[next] [prev] [prev-tail] [tail] [up]

3.1.82 \(\int \frac {(e x)^{-1+2 n}}{(a+b \text {csch}(c+d x^n))^2} \, dx\) [82]

Optimal. Leaf size=681 \[ \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )} \]

[Out]

1/2*(e*x)^(2*n)/a^2/e/n+b^2*(e*x)^(2*n)*ln(b+a*sinh(c+d*x^n))/a^2/(a^2+b^2)/d^2/e/n/(x^(2*n))+b^3*(e*x)^(2*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)-b^3*(e*x)^(2*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)+b^3*(e*x)^(2*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))-b^3*(e*x)^(2*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))-b^2*(e*x)^(2*n)*cosh(c+d*x^n)/a/(a^2+b^2)/d/e/n/(x^n)/(b+a*sinh(c+d*x^n))-2*
b*(e*x)^(2*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)^(2*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)^(2*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)+2*b*(e*x)^(2*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)

________________________________________________________________________________________

Rubi [A]
time = 0.86, antiderivative size = 681, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5549, 5545, 4276, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} -\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 e n \sqrt {a^2+b^2}}+\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2 e n \sqrt {a^2+b^2}}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (a \sinh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2+b^2\right )}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d e n \sqrt {a^2+b^2}}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d e n \sqrt {a^2+b^2}}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a d e n \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^n\right )+b\right )}+\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}-\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}+\frac {(e x)^{2 n}}{2 a^2 e n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + 2*n)/(a + b*Csch[c + d*x^n])^2,x]

[Out]

(e*x)^(2*n)/(2*a^2*e*n) + (b^3*(e*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*e*n*x^n) - (2*b*(e*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*e
*n*x^n) - (b^3*(e*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*e*n*x^n)
 + (2*b*(e*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*e*n*x^n) + (b^2*(
e*x)^(2*n)*Log[b + a*Sinh[c + d*x^n]])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*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*e*n*x^(2*n)) - (2*b*(e*x)^(2*n)*PolyLog[2, -(
(a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(2*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*e*n*x^(2*n)) + (2*b*(e*x)^(2*n)*PolyL
og[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) - (b^2*(e*x)^(2*n)*Co
sh[c + d*x^n])/(a*(a^2 + b^2)*d*e*n*x^n*(b + a*Sinh[c + d*x^n]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
 (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
 - Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
 Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
 e, n}, x] && EqQ[c*d, 1]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
 /; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
 FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

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*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 5545

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(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[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 5549

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac {x^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(a+b \text {csch}(c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \left (\frac {x}{a^2}+\frac {b^2 x}{a^2 (b+a \sinh (c+d x))^2}-\frac {2 b x}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(b+a \sinh (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}-\frac {\left (4 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\cosh (c+d x)}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a \left (a^2+b^2\right ) d e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac {\left (2 b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n}-\frac {\left (4 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \sqrt {a^2+b^2} e n}+\frac {\left (4 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \sqrt {a^2+b^2} e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {2 b x^{-n} (e x)^{2 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 x^{-n} (e x)^{2 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 {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac {\left (2 b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right )^{3/2} e n}-\frac {\left (2 b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{c+d x} x}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right )^{3/2} e n}+\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {a^2+b^2} d e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {a^2+b^2} d e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 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)^{2 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 {b^3 x^{-n} (e x)^{2 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)^{2 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 {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 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)^{2 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 {b^3 x^{-n} (e x)^{2 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)^{2 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 {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\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 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\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 {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}-\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}\\ &=\frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 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)^{2 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 {b^3 x^{-n} (e x)^{2 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)^{2 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 {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\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 {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\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 {b^3 x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\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 {2 b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-\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 {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 25.16, size = 3259, normalized size = 4.79 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*x)^(-1 + 2*n)/(a + b*Csch[c + d*x^n])^2,x]

[Out]

