Optimal. Leaf size=112 \[ -\frac {2}{27} a^2 c x^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {2 c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{9 a}-\frac {4 c \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{3 a}+\frac {14 c x}{9} \]
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Rubi [A] time = 0.26, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5681, 5718, 5654, 8} \[ -\frac {2}{27} a^2 c x^3+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {2 c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{9 a}-\frac {4 c \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{3 a}+\frac {14 c x}{9} \]
Antiderivative was successfully verified.
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Rule 8
Rule 5654
Rule 5681
Rule 5718
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2 \, dx &=\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{3} (2 c) \int \cosh ^{-1}(a x)^2 \, dx+\frac {1}{3} (2 a c) \int x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \, dx\\ &=\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2-\frac {1}{9} (2 c) \int \left (-1+a^2 x^2\right ) \, dx-\frac {1}{3} (4 a c) \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {2 c x}{9}-\frac {2}{27} a^2 c x^3-\frac {4 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 a}+\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac {1}{3} (4 c) \int 1 \, dx\\ &=\frac {14 c x}{9}-\frac {2}{27} a^2 c x^3-\frac {4 c \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 a}+\frac {2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac {2}{3} c x \cosh ^{-1}(a x)^2+\frac {1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.12, size = 73, normalized size = 0.65 \[ \frac {c \left (-2 a^3 x^3-9 a x \left (a^2 x^2-3\right ) \cosh ^{-1}(a x)^2+6 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 x^2-7\right ) \cosh ^{-1}(a x)+42 a x\right )}{27 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 95, normalized size = 0.85 \[ -\frac {2 \, a^{3} c x^{3} - 42 \, a c x + 9 \, {\left (a^{3} c x^{3} - 3 \, a c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a^{2} c x^{2} - 7 \, c\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{27 \, a} \]
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 = 0.07, size = 90, normalized size = 0.80 \[ -\frac {c \left (9 a^{3} x^{3} \mathrm {arccosh}\left (a x \right )^{2}-6 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, a^{2} x^{2}-27 a x \mathrm {arccosh}\left (a x \right )^{2}+42 \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )+2 x^{3} a^{3}-42 a x \right )}{27 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 76, normalized size = 0.68 \[ -\frac {2}{27} \, a^{2} c x^{3} + \frac {2}{9} \, {\left (\sqrt {a^{2} x^{2} - 1} c x^{2} - \frac {7 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right ) - \frac {1}{3} \, {\left (a^{2} c x^{3} - 3 \, c x\right )} \operatorname {arcosh}\left (a x\right )^{2} + \frac {14}{9} \, c x \]
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 \[ \int {\mathrm {acosh}\left (a\,x\right )}^2\,\left (c-a^2\,c\,x^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 105, normalized size = 0.94 \[ \begin {cases} - \frac {a^{2} c x^{3} \operatorname {acosh}^{2}{\left (a x \right )}}{3} - \frac {2 a^{2} c x^{3}}{27} + \frac {2 a c x^{2} \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{9} + c x \operatorname {acosh}^{2}{\left (a x \right )} + \frac {14 c x}{9} - \frac {14 c \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{9 a} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} c x}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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