3.153 \(\int \frac{x^2 (a+b \text{csch}^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}},x\right ) \]

[Out]

Unintegrable[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]

________________________________________________________________________________________

Rubi [A]  time = 0.107016, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]

[Out]

Defer[Int][(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=\int \frac{x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 4.44414, size = 0, normalized size = 0. \[ \int \frac{x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]

[Out]

Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]

________________________________________________________________________________________

Maple [A]  time = 0.448, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x)

[Out]

int(x^2*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x)

________________________________________________________________________________________

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(x^2*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} \operatorname{arcsch}\left (c x\right ) + a x^{2}\right )} \sqrt{e x^{2} + d}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

integral((b*x^2*arccsch(c*x) + a*x^2)*sqrt(e*x^2 + d)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*acsch(c*x))/(e*x**2+d)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x^2/(e*x^2 + d)^(3/2), x)