Optimal. Leaf size=116 \[ \frac {a+b \cosh ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}+\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {b \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {b c x}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.18, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5754, 5721, 5461, 4182, 2279, 2391, 39} \[ \frac {b \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {b \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}+\frac {a+b \cosh ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac {b c x}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 2279
Rule 2391
Rule 4182
Rule 5461
Rule 5721
Rule 5754
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx &=\frac {a+b \cosh ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d}\\ &=-\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {\operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {2 \operatorname {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {b \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {b \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 149, normalized size = 1.28 \[ \frac {\frac {a}{1-c^2 x^2}-a \log \left (1-c^2 x^2\right )+2 a \log (x)+b \left (\frac {\cosh ^{-1}(c x)}{1-c^2 x^2}-\text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+\text {Li}_2\left (e^{-2 \cosh ^{-1}(c x)}\right )+\frac {c x \sqrt {\frac {c x-1}{c x+1}}}{1-c x}-2 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )+2 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )}{2 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (c x\right ) + a}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 339, normalized size = 2.92 \[ \frac {a \ln \left (c x \right )}{d^{2}}-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{2 d^{2}}+\frac {a}{4 d^{2} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{2 d^{2}}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}\, c x}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,c^{2} x^{2}}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{2}}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2 d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {1}{c^{2} d^{2} x^{2} - d^{2}} + \frac {\log \left (c x + 1\right )}{d^{2}} + \frac {\log \left (c x - 1\right )}{d^{2}} - \frac {2 \, \log \relax (x)}{d^{2}}\right )} + b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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