text {csch}\left (c+d x^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a x^6}{6}-\frac {b x^4 \text {arctanh}\left (e^{c+d x^2}\right )}{d}+\frac {b \operatorname {PolyLog}\left (3,-e^{d x^2+c}\right )}{d^3}-\frac {b \operatorname {PolyLog}\left (3,e^{d x^2+c}\right )}{d^3}-\frac {b x^2 \operatorname {PolyLog}\left (2,-e^{d x^2+c}\right )}{d^2}+\frac {b x^2 \operatorname {PolyLog}\left (2,e^{d x^2+c}\right )}{d^2}\) |
Input:
Int[x^5*(a + b*Csch[c + d*x^2]),x]
Output:
(a*x^6)/6 - (b*x^4*ArcTanh[E^(c + d*x^2)])/d - (b*x^2*PolyLog[2, -E^(c + d *x^2)])/d^2 + (b*x^2*PolyLog[2, E^(c + d*x^2)])/d^2 + (b*PolyLog[3, -E^(c + d*x^2)])/d^3 - (b*PolyLog[3, E^(c + d*x^2)])/d^3
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]]
\[\int x^{5} \left (a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )\right )d x\]
Input:
int(x^5*(a+b*csch(d*x^2+c)),x)
Output:
int(x^5*(a+b*csch(d*x^2+c)),x)
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (95) = 190\).
Time = 0.10 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.01 \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\frac {a d^{3} x^{6} - 3 \, b d^{2} x^{4} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) + 6 \, b d x^{2} {\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) - 6 \, b d x^{2} {\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) + 3 \, b c^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 3 \, {\left (b d^{2} x^{4} - b c^{2}\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right ) - 6 \, b {\rm polylog}\left (3, \cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) + 6 \, b {\rm polylog}\left (3, -\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right )}{6 \, d^{3}} \] Input:
integrate(x^5*(a+b*csch(d*x^2+c)),x, algorithm="fricas")
Output:
1/6*(a*d^3*x^6 - 3*b*d^2*x^4*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) + 1) + 6*b*d*x^2*dilog(cosh(d*x^2 + c) + sinh(d*x^2 + c)) - 6*b*d*x^2*dilog(-cosh (d*x^2 + c) - sinh(d*x^2 + c)) + 3*b*c^2*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) - 1) + 3*(b*d^2*x^4 - b*c^2)*log(-cosh(d*x^2 + c) - sinh(d*x^2 + c) + 1) - 6*b*polylog(3, cosh(d*x^2 + c) + sinh(d*x^2 + c)) + 6*b*polylog(3, - cosh(d*x^2 + c) - sinh(d*x^2 + c)))/d^3
\[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\int x^{5} \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )\, dx \] Input:
integrate(x**5*(a+b*csch(d*x**2+c)),x)
Output:
Integral(x**5*(a + b*csch(c + d*x**2)), x)
\[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\int { {\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \] Input:
integrate(x^5*(a+b*csch(d*x^2+c)),x, algorithm="maxima")
Output:
1/6*a*x^6 + 2*b*integrate(x^5/(e^(d*x^2 + c) - e^(-d*x^2 - c)), x)
\[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\int { {\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \] Input:
integrate(x^5*(a+b*csch(d*x^2+c)),x, algorithm="giac")
Output:
integrate((b*csch(d*x^2 + c) + a)*x^5, x)
Timed out. \[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\int x^5\,\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right ) \,d x \] Input:
int(x^5*(a + b/sinh(c + d*x^2)),x)
Output:
int(x^5*(a + b/sinh(c + d*x^2)), x)
\[ \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx=\left (\int \mathrm {csch}\left (d \,x^{2}+c \right ) x^{5}d x \right ) b +\frac {a \,x^{6}}{6} \] Input:
int(x^5*(a+b*csch(d*x^2+c)),x)
Output:
(6*int(csch(c + d*x**2)*x**5,x)*b + a*x**6)/6