Optimal. Leaf size=102 \[ \frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}+\frac {b \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac {b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^3 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {5938, 5903,
4267, 2317, 2438, 75} \begin {gather*} \frac {2 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac {b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac {b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 2317
Rule 2438
Rule 4267
Rule 5903
Rule 5938
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{c^2}+\frac {b \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c d}\\ &=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}-\frac {\text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 d}\\ &=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}+\frac {b \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 d}-\frac {b \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 d}\\ &=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^3 d}\\ &=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 d}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}+\frac {b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^3 d}-\frac {b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 155, normalized size = 1.52 \begin {gather*} \frac {-2 a c x+2 b \sqrt {\frac {-1+c x}{1+c x}}+2 b c x \sqrt {\frac {-1+c x}{1+c x}}-2 b c x \cosh ^{-1}(c x)+b \cosh ^{-1}(c x)^2+2 b \cosh ^{-1}(c x) \log \left (1+e^{-\cosh ^{-1}(c x)}\right )-2 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )-a \log (1-c x)+a \log (1+c x)-2 b \text {PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )-2 b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 4.25, size = 187, normalized size = 1.83
method | result | size |
derivativedivides | \(\frac {-\frac {a c x}{d}-\frac {a \ln \left (c x -1\right )}{2 d}+\frac {a \ln \left (c x +1\right )}{2 d}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) c x}{d}+\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{3}}\) | \(187\) |
default | \(\frac {-\frac {a c x}{d}-\frac {a \ln \left (c x -1\right )}{2 d}+\frac {a \ln \left (c x +1\right )}{2 d}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) c x}{d}+\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}}{c^{3}}\) | \(187\) |
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 {a x^{2}}{c^{2} x^{2} - 1}\, dx + \int \frac {b x^{2} \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \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 = -0.01 \begin {gather*} \int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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