Integrand size = 16, antiderivative size = 125 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}+\frac {b x^4 \arctan \left (e^{c+d x^2}\right )}{d}-\frac {i b x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )}{d^2}+\frac {i b x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )}{d^2}+\frac {i b \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )}{d^3}-\frac {i b \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )}{d^3} \] Output:
1/6*a*x^6+b*x^4*arctan(exp(d*x^2+c))/d-I*b*x^2*polylog(2,-I*exp(d*x^2+c))/ d^2+I*b*x^2*polylog(2,I*exp(d*x^2+c))/d^2+I*b*polylog(3,-I*exp(d*x^2+c))/d ^3-I*b*polylog(3,I*exp(d*x^2+c))/d^3
Time = 0.18 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.14 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\frac {a x^6}{6}+\frac {i b \left (d^2 x^4 \log \left (1-i e^{c+d x^2}\right )-d^2 x^4 \log \left (1+i e^{c+d x^2}\right )-2 d x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x^2}\right )+2 d x^2 \operatorname {PolyLog}\left (2,i e^{c+d x^2}\right )+2 \operatorname {PolyLog}\left (3,-i e^{c+d x^2}\right )-2 \operatorname {PolyLog}\left (3,i e^{c+d x^2}\right )\right )}{2 d^3} \] Input:
Integrate[x^5*(a + b*Sech[c + d*x^2]),x]
Output:
(a*x^6)/6 + ((I/2)*b*(d^2*x^4*Log[1 - I*E^(c + d*x^2)] - d^2*x^4*Log[1 + I *E^(c + d*x^2)] - 2*d*x^2*PolyLog[2, (-I)*E^(c + d*x^2)] + 2*d*x^2*PolyLog [2, I*E^(c + d*x^2)] + 2*PolyLog[3, (-I)*E^(c + d*x^2)] - 2*PolyLog[3, I*E ^(c + d*x^2)]))/d^3
Time = 0.40 (sec) , antiderivative size = 125, 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.
\(\displaystyle \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a x^5+b x^5 \text {sech}\left (c+d x^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a x^6}{6}+\frac {b x^4 \arctan \left (e^{c+d x^2}\right )}{d}+\frac {i b \operatorname {PolyLog}\left (3,-i e^{d x^2+c}\right )}{d^3}-\frac {i b \operatorname {PolyLog}\left (3,i e^{d x^2+c}\right )}{d^3}-\frac {i b x^2 \operatorname {PolyLog}\left (2,-i e^{d x^2+c}\right )}{d^2}+\frac {i b x^2 \operatorname {PolyLog}\left (2,i e^{d x^2+c}\right )}{d^2}\) |
Input:
Int[x^5*(a + b*Sech[c + d*x^2]),x]
Output:
(a*x^6)/6 + (b*x^4*ArcTan[E^(c + d*x^2)])/d - (I*b*x^2*PolyLog[2, (-I)*E^( c + d*x^2)])/d^2 + (I*b*x^2*PolyLog[2, I*E^(c + d*x^2)])/d^2 + (I*b*PolyLo g[3, (-I)*E^(c + d*x^2)])/d^3 - (I*b*PolyLog[3, I*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]]
\[\int x^{5} \left (a +b \,\operatorname {sech}\left (d \,x^{2}+c \right )\right )d x\]
Input:
int(x^5*(a+b*sech(d*x^2+c)),x)
Output:
int(x^5*(a+b*sech(d*x^2+c)),x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (100) = 200\).
Time = 0.11 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.05 \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\frac {a d^{3} x^{6} + 6 i \, b d x^{2} {\rm Li}_2\left (i \, \cosh \left (d x^{2} + c\right ) + i \, \sinh \left (d x^{2} + c\right )\right ) - 6 i \, b d x^{2} {\rm Li}_2\left (-i \, \cosh \left (d x^{2} + c\right ) - i \, \sinh \left (d x^{2} + c\right )\right ) + 3 i \, b c^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + i\right ) - 3 i \, b c^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - i\right ) - 3 \, {\left (i \, b d^{2} x^{4} - i \, b c^{2}\right )} \log \left (i \, \cosh \left (d x^{2} + c\right ) + i \, \sinh \left (d x^{2} + c\right ) + 1\right ) - 3 \, {\left (-i \, b d^{2} x^{4} + i \, b c^{2}\right )} \log \left (-i \, \cosh \left (d x^{2} + c\right ) - i \, \sinh \left (d x^{2} + c\right ) + 1\right ) - 6 i \, b {\rm polylog}\left (3, i \, \cosh \left (d x^{2} + c\right ) + i \, \sinh \left (d x^{2} + c\right )\right ) + 6 i \, b {\rm polylog}\left (3, -i \, \cosh \left (d x^{2} + c\right ) - i \, \sinh \left (d x^{2} + c\right )\right )}{6 \, d^{3}} \] Input:
integrate(x^5*(a+b*sech(d*x^2+c)),x, algorithm="fricas")
Output:
1/6*(a*d^3*x^6 + 6*I*b*d*x^2*dilog(I*cosh(d*x^2 + c) + I*sinh(d*x^2 + c)) - 6*I*b*d*x^2*dilog(-I*cosh(d*x^2 + c) - I*sinh(d*x^2 + c)) + 3*I*b*c^2*lo g(cosh(d*x^2 + c) + sinh(d*x^2 + c) + I) - 3*I*b*c^2*log(cosh(d*x^2 + c) + sinh(d*x^2 + c) - I) - 3*(I*b*d^2*x^4 - I*b*c^2)*log(I*cosh(d*x^2 + c) + I*sinh(d*x^2 + c) + 1) - 3*(-I*b*d^2*x^4 + I*b*c^2)*log(-I*cosh(d*x^2 + c) - I*sinh(d*x^2 + c) + 1) - 6*I*b*polylog(3, I*cosh(d*x^2 + c) + I*sinh(d* x^2 + c)) + 6*I*b*polylog(3, -I*cosh(d*x^2 + c) - I*sinh(d*x^2 + c)))/d^3
\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\int x^{5} \left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )\, dx \] Input:
integrate(x**5*(a+b*sech(d*x**2+c)),x)
Output:
Integral(x**5*(a + b*sech(c + d*x**2)), x)
\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \] Input:
integrate(x^5*(a+b*sech(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)
\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d x^{2} + c\right ) + a\right )} x^{5} \,d x } \] Input:
integrate(x^5*(a+b*sech(d*x^2+c)),x, algorithm="giac")
Output:
integrate((b*sech(d*x^2 + c) + a)*x^5, x)
Timed out. \[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\int x^5\,\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right ) \,d x \] Input:
int(x^5*(a + b/cosh(c + d*x^2)),x)
Output:
int(x^5*(a + b/cosh(c + d*x^2)), x)
\[ \int x^5 \left (a+b \text {sech}\left (c+d x^2\right )\right ) \, dx=\left (\int \mathrm {sech}\left (d \,x^{2}+c \right ) x^{5}d x \right ) b +\frac {a \,x^{6}}{6} \] Input:
int(x^5*(a+b*sech(d*x^2+c)),x)
Output:
(6*int(sech(c + d*x**2)*x**5,x)*b + a*x**6)/6