3.4.53 \(\int \frac {1}{(c+a^2 c x^2) \sinh ^{-1}(a x)} \, dx\) [353]

Optimal. Leaf size=22 \[ \text {Int}\left (\frac {1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)},x\right ) \]

[Out]

Unintegrable(1/(a^2*c*x^2+c)/arcsinh(a*x),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 \frac {1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)} \, dx \end {gather*}

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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{2} c \,x^{2}+c \right ) \arcsinh \left (a x \right )}\, dx\]

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 = 0.00, 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(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 = 0.00, 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(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{2} x^{2} \operatorname {asinh}{\left (a x \right )} + \operatorname {asinh}{\left (a x \right )}}\, dx}{c} \end {gather*}

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 = 0.00, 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(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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{\mathrm {asinh}\left (a\,x\right )\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(asinh(a*x)*(c + a^2*c*x^2)),x)

[Out]

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

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