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

Optimal. Leaf size=21 \[ \text{Unintegrable}\left (\frac{1}{\left (a^2 c x^2+c\right ) \sinh ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0262787, 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 = {} \[ \int \frac{1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.34705, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({a}^{2}c{x}^{2}+c \right ){\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)/arcsinh(a*x),x)

[Out]

int(1/(a^2*c*x^2+c)/arcsinh(a*x),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 )} \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)/arcsinh(a*x),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} \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)/arcsinh(a*x),x, algorithm="giac")

[Out]

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