Optimal. Leaf size=22 \[ \text {Int}\left (\frac {1}{\left (a^2 c x^2+c\right )^2 \sinh ^{-1}(a x)},x\right ) \]
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Rubi [A] time = 0.03, 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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)} \, dx &=\int \frac {1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)} \, dx\\ \end {align*}
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Mathematica [A] time = 1.61, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \operatorname {arsinh}\left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \operatorname {arsinh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{2} \arcsinh \left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \operatorname {arsinh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{\mathrm {asinh}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{4} x^{4} \operatorname {asinh}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {asinh}{\left (a x \right )} + \operatorname {asinh}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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