Optimal. Leaf size=95 \[ -\frac {a+b \cosh ^{-1}(c x)}{d x}+\frac {2 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}+\frac {b c \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {b c \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d} \]
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Rubi [A] time = 0.14, antiderivative size = 95, 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 = {5746, 92, 205, 5694, 4182, 2279, 2391} \[ \frac {b c \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {b c \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {a+b \cosh ^{-1}(c x)}{d x}+\frac {2 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}+\frac {b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 2279
Rule 2391
Rule 4182
Rule 5694
Rule 5746
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{d x}+c^2 \int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx+\frac {(b c) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d x}-\frac {c \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac {\left (b c^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d x}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}+\frac {2 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {(b c) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}-\frac {(b c) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d x}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}+\frac {2 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {(b c) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{d x}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}+\frac {2 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {b c \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {b c \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 132, normalized size = 1.39 \[ \frac {-\frac {a+b \cosh ^{-1}(c x)}{x}-c \log \left (1-e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )+c \log \left (e^{\cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {b c \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+b c \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )-b c \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \operatorname {arcosh}\left (c x\right ) + a}{c^{2} d x^{4} - d x^{2}}, x\right ) \]
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 \[ \int -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 161, normalized size = 1.69 \[ -\frac {a}{d x}-\frac {c a \ln \left (c x -1\right )}{2 d}+\frac {c a \ln \left (c x +1\right )}{2 d}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{d x}+\frac {2 c b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {c b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {c b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {c b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d} \]
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 \[ \frac {1}{8} \, {\left (24 \, c^{3} \int \frac {x \log \left (c x - 1\right )}{4 \, {\left (c^{2} d x^{2} - d\right )}}\,{d x} - 4 \, c^{2} {\left (\frac {\log \left (c x + 1\right )}{c d} - \frac {\log \left (c x - 1\right )}{c d}\right )} - 8 \, c^{2} \int \frac {\log \left (c x - 1\right )}{4 \, {\left (c^{2} d x^{2} - d\right )}}\,{d x} - \frac {c x \log \left (c x + 1\right )^{2} + 2 \, c x \log \left (c x + 1\right ) \log \left (c x - 1\right ) - 4 \, {\left (c x \log \left (c x + 1\right ) - c x \log \left (c x - 1\right ) - 2\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{d x} + 8 \, \int \frac {c^{2} x \log \left (c x + 1\right ) - c^{2} x \log \left (c x - 1\right ) - 2 \, c}{2 \, {\left (c^{3} d x^{4} - c d x^{2} + {\left (c^{2} d x^{3} - d x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )}}\,{d x}\right )} b + \frac {1}{2} \, a {\left (\frac {c \log \left (c x + 1\right )}{d} - \frac {c \log \left (c x - 1\right )}{d} - \frac {2}{d x}\right )} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,\left (d-c^2\,d\,x^2\right )} \,d x \]
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 \[ - \frac {\int \frac {a}{c^{2} x^{4} - x^{2}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{4} - x^{2}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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