(b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Csch[c/2]*Csch[c + d*x^n]^2*Sech[c/2]*(b*Cosh[c] + a*Sinh[d*x^n])*(b + a*Sinh[
c + d*x^n]))/(2*a^2*(a^2 + b^2)*d*n*(a + b*Csch[c + d*x^n])^2) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Coth[c]*Csch[
c + d*x^n]^2*(b + a*Sinh[c + d*x^n])^2)/(a^2*(a^2 + b^2)*d*n*(a + b*Csch[c + d*x^n])^2) - (2*b^3*x^(1 - 2*n)*(
e*x)^(-1 + 2*n)*ArcTan[(a - b*Tanh[(c + d*x^n)/2])/Sqrt[-a^2 - b^2]]*Coth[c]*Csch[c + d*x^n]^2*(b + a*Sinh[c +
 d*x^n])^2)/(a^2*Sqrt[-a^2 - b^2]*(a^2 + b^2)*d^2*n*(a + b*Csch[c + d*x^n])^2) - (2*b*x^(1 - 2*n)*(e*x)^(-1 +
2*n)*Csch[c + d*x^n]^2*(((-I)*Pi*ArcTanh[(-a + b*Tanh[(c + d*x^n)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - (2*(
(-I)*c + Pi/2 - I*d*x^n)*ArcTanh[(((-I)*a + b)*Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - 2*((-I)*c
 + ArcCos[((-I)*b)/a])*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] + (ArcCos[((-
I)*b)/a] - (2*I)*(ArcTanh[(((-I)*a + b)*Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - ArcTanh[(((-I)*a
 - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]]))*Log[Sqrt[-a^2 - b^2]/(Sqrt[2]*Sqrt[(-I)*a]*E^((I/2
)*((-I)*c + Pi/2 - I*d*x^n))*Sqrt[b + a*Sinh[c + d*x^n]])] + (ArcCos[((-I)*b)/a] + (2*I)*(ArcTanh[(((-I)*a + b
)*Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]
)/Sqrt[-a^2 - b^2]]))*Log[(Sqrt[-a^2 - b^2]*E^((I/2)*((-I)*c + Pi/2 - I*d*x^n)))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b
+ a*Sinh[c + d*x^n]])] - (ArcCos[((-I)*b)/a] + (2*I)*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/S
qrt[-a^2 - b^2]])*Log[1 - (I*(b - I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*
x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] + (-ArcCos[((-I)*b)/a] + (2*I)
*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]])*Log[1 - (I*(b + I*Sqrt[-a^2 - b^2]
)*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I
)*c + Pi/2 - I*d*x^n)/2]))] + I*(PolyLog[2, (I*(b - I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-
I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] - PolyLog[2,
 (I*(b + I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b
+ Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))]))/Sqrt[-a^2 - b^2])*(b + a*Sinh[c + d*x^n])^2)/((a^2 +
b^2)*d^2*n*(a + b*Csch[c + d*x^n])^2) - (b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*Csch[c + d*x^n]^2*(((-I)*Pi*ArcTanh[
(-a + b*Tanh[(c + d*x^n)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - (2*((-I)*c + Pi/2 - I*d*x^n)*ArcTanh[(((-I)*a
 + b)*Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - 2*((-I)*c + ArcCos[((-I)*b)/a])*ArcTanh[(((-I)*a -
 b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] + (ArcCos[((-I)*b)/a] - (2*I)*(ArcTanh[(((-I)*a + b)*C
ot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/S
qrt[-a^2 - b^2]]))*Log[Sqrt[-a^2 - b^2]/(Sqrt[2]*Sqrt[(-I)*a]*E^((I/2)*((-I)*c + Pi/2 - I*d*x^n))*Sqrt[b + a*S
inh[c + d*x^n]])] + (ArcCos[((-I)*b)/a] + (2*I)*(ArcTanh[(((-I)*a + b)*Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[
-a^2 - b^2]] - ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]]))*Log[(Sqrt[-a^2 - b^
2]*E^((I/2)*((-I)*c + Pi/2 - I*d*x^n)))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b + a*Sinh[c + d*x^n]])] - (ArcCos[((-I)*b)
/a] + (2*I)*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]])*Log[1 - (I*(b - I*Sqrt[
-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^
2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] + (-ArcCos[((-I)*b)/a] + (2*I)*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2
 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]])*Log[1 - (I*(b + I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I
)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] + I*(PolyLog[
2, (I*(b - I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a +
b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] - PolyLog[2, (I*(b + I*Sqrt[-a^2 - b^2])*((-I)*a + b
- Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I
*d*x^n)/2]))]))/Sqrt[-a^2 - b^2])*(b + a*Sinh[c + d*x^n])^2)/(a^2*(a^2 + b^2)*d^2*n*(a + b*Csch[c + d*x^n])^2)
 + (x^(1 - n)*(e*x)^(-1 + 2*n)*Csch[c/2]*Csch[c + d*x^n]^2*Sech[c/2]*(-2*b^2*Cosh[c] + a^2*d*x^n*Sinh[c] + b^2
*d*x^n*Sinh[c])*(b + a*Sinh[c + d*x^n])^2)/(4*a^2*(a^2 + b^2)*d*n*(a + b*Csch[c + d*x^n])^2) - (b^2*x^(1 - 2*n
)*(e*x)^(-1 + 2*n)*Csch[c]*Csch[c + d*x^n]^2*(-(a*d*x^n*Cosh[c]) + a*Log[b + a*Cosh[d*x^n]*Sinh[c] + a*Cosh[c]
*Sinh[d*x^n]]*Sinh[c] + (2*a*b*ArcTan[(a*Cosh[c] + (-b + a*Sinh[c])*Tanh[(d*x^n)/2])/Sqrt[-b^2 - a^2*Cosh[c]^2
 + a^2*Sinh[c]^2]]*Cosh[c])/Sqrt[-b^2 - a^2*Cosh[c]^2 + a^2*Sinh[c]^2])*(b + a*Sinh[c + d*x^n])^2)/(a*(a^2 + b
^2)*d^2*n*(a + b*Csch[c + d*x^n])^2*(-(a^2*Cosh...

