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

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

[Out]

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

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Rubi [A]  time = 0.105802, 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 = {} \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 5.53719, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.491, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{{x}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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