Optimal. Leaf size=175 \[ -\frac {122 c \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {2}{27} a c x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {14}{3} c x \cosh ^{-1}(a x)-\frac {2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3 \]
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Rubi [A]
time = 0.30, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5897, 5879,
5915, 75, 5889, 5894, 12, 471} \begin {gather*} -\frac {2}{9} a^2 c x^3 \cosh ^{-1}(a x)+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {2}{27} a c x^2 \sqrt {a x-1} \sqrt {a x+1}-\frac {122 c \sqrt {a x-1} \sqrt {a x+1}}{27 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^3+\frac {14}{3} c x \cosh ^{-1}(a x)+\frac {c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{3 a}-\frac {2 c \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 471
Rule 5879
Rule 5889
Rule 5894
Rule 5897
Rule 5915
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^3 \, dx &=\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{3} (2 c) \int \cosh ^{-1}(a x)^3 \, dx+(a c) \int x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3-\frac {1}{3} (2 c) \int \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \, dx-(2 a c) \int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {2}{3} c x \cosh ^{-1}(a x)-\frac {2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+(4 c) \int \cosh ^{-1}(a x) \, dx+\frac {1}{3} (2 a c) \int \frac {x \left (-3+a^2 x^2\right )}{3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {14}{3} c x \cosh ^{-1}(a x)-\frac {2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac {1}{9} (2 a c) \int \frac {x \left (-3+a^2 x^2\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx-(4 a c) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {4 c \sqrt {-1+a x} \sqrt {1+a x}}{a}+\frac {2}{27} a c x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {14}{3} c x \cosh ^{-1}(a x)-\frac {2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3-\frac {1}{27} (14 a c) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {122 c \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {2}{27} a c x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {14}{3} c x \cosh ^{-1}(a x)-\frac {2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac {2 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a}+\frac {c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 109, normalized size = 0.62 \begin {gather*} \frac {c \left (2 \sqrt {-1+a x} \sqrt {1+a x} \left (-61+a^2 x^2\right )-6 a x \left (-21+a^2 x^2\right ) \cosh ^{-1}(a x)+9 \sqrt {-1+a x} \sqrt {1+a x} \left (-7+a^2 x^2\right ) \cosh ^{-1}(a x)^2-9 a x \left (-3+a^2 x^2\right ) \cosh ^{-1}(a x)^3\right )}{27 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.18, size = 140, normalized size = 0.80 \[-\frac {c \left (9 x^{3} a^{3} \mathrm {arccosh}\left (a x \right )^{3}-9 x^{2} a^{2} \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+6 \,\mathrm {arccosh}\left (a x \right ) x^{3} a^{3}-27 \mathrm {arccosh}\left (a x \right )^{3} x a -2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+63 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-126 a x \,\mathrm {arccosh}\left (a x \right )+122 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{27 a}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 124, normalized size = 0.71 \begin {gather*} \frac {1}{3} \, {\left (\sqrt {a^{2} x^{2} - 1} c x^{2} - \frac {7 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right )^{2} - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \operatorname {arcosh}\left (a x\right )^{3} + \frac {2}{27} \, {\left (\sqrt {a^{2} x^{2} - 1} c x^{2} - \frac {3 \, {\left (a^{2} c x^{3} - 21 \, c x\right )} \operatorname {arcosh}\left (a x\right )}{a} - \frac {61 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 140, normalized size = 0.80 \begin {gather*} -\frac {9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 9 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (a^{3} c x^{3} - 21 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{2} c x^{2} - 61 \, c\right )} \sqrt {a^{2} x^{2} - 1}}{27 \, 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 = 0.32, size = 160, normalized size = 0.91 \begin {gather*} \begin {cases} - \frac {a^{2} c x^{3} \operatorname {acosh}^{3}{\left (a x \right )}}{3} - \frac {2 a^{2} c x^{3} \operatorname {acosh}{\left (a x \right )}}{9} + \frac {a c x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{3} + \frac {2 a c x^{2} \sqrt {a^{2} x^{2} - 1}}{27} + c x \operatorname {acosh}^{3}{\left (a x \right )} + \frac {14 c x \operatorname {acosh}{\left (a x \right )}}{3} - \frac {7 c \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{3 a} - \frac {122 c \sqrt {a^{2} x^{2} - 1}}{27 a} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} c 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 = -0.01 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^3\,\left (c-a^2\,c\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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