Optimal. Leaf size=114 \[ -\frac {8 a \sqrt {a x-1} \sqrt {a x+1} \text {Int}\left (\frac {x}{\left (a^2 x^2-1\right )^3 \cosh ^{-1}(a x)^{3/2}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.23, 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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx \]
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 )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{(-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {2 \sqrt {-1+a x}}{3 a c^2 (1-a x)^2 (1+a x)^{3/2} \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}-\frac {\left (8 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^3 \cosh ^{-1}(a x)^{3/2}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 2.36, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [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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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.75, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}\,{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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