________________________________________________________________________________________

Maple [F]
time = 2.45, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{-1+2 n}}{\left (a +b \,\mathrm {csch}\left (c +d \,x^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+2*n)/(a+b*csch(c+d*x^n))^2,x)

[Out]

int((e*x)^(-1+2*n)/(a+b*csch(c+d*x^n))^2,x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="maxima")

[Out]

1/2*(4*a*b^2*e^(n*log(x) + 2*n) + (a^3*d*e^(2*n) + a*b^2*d*e^(2*n))*x^(2*n) - (a^3*d*e^(2*c + 2*n) + a*b^2*d*e
^(2*c + 2*n))*e^(2*d*x^n + 2*n*log(x)) - 2*(2*b^3*e^(n*log(x) + c + 2*n) + (a^2*b*d*e^(c + 2*n) + b^3*d*e^(c +
 2*n))*x^(2*n))*e^(d*x^n))/((a^5*d*n + a^3*b^2*d*n)*e - (a^5*d*n*e^(2*c) + a^3*b^2*d*n*e^(2*c))*e^(2*d*x^n + 1
) - 2*(a^4*b*d*n*e^c + a^2*b^3*d*n*e^c)*e^(d*x^n + 1)) - integrate(2*(a*b^2*e^(n*log(x) + 2*n) - (b^3*e^(n*log
(x) + c + 2*n) + (2*a^2*b*d*e^(c + 2*n) + b^3*d*e^(c + 2*n))*x^(2*n))*e^(d*x^n))/((a^5*d + a^3*b^2*d)*x*e - (a
^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*x*e^(2*d*x^n + 1) - 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*x*e^(d*x^n + 1)), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7404 vs. \(2 (645) = 1290\).
time = 0.49, size = 7404, normalized size = 10.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="fricas")

