3.327 \(\int \frac{x^m \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0917486, 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 = {} \[ \int \frac{x^m \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.485458, size = 0, normalized size = 0. \[ \int \frac{x^m \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.249, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m} \operatorname{arsinh}\left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*arcsinh(a*x)^2/sqrt(a^2*x^2 + 1), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*asinh(a*x)**2/(a**2*x**2+1)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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