Optimal. Leaf size=29 \[ \text {Int}\left (\frac {x}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]
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Rubi [A] time = 0.40, 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 = {} \[ \int \frac {x}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 7.16, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{a c^{4} x^{4} - 2 \, a c^{2} x^{2} + {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \operatorname {arcosh}\left (c x\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\, dx \]
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 \[ \int \frac {x}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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