[Out]

1/2*((((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(2*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*sinh(2*n - 1))*cosh(n*log(x
))^2 - 4*(a^3*b^2 + a*b^4)*c*cosh(2*n - 1) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(2*n - 1) + (a^5 + 2*a^3*b^2 +
 a*b^4)*d^2*sinh(2*n - 1))*sinh(n*log(x))^2 - 4*(a^3*b^2 + a*b^4)*c*sinh(2*n - 1) - 4*((a^3*b^2 + a*b^4)*d*cos
h(2*n - 1) + (a^3*b^2 + a*b^4)*d*sinh(2*n - 1))*cosh(n*log(x)) - 2*(2*(a^3*b^2 + a*b^4)*d*cosh(2*n - 1) + 2*(a
^3*b^2 + a*b^4)*d*sinh(2*n - 1) - ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(2*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2
*sinh(2*n - 1))*cosh(n*log(x)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - ((a^5 + 2*a^
3*b^2 + a*b^4)*d^2*cosh(2*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*sinh(2*n - 1))*cosh(n*log(x))^2 + 4*(a^3*b^2
+ a*b^4)*c*cosh(2*n - 1) + (((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(2*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*sinh(
2*n - 1))*cosh(n*log(x))^2 - 4*(a^3*b^2 + a*b^4)*c*cosh(2*n - 1) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(2*n - 1
) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*sinh(2*n - 1))*sinh(n*log(x))^2 - 4*(a^3*b^2 + a*b^4)*c*sinh(2*n - 1) - 4*((
a^3*b^2 + a*b^4)*d*cosh(2*n - 1) + (a^3*b^2 + a*b^4)*d*sinh(2*n - 1))*cosh(n*log(x)) - 2*(2*(a^3*b^2 + a*b^4)*
d*cosh(2*n - 1) + 2*(a^3*b^2 + a*b^4)*d*sinh(2*n - 1) - ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(2*n - 1) + (a^5 +
2*a^3*b^2 + a*b^4)*d^2*sinh(2*n - 1))*cosh(n*log(x)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x))
 + c)^2 - 2*((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(2*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*sinh(2*n - 1))*cosh(n
*log(x))*sinh(n*log(x)) - ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(2*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*sinh(2*
n - 1))*sinh(n*log(x))^2 + 4*(a^3*b^2 + a*b^4)*c*sinh(2*n - 1) + 2*(((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh(2*n -
1) + (a^4*b + 2*a^2*b^3 + b^5)*d^2*sinh(2*n - 1))*cosh(n*log(x))^2 - 4*(a^2*b^3 + b^5)*c*cosh(2*n - 1) + ((a^4
*b + 2*a^2*b^3 + b^5)*d^2*cosh(2*n - 1) + (a^4*b + 2*a^2*b^3 + b^5)*d^2*sinh(2*n - 1))*sinh(n*log(x))^2 - 4*(a
^2*b^3 + b^5)*c*sinh(2*n - 1) - 2*((a^2*b^3 + b^5)*d*cosh(2*n - 1) + (a^2*b^3 + b^5)*d*sinh(2*n - 1))*cosh(n*l
og(x)) - 2*((a^2*b^3 + b^5)*d*cosh(2*n - 1) + (a^2*b^3 + b^5)*d*sinh(2*n - 1) - ((a^4*b + 2*a^2*b^3 + b^5)*d^2
*cosh(2*n - 1) + (a^4*b + 2*a^2*b^3 + b^5)*d^2*sinh(2*n - 1))*cosh(n*log(x)))*sinh(n*log(x)))*cosh(d*cosh(n*lo
g(x)) + d*sinh(n*log(x)) + c) - 2*(((2*a^4*b + a^2*b^3)*cosh(2*n - 1) + (2*a^4*b + a^2*b^3)*sinh(2*n - 1))*sqr
t((a^2 + b^2)/a^2)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((2*a^4*b + a^2*b^3)*cosh(2*n - 1) + (2*a
^4*b + a^2*b^3)*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((2*a
