Optimal. Leaf size=258 \[ -\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \tanh ^{-1}(a x)}{6 a c^3}+\frac {3 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {3 \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3} \]
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Rubi [A]
time = 0.34, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5901, 5903,
4267, 2611, 2320, 6724, 5915, 35, 213, 41, 205} \begin {gather*} -\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {3 \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\cosh ^{-1}(a x)}{6 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}-\frac {5 \tanh ^{-1}(a x)}{6 a c^3}+\frac {3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 35
Rule 41
Rule 205
Rule 213
Rule 2320
Rule 2611
Rule 4267
Rule 5901
Rule 5903
Rule 5915
Rule 6724
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {a \int \frac {x \cosh ^{-1}(a x)}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{2 c^3}+\frac {3 \int \frac {\cosh ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac {\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{\left (-1+a^2 x^2\right )^2} \, dx}{6 c^3}+\frac {(3 a) \int \frac {x \cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{4 c^3}+\frac {3 \int \frac {\cosh ^{-1}(a x)^2}{c-a^2 c x^2} \, dx}{8 c^2}\\ &=-\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{12 c^3}+\frac {3 \int \frac {1}{-1+a^2 x^2} \, dx}{4 c^3}-\frac {3 \text {Subst}\left (\int x^2 \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}\\ &=-\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \tanh ^{-1}(a x)}{6 a c^3}+\frac {3 \text {Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}-\frac {3 \text {Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \tanh ^{-1}(a x)}{6 a c^3}+\frac {3 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \text {Subst}\left (\int \text {Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac {3 \text {Subst}\left (\int \text {Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \tanh ^{-1}(a x)}{6 a c^3}+\frac {3 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}\\ &=-\frac {x}{12 c^3 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac {3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {5 \tanh ^{-1}(a x)}{6 a c^3}+\frac {3 \cosh ^{-1}(a x) \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \cosh ^{-1}(a x) \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac {3 \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac {3 \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}\\ \end {align*}
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Mathematica [A]
time = 3.04, size = 319, normalized size = 1.24 \begin {gather*} -\frac {80 \cosh ^{-1}(a x) \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )+2 \left (-2+9 \cosh ^{-1}(a x)^2\right ) \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x) \text {csch}^4\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-3 \cosh ^{-1}(a x)^2 \text {csch}^4\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-160 \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )+72 \left (\cosh ^{-1}(a x)^2 \log \left (1-e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \log \left (1+e^{-\cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-e^{-\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,e^{-\cosh ^{-1}(a x)}\right )+2 \text {PolyLog}\left (3,-e^{-\cosh ^{-1}(a x)}\right )-2 \text {PolyLog}\left (3,e^{-\cosh ^{-1}(a x)}\right )\right )+2 \left (-2+9 \cosh ^{-1}(a x)^2\right ) \text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )+3 \cosh ^{-1}(a x)^2 \text {sech}^4\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-\frac {32 \cosh ^{-1}(a x) \sinh ^4\left (\frac {1}{2} \cosh ^{-1}(a x)\right )}{\left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3}-80 \cosh ^{-1}(a x) \tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )}{192 a c^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 3.49, size = 320, normalized size = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {18 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, x^{2} a^{2}+9 \mathrm {arccosh}\left (a x \right )^{2} x^{3} a^{3}-22 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-15 \mathrm {arccosh}\left (a x \right )^{2} a x -2 a^{3} x^{3}+2 a x}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \arctanh \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{3 c^{3}}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {3 \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\) | \(320\) |
default | \(\frac {-\frac {18 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}\, x^{2} a^{2}+9 \mathrm {arccosh}\left (a x \right )^{2} x^{3} a^{3}-22 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-15 \mathrm {arccosh}\left (a x \right )^{2} a x -2 a^{3} x^{3}+2 a x}{24 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}-\frac {5 \arctanh \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{3 c^{3}}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {3 \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{8 c^{3}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}-\frac {3 \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{4 c^{3}}}{a}\) | \(320\) |
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*} - \frac {\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \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 )}^2}{{\left (c-a^2\,c\,x^2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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