Optimal. Leaf size=241 \[ \frac {x \cosh ^{-1}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5899, 5913,
3797, 2221, 2611, 2320, 6724} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 5899
Rule 5913
Rule 6724
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.18, size = 145, normalized size = 0.60 \begin {gather*} \frac {x \cosh ^{-1}(a x)^3+\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (\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 (1+e^{\cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )+6 \text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )+6 \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )\right )}{a}}{c \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(547\) vs.
\(2(252)=504\).
time = 2.53, size = 548, normalized size = 2.27
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-\sqrt {a x -1}\, \sqrt {a x +1}+a x \right ) \mathrm {arccosh}\left (a x \right )^{3}}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{3}}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {6 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {6 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {6 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {6 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}\) | \(548\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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