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

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

[Out]

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

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 23.25, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e x^{5} + d x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arccsch(c*x) + a)/(e*x^5 + d*x^3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{3} \sqrt {e \,x^{2}+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {e \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {e x^{2} + d}}{d x^{2}}\right )} + b \int \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}{\sqrt {e x^{2} + d} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^3/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/x**3/(e*x**2+d)**(1/2),x)

[Out]

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

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