\(\int (e x)^{-1+n} (a+b \text {csch}(c+d x^n)) \, dx\) [72]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [C] (warning: unable to verify)
   Fricas [B] (verification not implemented)
   Sympy [F]
   Maxima [A] (verification not implemented)
   Giac [F]
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]

[Out]

a*(e*x)^n/e/n-b*(e*x)^n*arctanh(cosh(c+d*x^n))/d/e/n/(x^n)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {14, 5549, 5545, 3855} \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]

[In]

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

[Out]

(a*(e*x)^n)/(e*n) - (b*(e*x)^n*ArcTanh[Cosh[c + d*x^n]])/(d*e*n*x^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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.36 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {x^{-n} (e x)^n \left (a \left (c+d x^n\right )-b \log \left (\cosh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )+b \log \left (\sinh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )\right )}{d e n} \]

[In]

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

[Out]

((e*x)^n*(a*(c + d*x^n) - b*Log[Cosh[(c + d*x^n)/2]] + b*Log[Sinh[(c + d*x^n)/2]]))/(d*e*n*x^n)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.79 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.44

method result size
risch \(\frac {a x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{n}-\frac {2 \,\operatorname {arctanh}\left ({\mathrm e}^{c +d \,x^{n}}\right ) e^{n} b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{d e n}\) \(155\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (45) = 90\).

Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 4.00 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) \cosh \left (n \log \left (x\right )\right ) + a d \cosh \left (n \log \left (x\right )\right ) \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right ) - {\left (b \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + b \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 1\right ) + {\left (b \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + b \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - 1\right ) + {\left (a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + a d \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \sinh \left (n \log \left (x\right )\right )}{d n} \]

[In]

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

[Out]

(a*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + a*d*cosh(n*log(x))*sinh((n - 1)*log(e)) - (b*cosh((n - 1)*log(e)) +
 b*sinh((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) + (b*cosh((n - 1)*log(e)) + b*sinh((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) + (a*d*cosh((n - 1)*log(e)) + a*d*sinh((n - 1)*
log(e)))*sinh(n*log(x)))/(d*n)

Sympy [F]

\[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\int \left (e x\right )^{n - 1} \left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.67 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=-b {\left (\frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{d n} - \frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{d n}\right )} + \frac {\left (e x\right )^{n} a}{e n} \]

[In]

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

[Out]

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

Giac [F]

\[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\int { {\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{n - 1} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.49 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a\,x\,{\left (e\,x\right )}^{n-1}}{n}-\frac {2\,\mathrm {atan}\left (\frac {b\,x\,{\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,{\left (e\,x\right )}^{n-1}\,\sqrt {-d^2\,n^2\,x^{2\,n}}}{d\,n\,x^n\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {-d^2\,n^2\,x^{2\,n}}} \]

[In]

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

[Out]

(a*x*(e*x)^(n - 1))/n - (2*atan((b*x*exp(d*x^n)*exp(c)*(e*x)^(n - 1)*(-d^2*n^2*x^(2*n))^(1/2))/(d*n*x^n*(b^2*x
^2*(e*x)^(2*n - 2))^(1/2)))*(b^2*x^2*(e*x)^(2*n - 2))^(1/2))/(-d^2*n^2*x^(2*n))^(1/2)