3.2.53 \(\int \frac {x^2 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\) [153]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 2.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

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]

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Maple [A]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

(arcsinh(x*e^(1/2)/sqrt(d))*e^(-3/2) - x*e^(-1)/sqrt(x^2*e + d))*a + b*integrate(x^2*log(sqrt(1/(c^2*x^2) + 1)
+ 1/(c*x))/(x^2*e + d)^(3/2), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(x^2*e + d)/(x^4*e^2 + 2*d*x^2*e + d^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(3/2),x)

[Out]

int((x^2*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(3/2), x)

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