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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0418289, 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 = {} \[ \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.720636, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\it Arcsinh} \left ( ax \right ) }}}}\, 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)^(1/2),x)

[Out]

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

________________________________________________________________________________________

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}} \sqrt{\operatorname{arsinh}\left (a x\right )}}\,{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)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \sqrt{\operatorname{asinh}{\left (a x \right )}}}\, dx \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)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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}} \sqrt{\operatorname{arsinh}\left (a x\right )}}\,{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)^(1/2),x, algorithm="giac")

[Out]

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