Optimal. Leaf size=195 \[ \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e}+\frac {b \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {b \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \]
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Rubi [A] time = 0.26, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5800, 5562, 2190, 2279, 2391} \[ \frac {b \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {b \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 5562
Rule 5800
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{d+e x} \, dx &=\operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e}+\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c d-\sqrt {c^2 d^2-e^2}+e e^x} \, dx,x,\cosh ^{-1}(c x)\right )+\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c d+\sqrt {c^2 d^2-e^2}+e e^x} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {e e^x}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}-\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {e e^x}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{c d-\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{c d+\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {b \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {b \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 183, normalized size = 0.94 \[ \frac {-\left (a+b \cosh ^{-1}(c x)\right ) \left (a-2 b \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )-2 b \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )+b \cosh ^{-1}(c x)\right )+2 b^2 \text {Li}_2\left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )+2 b^2 \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{2 b e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (c x\right ) + a}{e x + d}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 314, normalized size = 1.61 \[ \frac {a \ln \left (c x e +c d \right )}{e}-\frac {b \mathrm {arccosh}\left (c x \right )^{2}}{2 e}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {b \dilog \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e -c d +\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e}+\frac {b \dilog \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) e +c d +\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e} \]
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 \[ b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e x + d}\,{d x} + \frac {a \log \left (e x + d\right )}{e} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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