Optimal. Leaf size=241 \[ -\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {a x-1} \sqrt {a x+1} \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.35, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5713, 5688, 5715, 3716, 2190, 2531, 2282, 6589} \[ -\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 5688
Rule 5713
Rule 5715
Rule 6589
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^3}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\left (3 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \cosh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x^2 \coth (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (6 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (6 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 145, normalized size = 0.60 \[ \frac {\frac {\sqrt {a x-1} \sqrt {a x+1} \left (-6 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )+6 \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )+6 \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^3-3 \cosh ^{-1}(a x)^2 \log \left (1-e^{\cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \log \left (e^{\cosh ^{-1}(a x)}+1\right )\right )}{a}+x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{3}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.39, size = 548, normalized size = 2.27 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-\sqrt {a x -1}\, \sqrt {a x +1}+a x \right ) \mathrm {arccosh}\left (a x \right )^{3}}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{3}}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {6 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {6 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {6 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {6 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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