3.80 \(\int \frac {(e x)^{-1+3 n}}{a+b \text {csch}(c+d x^n)} \, dx\)

Optimal. Leaf size=428 \[ \frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a d^3 e n \sqrt {a^2+b^2}}-\frac {2 b x^{-3 n} (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a d^3 e n \sqrt {a^2+b^2}}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a d^2 e n \sqrt {a^2+b^2}}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a d^2 e n \sqrt {a^2+b^2}}-\frac {b x^{-n} (e x)^{3 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a d e n \sqrt {a^2+b^2}}+\frac {b x^{-n} (e x)^{3 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a d e n \sqrt {a^2+b^2}}+\frac {(e x)^{3 n}}{3 a e n} \]

[Out]

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
*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(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)

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Rubi [A]  time = 0.84, antiderivative size = 428, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5441, 5437, 4191, 3322, 2264, 2190, 2531, 2282, 6589} \[ \frac {2 b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a d^3 e n \sqrt {a^2+b^2}}-\frac {2 b x^{-3 n} (e x)^{3 n} \text {PolyLog}\left (3,-\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}\right )}{a d^3 e n \sqrt {a^2+b^2}}-\frac {2 b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a d^2 e n \sqrt {a^2+b^2}}+\frac {2 b x^{-2 n} (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}\right )}{a d^2 e n \sqrt {a^2+b^2}}-\frac {b x^{-n} (e x)^{3 n} \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a d e n \sqrt {a^2+b^2}}+\frac {b x^{-n} (e x)^{3 n} \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a d e n \sqrt {a^2+b^2}}+\frac {(e x)^{3 n}}{3 a e n} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n]),x]

[Out]

(e*x)^(3*n)/(3*a*e*n) - (b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d*
e*n*x^n) + (b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d*e*n*x^n) - (2
*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) + (
2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) +
(2*b*(e*x)^(3*n)*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)) -
(2*b*(e*x)^(3*n)*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))

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2264

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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3322

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 4191

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 5437

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 5441

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]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac {x^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx}{e}\\ &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b \text {csch}(c+d x)} \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{a}-\frac {b x^2}{a (b+a \sinh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {\left (b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a e n}\\ &=\frac {(e x)^{3 n}}{3 a e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{\sqrt {a^2+b^2} e n}+\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{\sqrt {a^2+b^2} e n}\\ &=\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 {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \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 \sqrt {a^2+b^2} d e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int x \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 \sqrt {a^2+b^2} d e n}\\ &=\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} \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {a^2+b^2} d^2 e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^n\right )}{a \sqrt {a^2+b^2} d^2 e n}\\ &=\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} \text {Li}_2\left (-\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} \text {Li}_2\left (-\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 {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a \sqrt {a^2+b^2} d^3 e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a \sqrt {a^2+b^2} d^3 e n}\\ &=\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} \text {Li}_2\left (-\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} \text {Li}_2\left (-\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} \text {Li}_3\left (-\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} \text {Li}_3\left (-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3 e n}\\ \end {align*}

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Mathematica [F]  time = 6.99, size = 0, normalized size = 0.00 \[ \int \frac {(e x)^{-1+3 n}}{a+b \text {csch}\left (c+d x^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n]),x]

[Out]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n]), x]

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fricas [C]  time = 0.49, size = 1850, normalized size = 4.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n)),x, algorithm="fricas")

[Out]

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)*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*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)*cosh((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*cos
h(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) + 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)*log(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*b) - 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)*log(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*b) - 3*(a*b*d^2*sqrt((a^2 +
b^2)/a^2)*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - a*b*c^2*sqrt((a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e)) + (a
*b*d^2*sqrt((a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e)) + a*b*d^2*sqrt((a^2 + b^2)/a^2)*sinh((3*n - 1)*log(e)))*si
nh(n*log(x))^2 + (a*b*d^2*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x))^2 - a*b*c^2*sqrt((a^2 + b^2)/a^2))*sinh((3*n -
1)*log(e)) + 2*(a*b*d^2*sqrt((a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + a*b*d^2*sqrt((a^2 + b^2)
/a^2)*cosh(n*log(x))*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*log(-((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) + 3*(a*b*d^2*sqrt((a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - a*b*c^2*sqrt((a^2 + b^2)
/a^2)*cosh((3*n - 1)*log(e)) + (a*b*d^2*sqrt((a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e)) + a*b*d^2*sqrt((a^2 + b^2
)/a^2)*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a*b*d^2*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x))^2 - a*b*c^2*sq
rt((a^2 + b^2)/a^2))*sinh((3*n - 1)*log(e)) + 2*(a*b*d^2*sqrt((a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e))*cosh(n*l
og(x)) + a*b*d^2*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x))*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*log(((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) + 6*(a*b*sqrt((a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e)) + a*b*sqrt((a^2
+ b^2)/a^2)*sinh((3*n - 1)*log(e)))*polylog(3, ((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) - 6*(a*b*sqrt
((a^2 + b^2)/a^2)*cosh((3*n - 1)*log(e)) + a*b*sqrt((a^2 + b^2)/a^2)*sinh((3*n - 1)*log(e)))*polylog(3, -((a*s
qrt((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) + 3*((a^2 + b^2)*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 + (a^
2 + b^2)*d^3*cosh(n*log(x))^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))/((a^3 + a*b^2)*d^3*n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 \, n - 1}}{b \operatorname {csch}\left (d x^{n} + c\right ) + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n)),x, algorithm="giac")

[Out]

integrate((e*x)^(3*n - 1)/(b*csch(d*x^n + c) + a), x)

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maple [F]  time = 1.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{3 n -1}}{a +b \,\mathrm {csch}\left (c +d \,x^{n}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(3*n-1)/(a+b*csch(c+d*x^n)),x)

[Out]

int((e*x)^(3*n-1)/(a+b*csch(c+d*x^n)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, b e^{3 \, n} \int \frac {e^{\left (d x^{n} + 3 \, n \log \relax (x) + c\right )}}{a^{2} e x e^{\left (2 \, d x^{n} + 2 \, c\right )} + 2 \, a b e x e^{\left (d x^{n} + c\right )} - a^{2} e x}\,{d x} + \frac {e^{3 \, n - 1} x^{3 \, n}}{3 \, a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n)),x, algorithm="maxima")

[Out]

-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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{3\,n-1}}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x^n\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(3*n - 1)/(a + b/sinh(c + d*x^n)),x)

[Out]

int((e*x)^(3*n - 1)/(a + b/sinh(c + d*x^n)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 n - 1}}{a + b \operatorname {csch}{\left (c + d x^{n} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(-1+3*n)/(a+b*csch(c+d*x**n)),x)

[Out]

Integral((e*x)**(3*n - 1)/(a + b*csch(c + d*x**n)), x)

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