Integrand size = 18, antiderivative size = 922 \[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=-\frac {b^2 x^4}{2 a^2 \left (a^2+b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {b^3 x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {2 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b^3 x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b x^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {2 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 b \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {b^2 x^4 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )} \] Output:
-1/2*b^2*x^4/a^2/(a^2+b^2)/d+1/6*x^6/a^2+b^2*x^2*ln(1+a*exp(d*x^2+c)/(b-(a ^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2+1/2*b^3*x^4*ln(1+a*exp(d*x^2+c)/(b-(a^2+ b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d-b*x^4*ln(1+a*exp(d*x^2+c)/(b-(a^2+b^2)^ (1/2)))/a^2/(a^2+b^2)^(1/2)/d+b^2*x^2*ln(1+a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/ 2)))/a^2/(a^2+b^2)/d^2-1/2*b^3*x^4*ln(1+a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)) )/a^2/(a^2+b^2)^(3/2)/d+b*x^4*ln(1+a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/a^2 /(a^2+b^2)^(1/2)/d+b^2*polylog(2,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a^2/ (a^2+b^2)/d^3+b^3*x^2*polylog(2,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/a^2/( a^2+b^2)^(3/2)/d^2-2*b*x^2*polylog(2,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1/2)))/ a^2/(a^2+b^2)^(1/2)/d^2+b^2*polylog(2,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2))) /a^2/(a^2+b^2)/d^3-b^3*x^2*polylog(2,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/2)))/ a^2/(a^2+b^2)^(3/2)/d^2+2*b*x^2*polylog(2,-a*exp(d*x^2+c)/(b+(a^2+b^2)^(1/ 2)))/a^2/(a^2+b^2)^(1/2)/d^2-b^3*polylog(3,-a*exp(d*x^2+c)/(b-(a^2+b^2)^(1 /2)))/a^2/(a^2+b^2)^(3/2)/d^3+2*b*polylog(3,-a*exp(d*x^2+c)/(b-(a^2+b^2)^( 1/2)))/a^2/(a^2+b^2)^(1/2)/d^3+b^3*polylog(3,-a*exp(d*x^2+c)/(b+(a^2+b^2)^ (1/2)))/a^2/(a^2+b^2)^(3/2)/d^3-2*b*polylog(3,-a*exp(d*x^2+c)/(b+(a^2+b^2) ^(1/2)))/a^2/(a^2+b^2)^(1/2)/d^3-1/2*b^2*x^4*cosh(d*x^2+c)/a/(a^2+b^2)/d/( b+a*sinh(d*x^2+c))
Time = 3.75 (sec) , antiderivative size = 1502, normalized size of antiderivative = 1.63 \[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[x^5/(a + b*Csch[c + d*x^2])^2,x]
Output:
(Csch[c + d*x^2]^2*(b + a*Sinh[c + d*x^2])*((6*b^2*x^4*Csch[c]*(b*Cosh[c] + a*Sinh[d*x^2]))/((a^2 + b^2)*d) + 2*x^6*(b + a*Sinh[c + d*x^2]) - (6*b*E ^(2*c)*(2*b*d^2*E^(2*c)*Sqrt[(a^2 + b^2)*E^(2*c)]*x^4 + 2*b*d*Sqrt[(a^2 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^ (2*c)])] - 2*b*d*E^(2*c)*Sqrt[(a^2 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*a^2*d^2*E^c*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - b^2*d^2*E^c*x^4 *Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*a^2* d^2*E^(3*c)*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2 *c)])] + b^2*d^2*E^(3*c)*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^ 2 + b^2)*E^(2*c)])] + 2*b*d*Sqrt[(a^2 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2* c + d*x^2))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*b*d*E^(2*c)*Sqrt[(a^2 + b^2)*E^(2*c)]*x^2*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c + Sqrt[(a^2 + b^2) *E^(2*c)])] + 2*a^2*d^2*E^c*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c + Sqrt[ (a^2 + b^2)*E^(2*c)])] + b^2*d^2*E^c*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^ c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 2*a^2*d^2*E^(3*c)*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - b^2*d^2*E^(3*c)*x^4*Log[1 + (a*E^(2*c + d*x^2))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 2*(-1 + E^(2 *c))*(-(b*Sqrt[(a^2 + b^2)*E^(2*c)]) + 2*a^2*d*E^c*x^2 + b^2*d*E^c*x^2)*Po lyLog[2, -((a*E^(2*c + d*x^2))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - ...
Time = 2.38 (sec) , antiderivative size = 924, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5960, 3042, 4679, 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.
