Optimal. Leaf size=266 \[ -\frac {2}{343} a^6 c^3 x^7+\frac {234 a^4 c^3 x^5}{6125}-\frac {1514 a^2 c^3 x^3}{11025}+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac {2 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)}{49 a}-\frac {12 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac {16 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac {32 c^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{35 a}+\frac {4322 c^3 x}{3675} \]
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Rubi [A] time = 0.68, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ -\frac {2}{343} a^6 c^3 x^7+\frac {234 a^4 c^3 x^5}{6125}-\frac {1514 a^2 c^3 x^3}{11025}+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac {2 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)}{49 a}-\frac {12 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac {16 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac {32 c^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{35 a}+\frac {4322 c^3 x}{3675} \]
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 )^3 \cosh ^{-1}(a x)^2 \, dx &=\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac {1}{7} (6 c) \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^2 \, dx+\frac {1}{7} \left (2 a c^3\right ) \int x (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x) \, dx\\ &=\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac {1}{35} \left (24 c^2\right ) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2 \, dx-\frac {1}{49} \left (2 c^3\right ) \int \left (-1+a^2 x^2\right )^3 \, dx-\frac {1}{35} \left (12 a c^3\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx\\ &=-\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2-\frac {1}{49} \left (2 c^3\right ) \int \left (-1+3 a^2 x^2-3 a^4 x^4+a^6 x^6\right ) \, dx+\frac {1}{175} \left (12 c^3\right ) \int \left (-1+a^2 x^2\right )^2 \, dx+\frac {1}{35} \left (16 c^3\right ) \int \cosh ^{-1}(a x)^2 \, dx+\frac {1}{35} \left (16 a c^3\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \, dx\\ &=\frac {2 c^3 x}{49}-\frac {2}{49} a^2 c^3 x^3+\frac {6}{245} a^4 c^3 x^5-\frac {2}{343} a^6 c^3 x^7+\frac {16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac {1}{175} \left (12 c^3\right ) \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac {1}{105} \left (16 c^3\right ) \int \left (-1+a^2 x^2\right ) \, dx-\frac {1}{35} \left (32 a c^3\right ) \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {962 c^3 x}{3675}-\frac {1514 a^2 c^3 x^3}{11025}+\frac {234 a^4 c^3 x^5}{6125}-\frac {2}{343} a^6 c^3 x^7-\frac {32 c^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{35 a}+\frac {16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2+\frac {1}{35} \left (32 c^3\right ) \int 1 \, dx\\ &=\frac {4322 c^3 x}{3675}-\frac {1514 a^2 c^3 x^3}{11025}+\frac {234 a^4 c^3 x^5}{6125}-\frac {2}{343} a^6 c^3 x^7-\frac {32 c^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{35 a}+\frac {16 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac {12 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac {2 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)}{49 a}+\frac {16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.28, size = 125, normalized size = 0.47 \[ \frac {c^3 \left (-2250 a^7 x^7+14742 a^5 x^5-52990 a^3 x^3-11025 a x \left (5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right ) \cosh ^{-1}(a x)^2+210 \sqrt {a x-1} \sqrt {a x+1} \left (75 a^6 x^6-351 a^4 x^4+757 a^2 x^2-2161\right ) \cosh ^{-1}(a x)+453810 a x\right )}{385875 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 175, normalized size = 0.66 \[ -\frac {2250 \, a^{7} c^{3} x^{7} - 14742 \, a^{5} c^{3} x^{5} + 52990 \, a^{3} c^{3} x^{3} - 453810 \, a c^{3} x + 11025 \, {\left (5 \, a^{7} c^{3} x^{7} - 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} - 35 \, a c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 210 \, {\left (75 \, a^{6} c^{3} x^{6} - 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} - 2161 \, c^{3}\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{385875 \, a} \]
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 = 0.21, size = 188, normalized size = 0.71 \[ -\frac {c^{3} \left (55125 \mathrm {arccosh}\left (a x \right )^{2} a^{7} x^{7}-15750 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}-231525 \mathrm {arccosh}\left (a x \right )^{2} a^{5} x^{5}+73710 \,\mathrm {arccosh}\left (a x \right ) a^{4} x^{4} \sqrt {a x -1}\, \sqrt {a x +1}+2250 x^{7} a^{7}+385875 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{2}-158970 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{2} x^{2}-14742 x^{5} a^{5}-385875 a x \mathrm {arccosh}\left (a x \right )^{2}+453810 \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )+52990 x^{3} a^{3}-453810 a x \right )}{385875 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 178, normalized size = 0.67 \[ -\frac {2}{343} \, a^{6} c^{3} x^{7} + \frac {234}{6125} \, a^{4} c^{3} x^{5} - \frac {1514}{11025} \, a^{2} c^{3} x^{3} + \frac {4322}{3675} \, c^{3} x + \frac {2}{3675} \, {\left (75 \, \sqrt {a^{2} x^{2} - 1} a^{4} c^{3} x^{6} - 351 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{3} x^{4} + 757 \, \sqrt {a^{2} x^{2} - 1} c^{3} x^{2} - \frac {2161 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right ) - \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} - 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} - 35 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {acosh}\left (a\,x\right )}^2\,{\left (c-a^2\,c\,x^2\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.25, size = 243, normalized size = 0.91 \[ \begin {cases} - \frac {a^{6} c^{3} x^{7} \operatorname {acosh}^{2}{\left (a x \right )}}{7} - \frac {2 a^{6} c^{3} x^{7}}{343} + \frac {2 a^{5} c^{3} x^{6} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{49} + \frac {3 a^{4} c^{3} x^{5} \operatorname {acosh}^{2}{\left (a x \right )}}{5} + \frac {234 a^{4} c^{3} x^{5}}{6125} - \frac {234 a^{3} c^{3} x^{4} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{1225} - a^{2} c^{3} x^{3} \operatorname {acosh}^{2}{\left (a x \right )} - \frac {1514 a^{2} c^{3} x^{3}}{11025} + \frac {1514 a c^{3} x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{3675} + c^{3} x \operatorname {acosh}^{2}{\left (a x \right )} + \frac {4322 c^{3} x}{3675} - \frac {4322 c^{3} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{3675 a} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} c^{3} x}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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