Optimal. Leaf size=195 \[ \frac {2}{125} a^4 c^2 x^5-\frac {76}{675} a^2 c^2 x^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2-\frac {2 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {16 c^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{15 a}+\frac {298 c^2 x}{225} \]
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Rubi [A] time = 0.46, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5681, 5718, 194, 5654, 8} \[ \frac {2}{125} a^4 c^2 x^5-\frac {76}{675} a^2 c^2 x^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2-\frac {2 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {16 c^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{15 a}+\frac {298 c^2 x}{225} \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
Rule 5654
Rule 5681
Rule 5718
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^2 \, dx &=\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2 \, dx-\frac {1}{5} \left (2 a c^2\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx\\ &=-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{25} \left (2 c^2\right ) \int \left (-1+a^2 x^2\right )^2 \, dx+\frac {1}{15} \left (8 c^2\right ) \int \cosh ^{-1}(a x)^2 \, dx+\frac {1}{15} \left (8 a c^2\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \, dx\\ &=\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{25} \left (2 c^2\right ) \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac {1}{45} \left (8 c^2\right ) \int \left (-1+a^2 x^2\right ) \, dx-\frac {1}{15} \left (16 a c^2\right ) \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {58 c^2 x}{225}-\frac {76}{675} a^2 c^2 x^3+\frac {2}{125} a^4 c^2 x^5-\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{15 a}+\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{15} \left (16 c^2\right ) \int 1 \, dx\\ &=\frac {298 c^2 x}{225}-\frac {76}{675} a^2 c^2 x^3+\frac {2}{125} a^4 c^2 x^5-\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{15 a}+\frac {8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac {2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.23, size = 101, normalized size = 0.52 \[ \frac {c^2 \left (54 a^5 x^5-380 a^3 x^3+225 a x \left (3 a^4 x^4-10 a^2 x^2+15\right ) \cosh ^{-1}(a x)^2-30 \sqrt {a x-1} \sqrt {a x+1} \left (9 a^4 x^4-38 a^2 x^2+149\right ) \cosh ^{-1}(a x)+4470 a x\right )}{3375 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 142, normalized size = 0.73 \[ \frac {54 \, a^{5} c^{2} x^{5} - 380 \, a^{3} c^{2} x^{3} + 4470 \, a c^{2} x + 225 \, {\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 30 \, {\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{3375 \, a} \]
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 = 0.09, size = 140, normalized size = 0.72 \[ \frac {c^{2} \left (675 \mathrm {arccosh}\left (a x \right )^{2} a^{5} x^{5}-270 \,\mathrm {arccosh}\left (a x \right ) a^{4} x^{4} \sqrt {a x -1}\, \sqrt {a x +1}-2250 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{2}+1140 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{2} x^{2}+54 x^{5} a^{5}+3375 a x \mathrm {arccosh}\left (a x \right )^{2}-4470 \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )-380 x^{3} a^{3}+4470 a x \right )}{3375 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 134, normalized size = 0.69 \[ \frac {2}{125} \, a^{4} c^{2} x^{5} - \frac {76}{675} \, a^{2} c^{2} x^{3} + \frac {298}{225} \, c^{2} x - \frac {2}{225} \, {\left (9 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 38 \, \sqrt {a^{2} x^{2} - 1} c^{2} x^{2} + \frac {149 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right ) + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {acosh}\left (a\,x\right )}^2\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.90, size = 182, normalized size = 0.93 \[ \begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {acosh}^{2}{\left (a x \right )}}{5} + \frac {2 a^{4} c^{2} x^{5}}{125} - \frac {2 a^{3} c^{2} x^{4} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{25} - \frac {2 a^{2} c^{2} x^{3} \operatorname {acosh}^{2}{\left (a x \right )}}{3} - \frac {76 a^{2} c^{2} x^{3}}{675} + \frac {76 a c^{2} x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{225} + c^{2} x \operatorname {acosh}^{2}{\left (a x \right )} + \frac {298 c^{2} x}{225} - \frac {298 c^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{225 a} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} c^{2} x}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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