\(\int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx\) [33]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx=\text {Int}\left (\frac {x}{a+b \text {csch}^{-1}(c x)},x\right ) \]

[Out]

Unintegrable(x/(a+b*arccsch(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx=\int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx \]

[In]

Int[x/(a + b*ArcCsch[c*x]),x]

[Out]

Defer[Int][x/(a + b*ArcCsch[c*x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.90 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx=\int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx \]

[In]

Integrate[x/(a + b*ArcCsch[c*x]),x]

[Out]

Integrate[x/(a + b*ArcCsch[c*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \frac {x}{a +b \,\operatorname {arccsch}\left (c x \right )}d x\]

[In]

int(x/(a+b*arccsch(c*x)),x)

[Out]

int(x/(a+b*arccsch(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx=\int { \frac {x}{b \operatorname {arcsch}\left (c x\right ) + a} \,d x } \]

[In]

integrate(x/(a+b*arccsch(c*x)),x, algorithm="fricas")

[Out]

integral(x/(b*arccsch(c*x) + a), x)

Sympy [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx=\int \frac {x}{a + b \operatorname {acsch}{\left (c x \right )}}\, dx \]

[In]

integrate(x/(a+b*acsch(c*x)),x)

[Out]

Integral(x/(a + b*acsch(c*x)), x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx=\int { \frac {x}{b \operatorname {arcsch}\left (c x\right ) + a} \,d x } \]

[In]

integrate(x/(a+b*arccsch(c*x)),x, algorithm="maxima")

[Out]

integrate(x/(b*arccsch(c*x) + a), x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx=\int { \frac {x}{b \operatorname {arcsch}\left (c x\right ) + a} \,d x } \]

[In]

integrate(x/(a+b*arccsch(c*x)),x, algorithm="giac")

[Out]

integrate(x/(b*arccsch(c*x) + a), x)

Mupad [N/A]

Not integrable

Time = 5.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.50 \[ \int \frac {x}{a+b \text {csch}^{-1}(c x)} \, dx=\int \frac {x}{a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )} \,d x \]

[In]

int(x/(a + b*asinh(1/(c*x))),x)

[Out]

int(x/(a + b*asinh(1/(c*x))), x)