3.1.79 \(\int \frac {(e x)^{-1+2 n}}{a+b \text {csch}(c+d x^n)} \, dx\) [79]

Optimal. Leaf size=291 \[ \frac {(e x)^{2 n}}{2 a e n}-\frac {b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d e n}+\frac {b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d e n}-\frac {b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2 e n}+\frac {b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2 e n} \]

[Out]

1/2*(e*x)^(2*n)/a/e/n-b*(e*x)^(2*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)^(2*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)^(2*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)+b*(e*x)^(2*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)

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Rubi [A]
time = 0.39, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5549, 5545, 4276, 3403, 2296, 2221, 2317, 2438} \begin {gather*} -\frac {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 d^2 e n \sqrt {a^2+b^2}}+\frac {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 d^2 e n \sqrt {a^2+b^2}}-\frac {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 d e n \sqrt {a^2+b^2}}+\frac {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 d e n \sqrt {a^2+b^2}}+\frac {(e x)^{2 n}}{2 a e n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x)^(2*n)/(2*a*e*n) - (b*(e*x)^(2*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)^(2*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)^(2*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)) + (b*
(e*x)^(2*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))

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

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Mathematica [C] Result contains complex when optimal does not.
time = 3.29, size = 1181, normalized size = 4.06 \begin {gather*} \frac {(e x)^{2 n} \text {csch}\left (c+d x^n\right ) \left (1-\frac {2 b x^{-2 n} \left (-\frac {i \pi \tanh ^{-1}\left (\frac {-a+b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}-\frac {2 \left (c+i \text {ArcCos}\left (-\frac {i b}{a}\right )\right ) \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )+\left (-2 i c+\pi -2 i d x^n\right ) \tanh ^{-1}\left (\frac {(-i a+b) \tan \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )-\left (\text {ArcCos}\left (-\frac {i b}{a}\right )-2 \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {(a+i b) \left (a-i b+\sqrt {-a^2-b^2}\right ) \left (1+i \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {i b}{a}\right )+2 \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {i (a+i b) \left (-a+i b+\sqrt {-a^2-b^2}\right ) \left (i+\cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )}\right )+\left (\text {ArcCos}\left (-\frac {i b}{a}\right )+2 \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )-2 i \tanh ^{-1}\left (\frac {(-i a+b) \tan \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (-\frac {(-1)^{3/4} \sqrt {-a^2-b^2} e^{-\frac {c}{2}-\frac {d x^n}{2}}}{\sqrt {2} \sqrt {-i a} \sqrt {b+a \sinh \left (c+d x^n\right )}}\right )+\left (\text {ArcCos}\left (-\frac {i b}{a}\right )-2 \text {ArcTan}\left (\frac {(a-i b) \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )+2 i \tanh ^{-1}\left (\frac {(-i a+b) \tan \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {-a^2-b^2} e^{\frac {1}{2} \left (c+d x^n\right )}}{\sqrt {2} \sqrt {-i a} \sqrt {b+a \sinh \left (c+d x^n\right )}}\right )+i \left (\text {PolyLog}\left (2,\frac {\left (i b+\sqrt {-a^2-b^2}\right ) \left (a+i b-i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (b+i \sqrt {-a^2-b^2}\right ) \left (i a-b+\sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )}{a \left (a+i b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )}\right )\right )}{\sqrt {-a^2-b^2}}\right )}{d^2}\right ) \left (b+a \sinh \left (c+d x^n\right )\right )}{2 a e n \left (a+b \text {csch}\left (c+d x^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^(2*n)*Csch[c + d*x^n]*(1 - (2*b*(((-I)*Pi*ArcTanh[(-a + b*Tanh[(c + d*x^n)/2])/Sqrt[a^2 + b^2]])/Sqrt[a
^2 + b^2] - (2*(c + I*ArcCos[((-I)*b)/a])*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4])/Sqrt[-a^2 - b
^2]] + ((-2*I)*c + Pi - (2*I)*d*x^n)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2*I)*d*x^n)/4])/Sqrt[-a^2 - b^
2]] - (ArcCos[((-I)*b)/a] - 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4])/Sqrt[-a^2 - b^2]])*Log[((
a + I*b)*(a - I*b + Sqrt[-a^2 - b^2])*(1 + I*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4]))/(a*(a + I*b + I*Sqrt[-a^2 -
b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4]))] - (ArcCos[((-I)*b)/a] + 2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (
2*I)*d*x^n)/4])/Sqrt[-a^2 - b^2]])*Log[(I*(a + I*b)*(-a + I*b + Sqrt[-a^2 - b^2])*(I + Cot[((2*I)*c + Pi + (2*
I)*d*x^n)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4]))] + (ArcCos[((-I)*b)/a] +
2*ArcTan[((a - I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4])/Sqrt[-a^2 - b^2]] - (2*I)*ArcTanh[(((-I)*a + b)*Tan[
((2*I)*c + Pi + (2*I)*d*x^n)/4])/Sqrt[-a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[-a^2 - b^2]*E^(-1/2*c - (d*x^n)/2))
/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b + a*Sinh[c + d*x^n]]))] + (ArcCos[((-I)*b)/a] - 2*ArcTan[((a - I*b)*Cot[((2*I)*c
+ Pi + (2*I)*d*x^n)/4])/Sqrt[-a^2 - b^2]] + (2*I)*ArcTanh[(((-I)*a + b)*Tan[((2*I)*c + Pi + (2*I)*d*x^n)/4])/
Sqrt[-a^2 - b^2]])*Log[((-1)^(1/4)*Sqrt[-a^2 - b^2]*E^((c + d*x^n)/2))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b + a*Sinh[c
+ d*x^n]])] + I*(PolyLog[2, ((I*b + Sqrt[-a^2 - b^2])*(a + I*b - I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)
*d*x^n)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4]))] - PolyLog[2, ((b + I*Sqrt
[-a^2 - b^2])*(I*a - b + Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4]))/(a*(a + I*b + I*Sqrt[-a^2 - b^
2]*Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4]))]))/Sqrt[-a^2 - b^2]))/(d^2*x^(2*n)))*(b + a*Sinh[c + d*x^n]))/(2*a*e*
n*(a + b*Csch[c + d*x^n]))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.98, size = 577, normalized size = 1.98

