Integrand size = 24, antiderivative size = 428 \[ \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx=\frac {(e x)^{3 n}}{3 a e n}-\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d e n}+\frac {b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d e n}-\frac {2 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3 e n} \]
1/3*(e*x)^(3*n)/a/e/n-b*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(a^2+b^2)^(1/2) ))/a/d/e/n/(x^n)/(a^2+b^2)^(1/2)+b*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b+(a^2 +b^2)^(1/2)))/a/d/e/n/(x^n)/(a^2+b^2)^(1/2)-2*b*(e*x)^(3*n)*polylog(2,-a*e xp(c+d*x^n)/(b-(a^2+b^2)^(1/2)))/a/d^2/e/n/(x^(2*n))/(a^2+b^2)^(1/2)+2*b*( e*x)^(3*n)*polylog(2,-a*exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a/d^2/e/n/(x^(2* n))/(a^2+b^2)^(1/2)+2*b*(e*x)^(3*n)*polylog(3,-a*exp(c+d*x^n)/(b-(a^2+b^2) ^(1/2)))/a/d^3/e/n/(x^(3*n))/(a^2+b^2)^(1/2)-2*b*(e*x)^(3*n)*polylog(3,-a* exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a/d^3/e/n/(x^(3*n))/(a^2+b^2)^(1/2)
\[ \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx=\int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx \]
Time = 1.14 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5964, 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.
| \(\displaystyle \int \frac {(e x)^{3 n-1}}{a+b \text {csch}\left (c+d x^n\right )} \, dx\) |
| \(\Big \downarrow \) 5964 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{3 n-1}}{a+b \text {csch}\left (d x^n+c\right )}dx}{e}\) |
| \(\Big \downarrow \) 5960 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{2 n}}{a+b \text {csch}\left (d x^n+c\right )}dx^n}{e n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{2 n}}{a+i b \csc \left (i d x^n+i c\right )}dx^n}{e n}\) |
| \(\Big \downarrow \) 4679 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \left (\frac {x^{2 n}}{a}-\frac {b x^{2 n}}{a \left (b+a \sinh \left (d x^n+c\right )\right )}\right )dx^n}{e n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (\frac {2 b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a d^3 \sqrt {a^2+b^2}}-\frac {2 b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a d^3 \sqrt {a^2+b^2}}-\frac {2 b x^n \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a d^2 \sqrt {a^2+b^2}}+\frac {2 b x^n \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a d^2 \sqrt {a^2+b^2}}-\frac {b x^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a d \sqrt {a^2+b^2}}+\frac {b x^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a d \sqrt {a^2+b^2}}+\frac {x^{3 n}}{3 a}\right )}{e n}\) |
((e*x)^(3*n)*(x^(3*n)/(3*a) - (b*x^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sq rt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (b*x^(2*n)*Log[1 + (a*E^(c + d*x^ n))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (2*b*x^n*PolyLog[2, -( (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (2*b* x^n*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (2*b*PolyLog[3, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/( a*Sqrt[a^2 + b^2]*d^3) - (2*b*PolyLog[3, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3)))/(e*n*x^(3*n))
3.1.80.3.1 Defintions of rubi rules used
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[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x_Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m* (a + b*Csch[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
\[\int \frac {\left (e x \right )^{-1+3 n}}{a +b \,\operatorname {csch}\left (c +d \,x^{n}\right )}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1850 vs. \(2 (402) = 804\).
Time = 0.31 (sec) , antiderivative size = 1850, normalized size of antiderivative = 4.32 \[ \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx=\text {Too large to display} \]
1/3*((a^2 + b^2)*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^3 + (a^2 + b^2) *d^3*cosh(n*log(x))^3*sinh((3*n - 1)*log(e)) + ((a^2 + b^2)*d^3*cosh((3*n - 1)*log(e)) + (a^2 + b^2)*d^3*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^3 + 3*((a^2 + b^2)*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a^2 + b^2)*d^3 *cosh(n*log(x))*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 - 6*(a*b*d*sqrt(( a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + a*b*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x))*sinh((3*n - 1)*log(e)) + (a*b*d*sqrt((a^2 + b^2)/ a^2)*cosh((3*n - 1)*log(e)) + a*b*d*sqrt((a^2 + b^2)/a^2)*sinh((3*n - 1)*l og(e)))*sinh(n*log(x)))*dilog(((a*sqrt((a^2 + b^2)/a^2) + b)*cosh(d*cosh(n *log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt((a^2 + b^2)/a^2) + b)*sinh(d*co sh(n*log(x)) + d*sinh(n*log(x)) + c) - a)/a + 1) + 6*(a*b*d*sqrt((a^2 + b^ 2)/a^2)*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + a*b*d*sqrt((a^2 + b^2)/a^2 )*cosh(n*log(x))*sinh((3*n - 1)*log(e)) + (a*b*d*sqrt((a^2 + b^2)/a^2)*cos h((3*n - 1)*log(e)) + a*b*d*sqrt((a^2 + b^2)/a^2)*sinh((3*n - 1)*log(e)))* sinh(n*log(x)))*dilog(-((a*sqrt((a^2 + b^2)/a^2) - b)*cosh(d*cosh(n*log(x) ) + d*sinh(n*log(x)) + c) + (a*sqrt((a^2 + b^2)/a^2) - b)*sinh(d*cosh(n*lo g(x)) + d*sinh(n*log(x)) + c) + a)/a + 1) + 3*(a*b*c^2*sqrt((a^2 + b^2)/a^ 2)*cosh((3*n - 1)*log(e)) + a*b*c^2*sqrt((a^2 + b^2)/a^2)*sinh((3*n - 1)*l og(e)))*log(2*a*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*a*sinh(d *cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2...
\[ \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx=\int \frac {\left (e x\right )^{3 n - 1}}{a + b \operatorname {csch}{\left (c + d x^{n} \right )}}\, dx \]
\[ \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{b \operatorname {csch}\left (d x^{n} + c\right ) + a} \,d x } \]
-2*b*e^(3*n)*integrate(e^(d*x^n + 3*n*log(x) + c)/(a^2*e*x*e^(2*d*x^n + 2* c) + 2*a*b*e*x*e^(d*x^n + c) - a^2*e*x), x) + 1/3*e^(3*n - 1)*x^(3*n)/(a*n )
\[ \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{b \operatorname {csch}\left (d x^{n} + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx=\int \frac {{\left (e\,x\right )}^{3\,n-1}}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x^n\right )}} \,d x \]