Optimal. Leaf size=120 \[ -\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c d^2}+\frac {b \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 c d^2}-\frac {b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 c d^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5901, 5903,
4267, 2317, 2438, 75} \begin {gather*} \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{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 d^2}+\frac {b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 c d^2}-\frac {b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 c d^2}-\frac {b}{2 c d^2 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 2317
Rule 2438
Rule 4267
Rule 5901
Rule 5903
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {\text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{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 d^2}+\frac {b \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c d^2}-\frac {b \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{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 d^2}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c d^2}-\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c d^2}\\ &=-\frac {b}{2 c d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{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 d^2}+\frac {b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 c d^2}-\frac {b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 c d^2}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 189, normalized size = 1.58 \begin {gather*} \frac {\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 \cosh ^{-1}(c x) \left (c x+\left (-1+c^2 x^2\right ) \log \left (1-e^{\cosh ^{-1}(c x)}\right )+\left (1-c^2 x^2\right ) \log \left (1+e^{\cosh ^{-1}(c x)}\right )\right )+\left (a-a c^2 x^2\right ) \log (1-c x)-a \log (1+c x)+a c^2 x^2 \log (1+c x)}{-1+c^2 x^2}+2 b \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-2 b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{4 c d^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 4.25, size = 230, normalized size = 1.92
method | result | size |
derivativedivides | \(\frac {-\frac {a}{4 d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{4 d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}}{c}\) | \(230\) |
default | \(\frac {-\frac {a}{4 d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{4 d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{4 d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}}{c}\) | \(230\) |
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}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \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 {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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