method	result	size
risch	\(\frac {x \,{\mathrm e}^{\frac {\left (-1+2 n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{2 a n}-\frac {2 b \,{\mathrm e}^{-i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )} {\mathrm e}^{i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}} {\mathrm e}^{i \pi n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}} {\mathrm e}^{-i \pi n \mathrm {csgn}\left (i e x \right )^{3}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (i e x \right )^{3}}{2}} e^{2 n} {\mathrm e}^{c} \left (\frac {x^{n} d \left (\ln \left (\frac {a \,{\mathrm e}^{2 c +d \,x^{n}}+{\mathrm e}^{c} b -\sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}{{\mathrm e}^{c} b -\sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}\right )-\ln \left (\frac {a \,{\mathrm e}^{2 c +d \,x^{n}}+{\mathrm e}^{c} b +\sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}{{\mathrm e}^{c} b +\sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}\right )\right )}{2 \sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}+\frac {\dilog \left (\frac {a \,{\mathrm e}^{2 c +d \,x^{n}}+{\mathrm e}^{c} b -\sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}{{\mathrm e}^{c} b -\sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}\right )-\dilog \left (\frac {a \,{\mathrm e}^{2 c +d \,x^{n}}+{\mathrm e}^{c} b +\sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}{{\mathrm e}^{c} b +\sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}\right )}{2 \sqrt {b^{2} {\mathrm e}^{2 c}+a^{2} {\mathrm e}^{2 c}}}\right )}{a e n \,d^{2}}\)	\(577\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a/n*x*exp(1/2*(-1+2*n)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*Pi*csgn(I*e)*csgn(I*e*x)^2+I*Pi*csgn(I*x)*
csgn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+2*ln(x)+2*ln(e)))-2/a*b*exp(-I*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(I*Pi
*n*csgn(I*e)*csgn(I*e*x)^2)*exp(I*Pi*n*csgn(I*x)*csgn(I*e*x)^2)*exp(-I*Pi*n*csgn(I*e*x)^3)*exp(1/2*I*Pi*csgn(I
*e)*csgn(I*x)*csgn(I*e*x))*exp(-1/2*I*Pi*csgn(I*e)*csgn(I*e*x)^2)*exp(-1/2*I*Pi*csgn(I*x)*csgn(I*e*x)^2)*exp(1
/2*I*Pi*csgn(I*e*x)^3)*(e^n)^2/e*exp(c)/n/d^2*(1/2*x^n*d*(ln((a*exp(2*c+d*x^n)+exp(c)*b-(b^2*exp(2*c)+a^2*exp(
2*c))^(1/2))/(exp(c)*b-(b^2*exp(2*c)+a^2*exp(2*c))^(1/2)))-ln((a*exp(2*c+d*x^n)+exp(c)*b+(b^2*exp(2*c)+a^2*exp
(2*c))^(1/2))/(exp(c)*b+(b^2*exp(2*c)+a^2*exp(2*c))^(1/2))))/(b^2*exp(2*c)+a^2*exp(2*c))^(1/2)+1/2*(dilog((a*e
xp(2*c+d*x^n)+exp(c)*b-(b^2*exp(2*c)+a^2*exp(2*c))^(1/2))/(exp(c)*b-(b^2*exp(2*c)+a^2*exp(2*c))^(1/2)))-dilog(
(a*exp(2*c+d*x^n)+exp(c)*b+(b^2*exp(2*c)+a^2*exp(2*c))^(1/2))/(exp(c)*b+(b^2*exp(2*c)+a^2*exp(2*c))^(1/2))))/(
b^2*exp(2*c)+a^2*exp(2*c))^(1/2))

