Optimal. Leaf size=605 \[ \frac{65 a^3 c^2 x^4 \sqrt{c-a^2 c x^2}}{2304 \sqrt{a x-1} \sqrt{a x+1}}-\frac{865 a c^2 x^2 \sqrt{c-a^2 c x^2}}{2304 \sqrt{a x-1} \sqrt{a x+1}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2}}{216 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 a c^2 x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{245}{384} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{1}{36} c^2 x (1-a x)^2 (a x+1)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{65}{576} c^2 x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{5 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{5 c^2 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{115 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{768 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{6} x \left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^3+\frac{5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^3 \]
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Rubi [A] time = 1.49667, antiderivative size = 636, normalized size of antiderivative = 1.05, number of steps used = 25, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {5713, 5685, 5683, 5676, 5662, 5759, 30, 5716, 14, 261} \[ \frac{65 a^3 c^2 x^4 \sqrt{c-a^2 c x^2}}{2304 \sqrt{a x-1} \sqrt{a x+1}}-\frac{865 a c^2 x^2 \sqrt{c-a^2 c x^2}}{2304 \sqrt{a x-1} \sqrt{a x+1}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2}}{216 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 a c^2 x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{1}{6} c^2 x (1-a x)^2 (a x+1)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{5}{24} c^2 x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{245}{384} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{1}{36} c^2 x (1-a x)^2 (a x+1)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{65}{576} c^2 x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{5 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{5 c^2 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{115 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{768 a \sqrt{a x-1} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5685
Rule 5683
Rule 5676
Rule 5662
Rule 5759
Rule 30
Rule 5716
Rule 14
Rule 261
Rubi steps
\begin{align*} \int \left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^3 \, dx &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^3 \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{\left (5 c^2 \sqrt{c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^3 \, dx}{6 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right )^2 \cosh ^{-1}(a x)^2 \, dx}{2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5}{24} c^2 x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x) \, dx}{6 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (5 c^2 \sqrt{c-a^2 c x^2}\right ) \int \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (5 a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x)^2 \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{5 c^2 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{5}{24} c^2 x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{\left (5 c^2 \sqrt{c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx}{36 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (5 c^2 \sqrt{c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (5 c^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{\cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right )^2 \, dx}{36 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^2 \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2}}{216 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{65}{576} c^2 x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{15 a c^2 x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c^2 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{5}{24} c^2 x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{5 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (5 c^2 \sqrt{c-a^2 c x^2}\right ) \int \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \, dx}{48 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 c^2 \sqrt{c-a^2 c x^2}\right ) \int \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \, dx}{64 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (5 a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \, dx}{144 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (5 a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \, dx}{64 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 a^2 c^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2}}{216 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{245}{384} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{65}{576} c^2 x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{15 a c^2 x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c^2 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{5}{24} c^2 x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{5 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (5 c^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{96 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 c^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{128 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 c^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (5 a c^2 \sqrt{c-a^2 c x^2}\right ) \int \left (-x+a^2 x^3\right ) \, dx}{144 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (5 a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \, dx}{96 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (5 a c^2 \sqrt{c-a^2 c x^2}\right ) \int \left (-x+a^2 x^3\right ) \, dx}{64 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \, dx}{128 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 a c^2 \sqrt{c-a^2 c x^2}\right ) \int x \, dx}{32 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{865 a c^2 x^2 \sqrt{c-a^2 c x^2}}{2304 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{65 a^3 c^2 x^4 \sqrt{c-a^2 c x^2}}{2304 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2}}{216 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{245}{384} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{65}{576} c^2 x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{115 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{768 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 a c^2 x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c^2 \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{c^2 \left (1-a^2 x^2\right )^3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{5}{24} c^2 x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{5 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 1.15302, size = 189, normalized size = 0.31 \[ \frac{c^2 \sqrt{c-a^2 c x^2} \left (-4320 \cosh ^{-1}(a x)^4-72 \left (270 \cosh \left (2 \cosh ^{-1}(a x)\right )-27 \cosh \left (4 \cosh ^{-1}(a x)\right )+2 \cosh \left (6 \cosh ^{-1}(a x)\right )\right ) \cosh ^{-1}(a x)^2-9720 \cosh \left (2 \cosh ^{-1}(a x)\right )+243 \cosh \left (4 \cosh ^{-1}(a x)\right )-8 \cosh \left (6 \cosh ^{-1}(a x)\right )+288 \cosh ^{-1}(a x)^3 \left (45 \sinh \left (2 \cosh ^{-1}(a x)\right )-9 \sinh \left (4 \cosh ^{-1}(a x)\right )+\sinh \left (6 \cosh ^{-1}(a x)\right )\right )+12 \cosh ^{-1}(a x) \left (1620 \sinh \left (2 \cosh ^{-1}(a x)\right )-81 \sinh \left (4 \cosh ^{-1}(a x)\right )+4 \sinh \left (6 \cosh ^{-1}(a x)\right )\right )\right )}{55296 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.259, size = 887, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \operatorname{arcosh}\left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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