3.73 \(\int (e x)^{-1+2 n} (a+b \text{csch}(c+d x^n)) \, dx\)

Optimal. Leaf size=124 \[ -\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-e^{c+d x^n}\right )}{d^2 e n}+\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,e^{c+d x^n}\right )}{d^2 e n}+\frac{a (e x)^{2 n}}{2 e n}-\frac{2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.109991, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {14, 5441, 5437, 4182, 2279, 2391} \[ -\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-e^{c+d x^n}\right )}{d^2 e n}+\frac{b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,e^{c+d x^n}\right )}{d^2 e n}+\frac{a (e x)^{2 n}}{2 e n}-\frac{2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 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 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2279

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.144385, size = 175, normalized size = 1.41 \[ \frac{x^{-2 n} (e x)^{2 n} \left (2 b \text{PolyLog}\left (2,-e^{-c-d x^n}\right )-2 b \text{PolyLog}\left (2,e^{-c-d x^n}\right )+a d^2 x^{2 n}+2 b d x^n \log \left (1-e^{-c-d x^n}\right )-2 b d x^n \log \left (e^{-c-d x^n}+1\right )+2 b c \log \left (1-e^{-c-d x^n}\right )-2 b c \log \left (e^{-c-d x^n}+1\right )-2 b c \log \left (\tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )\right )\right )}{2 d^2 e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.218, size = 326, normalized size = 2.6 \begin{align*}{\frac{ax}{2\,n}{{\rm e}^{-{\frac{ \left ( -1+2\,n \right ) \left ( i \left ({\it csgn} \left ( iex \right ) \right ) ^{3}\pi -i\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}+i{\it csgn} \left ( iex \right ){\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ) \pi -2\,\ln \left ( x \right ) -2\,\ln \left ( e \right ) \right ) }{2}}}}}+2\,{\frac{{{\rm e}^{c}}b{{\rm e}^{-i\pi \,n{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{i\pi \,n{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i\pi \,n{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i\pi \,n \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}}{{\rm e}^{i/2\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) }}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{-i/2\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}}}{{\rm e}^{i/2\pi \, \left ({\it csgn} \left ( iex \right ) \right ) ^{3}}} \left ({e}^{n} \right ) ^{2} \left ( 1/2\,d{x}^{n} \left ( \ln \left ( 1-{{\rm e}^{c+d{x}^{n}}} \right ) -\ln \left ({{\rm e}^{c+d{x}^{n}}}+1 \right ) \right ){{\rm e}^{-c}}+1/2\, \left ({\it dilog} \left ( 1-{{\rm e}^{c+d{x}^{n}}} \right ) -{\it dilog} \left ({{\rm e}^{c+d{x}^{n}}}+1 \right ) \right ){{\rm e}^{-c}} \right ) }{en{d}^{2}}} \end{align*}

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)

[Out]

1/2*a/n*x*exp(-1/2*(-1+2*n)*(I*csgn(I*e*x)^3*Pi-I*Pi*csgn(I*e)*csgn(I*e*x)^2-I*Pi*csgn(I*x)*csgn(I*e*x)^2+I*cs
gn(I*e*x)*csgn(I*e)*csgn(I*x)*Pi-2*ln(x)-2*ln(e)))+2*b*exp(c)*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*c
sgn(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/n/d^2*(1/2*d*x^n*(ln(1-exp(c+d*x^n))-ln(exp(c+d*x^n)+1))*exp(-c)+1/2*(di
log(1-exp(c+d*x^n))-dilog(exp(c+d*x^n)+1))*exp(-c))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.10202, size = 1840, normalized size = 14.84 \begin{align*} \text{result too large to display} \end{align*}

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*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 + a*d^2*cosh(n*log(x))^2*sinh((2*n - 1)*log(e)) + (a*d^2*co
sh((2*n - 1)*log(e)) + a*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + 2*(b*cosh((2*n - 1)*log(e)) + b*sinh((
2*n - 1)*log(e)))*dilog(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x
)) + c)) - 2*(b*cosh((2*n - 1)*log(e)) + b*sinh((2*n - 1)*log(e)))*dilog(-cosh(d*cosh(n*log(x)) + d*sinh(n*log
(x)) + c) - sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) - 2*(b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + b*
d*cosh(n*log(x))*sinh((2*n - 1)*log(e)) + (b*d*cosh((2*n - 1)*log(e)) + b*d*sinh((2*n - 1)*log(e)))*sinh(n*log
(x)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) -
2*(b*c*cosh((2*n - 1)*log(e)) + b*c*sinh((2*n - 1)*log(e)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)
+ sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 1) + 2*(b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + b*c*cos
h((2*n - 1)*log(e)) + (b*d*cosh(n*log(x)) + b*c)*sinh((2*n - 1)*log(e)) + (b*d*cosh((2*n - 1)*log(e)) + b*d*si
nh((2*n - 1)*log(e)))*sinh(n*log(x)))*log(-cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - sinh(d*cosh(n*log(x
)) + d*sinh(n*log(x)) + c) + 1) + 2*(a*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + a*d^2*cosh(n*log(x))*sinh((
2*n - 1)*log(e)))*sinh(n*log(x)))/(d^2*n)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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