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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)),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (271) = 542\).
time = 0.47, size = 977, normalized size = 3.36 \begin {gather*} \frac {{\left ({\left (a^{2} + b^{2}\right )} d^{2} \cosh \left (2 \, n - 1\right ) + {\left (a^{2} + b^{2}\right )} d^{2} \sinh \left (2 \, n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right )^{2} + 2 \, {\left ({\left (a^{2} + b^{2}\right )} d^{2} \cosh \left (2 \, n - 1\right ) + {\left (a^{2} + b^{2}\right )} d^{2} \sinh \left (2 \, n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) \sinh \left (n \log \left (x\right )\right ) + {\left ({\left (a^{2} + b^{2}\right )} d^{2} \cosh \left (2 \, n - 1\right ) + {\left (a^{2} + b^{2}\right )} d^{2} \sinh \left (2 \, n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )^{2} - 2 \, {\left (a b \cosh \left (2 \, n - 1\right ) + a b \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm Li}_2\left (\frac {{\left (a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + b\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + {\left (a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + b\right )} \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - a}{a} + 1\right ) + 2 \, {\left (a b \cosh \left (2 \, n - 1\right ) + a b \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} {\rm Li}_2\left (-\frac {{\left (a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - b\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + {\left (a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - b\right )} \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + a}{a} + 1\right ) - 2 \, {\left (a b c \cosh \left (2 \, n - 1\right ) + a b c \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 2 \, a \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) + 2 \, {\left (a b c \cosh \left (2 \, n - 1\right ) + a b c \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 2 \, a \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - 2 \, a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + 2 \, b\right ) - 2 \, {\left ({\left (a b d \cosh \left (2 \, n - 1\right ) + a b d \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \cosh \left (n \log \left (x\right )\right ) + {\left (a b d \cosh \left (2 \, n - 1\right ) + a b d \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \sinh \left (n \log \left (x\right )\right ) + {\left (a b c \cosh \left (2 \, n - 1\right ) + a b c \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}}\right )} \log \left (-\frac {{\left (a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + b\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + {\left (a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} + b\right )} \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - a}{a}\right ) + 2 \, {\left ({\left (a b d \cosh \left (2 \, n - 1\right ) + a b d \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \cosh \left (n \log \left (x\right )\right ) + {\left (a b d \cosh \left (2 \, n - 1\right ) + a b d \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} \sinh \left (n \log \left (x\right )\right ) + {\left (a b c \cosh \left (2 \, n - 1\right ) + a b c \sinh \left (2 \, n - 1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{a^{2}}}\right )} \log \left (\frac {{\left (a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - b\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + {\left (a \sqrt {\frac {a^{2} + b^{2}}{a^{2}}} - b\right )} \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + a}{a}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d^{2} n} \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)),x, algorithm="fricas")

[Out]

1/2*(((a^2 + b^2)*d^2*cosh(2*n - 1) + (a^2 + b^2)*d^2*sinh(2*n - 1))*cosh(n*log(x))^2 + 2*((a^2 + b^2)*d^2*cos
h(2*n - 1) + (a^2 + b^2)*d^2*sinh(2*n - 1))*cosh(n*log(x))*sinh(n*log(x)) + ((a^2 + b^2)*d^2*cosh(2*n - 1) + (
a^2 + b^2)*d^2*sinh(2*n - 1))*sinh(n*log(x))^2 - 2*(a*b*cosh(2*n - 1) + a*b*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*b*cosh(2*n - 1) + a*b*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*b*c*cosh(2*n - 1)
+ a*b*c*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2)*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) + 2*(a*b*c*cosh(2*n - 1) + a*b*c*
sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2)*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) - 2*((a*b*d*cosh(2*n - 1) + a*b*d*sinh(2*n
- 1))*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + (a*b*d*cosh(2*n - 1) + a*b*d*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2
)*sinh(n*log(x)) + (a*b*c*cosh(2*n - 1) + a*b*c*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*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) + 2*((a*b*d*cosh(2*n - 1) + a*b*d*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2)*cosh(
n*log(x)) + (a*b*d*cosh(2*n - 1) + a*b*d*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2)*sinh(n*log(x)) + (a*b*c*cosh(2*n
- 1) + a*b*c*sinh(2*n - 1))*sqrt((a^2 + b^2)/a^2))*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))/
((a^3 + a*b^2)*d^2*n)

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

[Out]

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

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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)),x, algorithm="giac")

[Out]

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

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

[Out]

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

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