^3*b^2 + a*b^4)*cosh(2*n - 1) + (2*a^3*b^2 + a*b^4)*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2)*cosh(d*cosh(n*log(x))
 + d*sinh(n*log(x)) + c) + 2*(((2*a^4*b + a^2*b^3)*cosh(2*n - 1) + (2*a^4*b + a^2*b^3)*sinh(2*n - 1))*sqrt((a^
2 + b^2)/a^2)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((2*a^3*b^2 + a*b^4)*cosh(2*n - 1) + (2*a^3*b^2
+ a*b^4)*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - ((2*a^4*b + a^2
*b^3)*cosh(2*n - 1) + (2*a^4*b + a^2*b^3)*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*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*log(x)) + d
*sinh(n*log(x)) + c) - a)/a + 1) + 2*(((2*a^4*b + a^2*b^3)*cosh(2*n - 1) + (2*a^4*b + a^2*b^3)*sinh(2*n - 1))*
sqrt((a^2 + b^2)/a^2)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((2*a^4*b + a^2*b^3)*cosh(2*n - 1) + (
2*a^4*b + a^2*b^3)*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((
2*a^3*b^2 + a*b^4)*cosh(2*n - 1) + (2*a^3*b^2 + a*b^4)*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2)*cosh(d*cosh(n*log(
x)) + d*sinh(n*log(x)) + c) + 2*(((2*a^4*b + a^2*b^3)*cosh(2*n - 1) + (2*a^4*b + a^2*b^3)*sinh(2*n - 1))*sqrt(
(a^2 + b^2)/a^2)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((2*a^3*b^2 + a*b^4)*cosh(2*n - 1) + (2*a^3*b
^2 + a*b^4)*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - ((2*a^4*b +
a^2*b^3)*cosh(2*n - 1) + (2*a^4*b + a^2*b^3)*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*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*log(x))
 + d*sinh(n*log(x)) + c) + a)/a + 1) + 2*(((a^3*b^2 + a*b^4)*cosh(2*n - 1) + (a^3*b^2 + a*b^4)*sinh(2*n - 1) -
 ((2*a^4*b + a^2*b^3)*c*cosh(2*n - 1) + (2*a^4*b + a^2*b^3)*c*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*cosh(d*cos
h(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((a^3*b^2 + a*b^4)*cosh(2*n - 1) + (a^3*b^2 + a*b^4)*sinh(2*n - 1) - (
(2*a^4*b + a^2*b^3)*c*cosh(2*n - 1) + (2*a^4*b + a^2*b^3)*c*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*sinh(d*cosh(
n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((a^2*b^3 + b^5)*cosh(2*n - 1) + (a^2*b^3 + b^5)*sinh(2*n - 1) - ((2*a
^3*b^2 + a*b^4)*c*cosh(2*n - 1) + (2*a^3*b^2 + a*b^4)*c*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*cosh(d*cosh(n*lo
g(x)) + d*sinh(n*log(x)) + c) - (a^3*b^2 + a*b^...

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{2 n - 1}}{\left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(-1+2*n)/(a+b*csch(c+d*x**n))**2,x)

[Out]

Integral((e*x)**(2*n - 1)/(a + b*csch(c + d*x**n))**2, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="giac")

[Out]

integrate((e*x)^(2*n - 1)/(b*csch(d*x^n + c) + a)^2, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{2\,n-1}}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,x^n\right )}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(2*n - 1)/(a + b/sinh(c + d*x^n))^2,x)

[Out]

int((e*x)^(2*n - 1)/(a + b/sinh(c + d*x^n))^2, x)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]