3.2.24 \(\int \sqrt {d+e x^2} (a+b \text {csch}^{-1}(c x)) \, dx\) [124]

Optimal. Leaf size=23 \[ \text {Int}\left (\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ),x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.58, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.17, size = 0, normalized size = 0.00 \[\int \left (a +b \,\mathrm {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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