\(\int \frac {(f x)^m (a+b \text {csch}^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\) [172]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}},x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 25, 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 = {} \[ \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]

[In]

Int[((f*x)^m*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]

[Out]

Defer[Int][((f*x)^m*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.55 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]

[In]

Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]

[Out]

Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2], x]

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]

[In]

int((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x)

[Out]

int((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt {e x^{2} + d}} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral((b*arccsch(c*x) + a)*(f*x)^m/sqrt(e*x^2 + d), x)

Sympy [N/A]

Not integrable

Time = 23.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]

[In]

integrate((f*x)**m*(a+b*acsch(c*x))/(e*x**2+d)**(1/2),x)

[Out]

Integral((f*x)**m*(a + b*acsch(c*x))/sqrt(d + e*x**2), x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt {e x^{2} + d}} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*arccsch(c*x) + a)*(f*x)^m/sqrt(e*x^2 + d), x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{\sqrt {e x^{2} + d}} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*(f*x)^m/sqrt(e*x^2 + d), x)

Mupad [N/A]

Not integrable

Time = 5.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {(f x)^m \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]

[In]

int(((f*x)^m*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(1/2),x)

[Out]

int(((f*x)^m*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(1/2), x)