Optimal. Leaf size=58 \[ -\frac {a \text {Int}\left (\frac {x}{\left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)},x\right )}{c}-\frac {1}{a c \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.10, 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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^2} \, dx \]
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 ) \sinh ^{-1}(a x)^2} \, dx &=-\frac {1}{a c \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}-\frac {a \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)} \, dx}{c}\\ \end {align*}
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Mathematica [A] time = 1.53, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{2} c x^{2} + c\right )} \operatorname {arsinh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )} \operatorname {arsinh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} c \,x^{2}+c \right ) \arcsinh \left (a x \right )^{2}}\, dx \]
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 \[ -\frac {a x + \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} c x + a c\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )} - \int \frac {a^{4} x^{4} + {\left (a^{2} x^{2} + 1\right )}^{2} + {\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {a^{2} x^{2} + 1} - 1}{{\left (a^{6} c x^{6} + 3 \, a^{4} c x^{4} + 3 \, a^{2} c x^{2} + {\left (a^{4} c x^{4} + a^{2} c x^{2}\right )} {\left (a^{2} x^{2} + 1\right )} + 2 \, {\left (a^{5} c x^{5} + 2 \, a^{3} c x^{3} + a c x\right )} \sqrt {a^{2} x^{2} + 1} + c\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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 \[ \frac {\int \frac {1}{a^{2} x^{2} \operatorname {asinh}^{2}{\left (a x \right )} + \operatorname {asinh}^{2}{\left (a x \right )}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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