Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\left (1-c^2 x^2\right )^{3/2} x^m}{a+b \cosh ^{-1}(c x)},x\right ) \]
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Rubi [A] time = 0.54, 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 = {} \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx &=-\frac {\sqrt {1-c^2 x^2} \int \frac {x^m (-1+c x)^{3/2} (1+c x)^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} x^{m}}{b \operatorname {arcosh}\left (c x\right ) + a}, x\right ) \]
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 = 1.17, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{a +b \,\mathrm {arccosh}\left (c x \right )}\, dx \]
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 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m}}{b \operatorname {arcosh}\left (c x\right ) + a}\,{d x} \]
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 \[ \int \frac {x^m\,{\left (1-c^2\,x^2\right )}^{3/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]
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 \[ \int \frac {x^{m} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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