Integrand size = 24, antiderivative size = 1284 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{3 n}}{3 a^2 e n}+\frac {b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}-\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 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^2 \sqrt {-a^2+b^2} d e n}-\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 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^2 \sqrt {-a^2+b^2} d e n}-\frac {2 b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}+\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 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^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}+\frac {4 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^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {2 b^3 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^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 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^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )} \]
1/3*(e*x)^(3*n)/a^2/e/n+b^2*(e*x)^(3*n)/a^2/(a^2-b^2)/d/e/n/(x^n)-2*b^2*(e *x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n/ (x^(2*n))+b^3*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/(- a^2+b^2)^(3/2)/d/e/n/(x^n)-2*b^2*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b+(-a^2+ b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n/(x^(2*n))-b^3*(e*x)^(3*n)*ln(1+a*exp(c+ d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)-2*b^2*(e*x)^ (3*n)*polylog(2,-a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/ n/(x^(3*n))+2*b^3*(e*x)^(3*n)*polylog(2,-a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2 )))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))-2*b^2*(e*x)^(3*n)*polylog(2,-a* exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/n/(x^(3*n))-2*b^3*( e*x)^(3*n)*polylog(2,-a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^ (3/2)/d^2/e/n/(x^(2*n))-2*b^3*(e*x)^(3*n)*polylog(3,-a*exp(c+d*x^n)/(b-(-a ^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))+2*b^3*(e*x)^(3*n)*p olylog(3,-a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3/e/ n/(x^(3*n))+b^2*(e*x)^(3*n)*sinh(c+d*x^n)/a/(a^2-b^2)/d/e/n/(x^n)/(b+a*cos h(c+d*x^n))-2*b*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/ d/e/n/(x^n)/(-a^2+b^2)^(1/2)+2*b*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b+(-a^2+ b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)-4*b*(e*x)^(3*n)*polylog(2,-a *exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2) +4*b*(e*x)^(3*n)*polylog(2,-a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/d^...
\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx \]
Time = 2.27 (sec) , antiderivative size = 1019, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5963, 5959, 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.
| \(\displaystyle \int \frac {(e x)^{3 n-1}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx\) |
| \(\Big \downarrow \) 5963 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{3 n-1}}{\left (a+b \text {sech}\left (d x^n+c\right )\right )^2}dx}{e}\) |
| \(\Big \downarrow \) 5959 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{2 n}}{\left (a+b \text {sech}\left (d x^n+c\right )\right )^2}dx^n}{e n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{2 n}}{\left (a+b \csc \left (i d x^n+i c+\frac {\pi }{2}\right )\right )^2}dx^n}{e n}\) |
| \(\Big \downarrow \) 4679 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \left (-\frac {2 b x^{2 n}}{a^2 \left (b+a \cosh \left (d x^n+c\right )\right )}+\frac {x^{2 n}}{a^2}+\frac {b^2 x^{2 n}}{a^2 \left (b+a \cosh \left (d x^n+c\right )\right )^2}\right )dx^n}{e n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (-\frac {2 b^2 \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {b^2-a^2}}+1\right ) x^n}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b^2 \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {b^2-a^2}}+1\right ) x^n}{a^2 \left (a^2-b^2\right ) d^2}-\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right ) x^n}{a^2 \sqrt {b^2-a^2} d^2}+\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right ) x^n}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right ) x^n}{a^2 \sqrt {b^2-a^2} d^2}-\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right ) x^n}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {2 b \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{2 n}}{a^2 \sqrt {b^2-a^2} d}+\frac {b^3 \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {2 b \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{2 n}}{a^2 \sqrt {b^2-a^2} d}-\frac {b^3 \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {b^2 \sinh \left (d x^n+c\right ) x^{2 n}}{a \left (a^2-b^2\right ) d \left (b+a \cosh \left (d x^n+c\right )\right )}+\frac {b^2 x^{2 n}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^{3 n}}{3 a^2}-\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^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^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {4 b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}-\frac {2 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {4 b \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}+\frac {2 b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}\right )}{e n}\) |
((e*x)^(3*n)*((b^2*x^(2*n))/(a^2*(a^2 - b^2)*d) + x^(3*n)/(3*a^2) - (2*b^2 *x^n*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d ^2) + (b^3*x^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2])])/(a^2 *(-a^2 + b^2)^(3/2)*d) - (2*b*x^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[ -a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (2*b^2*x^n*Log[1 + (a*E^(c + d*x ^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*(a^2 - b^2)*d^2) - (b^3*x^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ( 2*b*x^(2*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[- a^2 + b^2]*d) - (2*b^2*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2 ]))])/(a^2*(a^2 - b^2)*d^3) + (2*b^3*x^n*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (4*b*x^n*PolyLog[2, -((a*E^(c + d*x^n))/(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^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(a^ 2 - b^2)*d^3) - (2*b^3*x^n*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (4*b*x^n*PolyLog[2, -((a*E^(c + d *x^n))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - (2*b^3*PolyL og[3, -((a*E^(c + d*x^n))/(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^n))/(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^n))/(b + Sqrt[ -a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3) - (4*b*PolyLog[3, -((a*E^(...
3.1.84.3.1 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[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sech[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[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m* (a + b*Sech[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
\[\int \frac {\left (e x \right )^{-1+3 n}}{{\left (a +b \,\operatorname {sech}\left (c +d \,x^{n}\right )\right )}^{2}}d x\]
Timed out. \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{3 n - 1}}{\left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
-1/3*(6*a*b^2*e^(3*n)*x^(2*n) - (a^3*d*e^(3*n) - a*b^2*d*e^(3*n))*x^(3*n) - (a^3*d*e^(3*n)*e^(2*c) - a*b^2*d*e^(3*n)*e^(2*c))*e^(2*d*x^n + 3*n*log(x )) + 2*(3*b^3*e^(3*n)*e^(2*n*log(x) + c) - (a^2*b*d*e^(3*n)*e^c - b^3*d*e^ (3*n)*e^c)*x^(3*n))*e^(d*x^n))/(a^5*d*e*n - a^3*b^2*d*e*n + (a^5*d*e*n*e^( 2*c) - a^3*b^2*d*e*n*e^(2*c))*e^(2*d*x^n) + 2*(a^4*b*d*e*n*e^c - a^2*b^3*d *e*n*e^c)*e^(d*x^n)) - integrate(-2*(2*a*b^2*e^(3*n)*x^(2*n) + (2*b^3*e^(3 *n)*e^(2*n*log(x) + c) - (2*a^2*b*d*e^(3*n)*e^c - b^3*d*e^(3*n)*e^c)*x^(3* n))*e^(d*x^n))/((a^5*d*e*e^(2*c) - a^3*b^2*d*e*e^(2*c))*x*e^(2*d*x^n) + 2* (a^4*b*d*e*e^c - a^2*b^3*d*e*e^c)*x*e^(d*x^n) + (a^5*d*e - a^3*b^2*d*e)*x) , x)
\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{3\,n-1}}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )}^2} \,d x \]