Optimal. Leaf size=124 \[ -\frac {\tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d^2}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac {b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}-\frac {b}{2 c^3 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.14, antiderivative size = 124, 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 = {5750, 74, 5694, 4182, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac {b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^3 d^2}-\frac {b}{2 c^3 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 74
Rule 2279
Rule 2391
Rule 4182
Rule 5694
Rule 5750
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}-\frac {\int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=-\frac {b}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3 d^2}\\ &=-\frac {b}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3 d^2}+\frac {b \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3 d^2}\\ &=-\frac {b}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d^2}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}\\ &=-\frac {b}{2 c^3 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^3 d^2}-\frac {b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}+\frac {b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 c^3 d^2}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 206, normalized size = 1.66 \[ \frac {-\frac {2 a c x}{c^2 x^2-1}+a \log (1-c x)-a \log (c x+1)-2 b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )+2 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )+\frac {b c x \sqrt {\frac {c x-1}{c x+1}}}{1-c x}+\frac {b \sqrt {\frac {c x-1}{c x+1}}}{1-c x}+b \sqrt {\frac {c x-1}{c x+1}}+\frac {b \cosh ^{-1}(c x)}{1-c x}-\frac {b \cosh ^{-1}(c x)}{c x+1}+2 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )-2 b \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )}{4 c^3 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \operatorname {arcosh}\left (c x\right ) + a x^{2}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 255, normalized size = 2.06 \[ -\frac {a}{4 c^{3} d^{2} \left (c x -1\right )}+\frac {a \ln \left (c x -1\right )}{4 c^{3} d^{2}}-\frac {a}{4 c^{3} d^{2} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{4 c^{3} d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) x}{2 c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c^{3} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{3} d^{2}}-\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{3} d^{2}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{3} d^{2}}+\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 c^{3} d^{2}} \]
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 \[ -\frac {1}{64} \, {\left (192 \, c^{3} \int \frac {x^{3} \log \left (c x - 1\right )}{8 \, {\left (c^{6} d^{2} x^{4} - 2 \, c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}}\,{d x} + 8 \, c^{2} {\left (\frac {2 \, x}{c^{6} d^{2} x^{2} - c^{4} d^{2}} + \frac {\log \left (c x + 1\right )}{c^{5} d^{2}} - \frac {\log \left (c x - 1\right )}{c^{5} d^{2}}\right )} - 64 \, c^{2} \int \frac {x^{2} \log \left (c x - 1\right )}{8 \, {\left (c^{6} d^{2} x^{4} - 2 \, c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}}\,{d x} + 3 \, {\left (c {\left (\frac {2}{c^{6} d^{2} x - c^{5} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{5} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{5} d^{2}}\right )} + \frac {4 \, \log \left (c x - 1\right )}{c^{6} d^{2} x^{2} - c^{4} d^{2}}\right )} c - \frac {4 \, {\left ({\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right )^{2} + 2 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) \log \left (c x - 1\right ) - 4 \, {\left (2 \, c x + {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) - {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )}}{c^{5} d^{2} x^{2} - c^{3} d^{2}} + 64 \, \int \frac {2 \, c x + {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) - {\left (c^{2} x^{2} - 1\right )} \log \left (c x - 1\right )}{4 \, {\left (c^{7} d^{2} x^{5} - 2 \, c^{5} d^{2} x^{3} + c^{3} d^{2} x + {\left (c^{6} d^{2} x^{4} - 2 \, c^{4} d^{2} x^{2} + c^{2} d^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )}}\,{d x} + 64 \, \int \frac {\log \left (c x - 1\right )}{8 \, {\left (c^{6} d^{2} x^{4} - 2 \, c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}}\,{d x}\right )} b - \frac {1}{4} \, a {\left (\frac {2 \, x}{c^{4} d^{2} x^{2} - c^{2} d^{2}} + \frac {\log \left (c x + 1\right )}{c^{3} d^{2}} - \frac {\log \left (c x - 1\right )}{c^{3} d^{2}}\right )} \]
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 \[ \int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a x^{2}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{2} \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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