\(\displaystyle \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 5960 |
\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+b \text {csch}\left (d x^2+c\right )\right )^2}dx^2\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+i b \csc \left (i d x^2+i c\right )\right )^2}dx^2\) |
\(\Big \downarrow \) 4679 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {2 b x^4}{a^2 \left (b+a \sinh \left (d x^2+c\right )\right )}+\frac {x^4}{a^2}+\frac {b^2 x^4}{a^2 \left (b+a \sinh \left (d x^2+c\right )\right )^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {x^6}{3 a^2}-\frac {2 b \log \left (\frac {e^{d x^2+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^4}{a^2 \sqrt {a^2+b^2} d}+\frac {b^3 \log \left (\frac {e^{d x^2+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^4}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {2 b \log \left (\frac {e^{d x^2+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^4}{a^2 \sqrt {a^2+b^2} d}-\frac {b^3 \log \left (\frac {e^{d x^2+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^4}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^2 x^4}{a^2 \left (a^2+b^2\right ) d}-\frac {b^2 \cosh \left (d x^2+c\right ) x^4}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (d x^2+c\right )\right )}+\frac {2 b^2 \log \left (\frac {e^{d x^2+c} a}{b-\sqrt {a^2+b^2}}+1\right ) x^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^2 \log \left (\frac {e^{d x^2+c} a}{b+\sqrt {a^2+b^2}}+1\right ) x^2}{a^2 \left (a^2+b^2\right ) d^2}-\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right ) x^2}{a^2 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right ) x^2}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right ) x^2}{a^2 \sqrt {a^2+b^2} d^2}-\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right ) x^2}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {4 b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {2 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {4 b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {2 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}\right )\) |
Input:
Int[x^5/(a + b*Csch[c + d*x^2])^2,x]
Output:
(-((b^2*x^4)/(a^2*(a^2 + b^2)*d)) + x^6/(3*a^2) + (2*b^2*x^2*Log[1 + (a*E^ (c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2) + (b^3*x^4*Log[ 1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (2*b*x^4*Log[1 + (a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (2*b^2*x^2*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(a ^2*(a^2 + b^2)*d^2) - (b^3*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b ^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (2*b*x^4*Log[1 + (a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (2*b^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (2*b^3*x^2*Poly Log[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2) *d^2) - (4*b*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/( a^2*Sqrt[a^2 + b^2]*d^2) + (2*b^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt [a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (2*b^3*x^2*PolyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (4*b*x^2*Po lyLog[2, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2] *d^2) - (2*b^3*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[a^2 + b^2]))])/(a^ 2*(a^2 + b^2)^(3/2)*d^3) + (4*b*PolyLog[3, -((a*E^(c + d*x^2))/(b - Sqrt[a ^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (2*b^3*PolyLog[3, -((a*E^(c + d* x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (4*b*PolyLog[ 3, -((a*E^(c + d*x^2))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^...
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt Q[m, 0]
Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x] )^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
\[\int \frac {x^{5}}{{\left (a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )\right )}^{2}}d x\]
Input:
int(x^5/(a+b*csch(d*x^2+c))^2,x)
Output:
int(x^5/(a+b*csch(d*x^2+c))^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 3756 vs. \(2 (834) = 1668\).
Time = 0.14 (sec) , antiderivative size = 3756, normalized size of antiderivative = 4.07 \[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(x^5/(a+b*csch(d*x^2+c))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{5}}{\left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \] Input:
integrate(x**5/(a+b*csch(d*x**2+c))**2,x)
Output:
Integral(x**5/(a + b*csch(c + d*x**2))**2, x)
\[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(x^5/(a+b*csch(d*x^2+c))^2,x, algorithm="maxima")
Output:
-1/6*((a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x^6*e^(2*d*x^2) - 6*a*b^2*x^4 - (a ^3*d + a*b^2*d)*x^6 + 2*(3*b^3*x^4*e^c + (a^2*b*d*e^c + b^3*d*e^c)*x^6)*e^ (d*x^2))/(a^5*d + a^3*b^2*d - (a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*e^(2*d*x ^2) - 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*e^(d*x^2)) - integrate(2*(2*a*b^2*x^ 2 - (2*b^3*x^2*e^c + (2*a^2*b*d*e^c + b^3*d*e^c)*x^4)*e^(d*x^2))*x/(a^5*d + a^3*b^2*d - (a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*e^(2*d*x^2) - 2*(a^4*b*d *e^c + a^2*b^3*d*e^c)*e^(d*x^2)), x)
\[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate(x^5/(a+b*csch(d*x^2+c))^2,x, algorithm="giac")
Output:
integrate(x^5/(b*csch(d*x^2 + c) + a)^2, x)
Timed out. \[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^5}{{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2} \,d x \] Input:
int(x^5/(a + b/sinh(c + d*x^2))^2,x)
Output:
int(x^5/(a + b/sinh(c + d*x^2))^2, x)
\[ \int \frac {x^5}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{5}}{\mathrm {csch}\left (d \,x^{2}+c \right )^{2} b^{2}+2 \,\mathrm {csch}\left (d \,x^{2}+c \right ) a b +a^{2}}d x \] Input:
int(x^5/(a+b*csch(d*x^2+c))^2,x)
Output:
int(x**5/(csch(c + d*x**2)**2*b**2 + 2*csch(c + d*x**2)*a*b + a**2),x)