Optimal. Leaf size=388 \[ -\frac {488 c^2 \sqrt {-1+a x} \sqrt {1+a x}}{135 a}+\frac {8}{135} a c^2 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {298}{75} c^2 x \cosh ^{-1}(a x)-\frac {76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3 \]
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Rubi [A]
time = 0.60, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5897, 5879,
5915, 75, 5889, 5894, 12, 471, 200, 534, 1261, 712} \begin {gather*} \frac {6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac {76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac {6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {8}{135} a c^2 x^2 \sqrt {a x-1} \sqrt {a x+1}-\frac {488 c^2 \sqrt {a x-1} \sqrt {a x+1}}{135 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac {298}{75} c^2 x \cosh ^{-1}(a x)-\frac {3 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {4 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac {8 c^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{5 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 200
Rule 471
Rule 534
Rule 712
Rule 1261
Rule 5879
Rule 5889
Rule 5894
Rule 5897
Rule 5915
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^3 \, dx &=\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^3 \, dx-\frac {1}{5} \left (3 a c^2\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2 \, dx\\ &=-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{25} \left (6 c^2\right ) \int \left (-1+a^2 x^2\right )^2 \cosh ^{-1}(a x) \, dx+\frac {1}{15} \left (8 c^2\right ) \int \cosh ^{-1}(a x)^3 \, dx+\frac {1}{5} \left (4 a c^2\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac {6}{25} c^2 x \cosh ^{-1}(a x)-\frac {4}{25} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac {1}{15} \left (8 c^2\right ) \int \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \, dx-\frac {1}{25} \left (6 a c^2\right ) \int \frac {x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt {-1+a x} \sqrt {1+a x}} \, dx-\frac {1}{5} \left (8 a c^2\right ) \int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {58}{75} c^2 x \cosh ^{-1}(a x)-\frac {76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{5} \left (16 c^2\right ) \int \cosh ^{-1}(a x) \, dx-\frac {1}{125} \left (2 a c^2\right ) \int \frac {x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx+\frac {1}{15} \left (8 a c^2\right ) \int \frac {x \left (-3+a^2 x^2\right )}{3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {298}{75} c^2 x \cosh ^{-1}(a x)-\frac {76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac {1}{45} \left (8 a c^2\right ) \int \frac {x \left (-3+a^2 x^2\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx-\frac {1}{5} \left (16 a c^2\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx-\frac {\left (2 a c^2 \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt {-1+a^2 x^2}} \, dx}{125 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a}+\frac {8}{135} a c^2 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {298}{75} c^2 x \cosh ^{-1}(a x)-\frac {76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac {1}{135} \left (56 a c^2\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx-\frac {\left (a c^2 \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {15-10 a^2 x+3 a^4 x^2}{\sqrt {-1+a^2 x}} \, dx,x,x^2\right )}{125 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {488 c^2 \sqrt {-1+a x} \sqrt {1+a x}}{135 a}+\frac {8}{135} a c^2 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {298}{75} c^2 x \cosh ^{-1}(a x)-\frac {76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac {\left (a c^2 \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {-1+a^2 x}}-4 \sqrt {-1+a^2 x}+3 \left (-1+a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{125 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {488 c^2 \sqrt {-1+a x} \sqrt {1+a x}}{135 a}+\frac {8}{135} a c^2 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {298}{75} c^2 x \cosh ^{-1}(a x)-\frac {76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac {6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac {8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 147, normalized size = 0.38 \begin {gather*} \frac {c^2 \left (-2 \sqrt {-1+a x} \sqrt {1+a x} \left (31841-842 a^2 x^2+81 a^4 x^4\right )+30 a x \left (2235-190 a^2 x^2+27 a^4 x^4\right ) \cosh ^{-1}(a x)-225 \sqrt {-1+a x} \sqrt {1+a x} \left (149-38 a^2 x^2+9 a^4 x^4\right ) \cosh ^{-1}(a x)^2+1125 a x \left (15-10 a^2 x^2+3 a^4 x^4\right ) \cosh ^{-1}(a x)^3\right )}{16875 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (-a^{2} c \,x^{2}+c \right )^{2} \mathrm {arccosh}\left (a x \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 210, normalized size = 0.54 \begin {gather*} -\frac {1}{75} \, {\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 )^{2} + \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 )^{3} - \frac {2}{16875} \, {\left (81 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 842 \, \sqrt {a^{2} x^{2} - 1} c^{2} x^{2} - \frac {15 \, {\left (27 \, a^{4} c^{2} x^{5} - 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right )}{a} + \frac {31841 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 204, normalized size = 0.53 \begin {gather*} \frac {1125 \, {\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 )^{3} - 225 \, {\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 )^{2} + 30 \, {\left (27 \, a^{5} c^{2} x^{5} - 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (81 \, a^{4} c^{2} x^{4} - 842 \, a^{2} c^{2} x^{2} + 31841 \, c^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{16875 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.74, size = 274, normalized size = 0.71 \begin {gather*} \begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {acosh}^{3}{\left (a x \right )}}{5} + \frac {6 a^{4} c^{2} x^{5} \operatorname {acosh}{\left (a x \right )}}{125} - \frac {3 a^{3} c^{2} x^{4} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{25} - \frac {6 a^{3} c^{2} x^{4} \sqrt {a^{2} x^{2} - 1}}{625} - \frac {2 a^{2} c^{2} x^{3} \operatorname {acosh}^{3}{\left (a x \right )}}{3} - \frac {76 a^{2} c^{2} x^{3} \operatorname {acosh}{\left (a x \right )}}{225} + \frac {38 a c^{2} x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{75} + \frac {1684 a c^{2} x^{2} \sqrt {a^{2} x^{2} - 1}}{16875} + c^{2} x \operatorname {acosh}^{3}{\left (a x \right )} + \frac {298 c^{2} x \operatorname {acosh}{\left (a x \right )}}{75} - \frac {149 c^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{75 a} - \frac {63682 c^{2} \sqrt {a^{2} x^{2} - 1}}{16875 a} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} c^{2} x}{8} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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