3.505 \(\int \frac{1}{(c+a^2 c x^2)^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{4 a \sqrt{a^2 x^2+1} \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2+1\right )^2 \sqrt{\sinh ^{-1}(a x)}},x\right )}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{a \left (a^2 c x^2+c\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[1 + a^2*x^2])/(a*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]]) - (4*a*Sqrt[1 + a^2*x^2]*Unintegrable[x/((
1 + a^2*x^2)^2*Sqrt[ArcSinh[a*x]]), x])/(c*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.089694, 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 = {} \[ \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2)),x]

[Out]

(-2*Sqrt[1 + a^2*x^2])/(a*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]]) - (4*a*Sqrt[1 + a^2*x^2]*Defer[Int][x/((1
+ a^2*x^2)^2*Sqrt[ArcSinh[a*x]]), x])/(c*Sqrt[c + a^2*c*x^2])

Rubi steps

\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 \sqrt{1+a^2 x^2}}{a \left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}}-\frac{\left (4 a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^2 \sqrt{\sinh ^{-1}(a x)}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.710224, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2)),x]

[Out]

Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2)), x]

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Maple [A]  time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(3/2),x)

[Out]

int(1/(a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(3/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a^2*c*x^2 + c)^(3/2)*arcsinh(a*x)^(3/2)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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(1/(a**2*c*x**2+c)**(3/2)/asinh(a*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*c*x^2+c)^(3/2)/arcsinh(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a^2*c*x^2 + c)^(3/2)*arcsinh(a*x)^(3/2)), x)