Optimal. Leaf size=489 \[ -\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 d}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b d}+\frac {\log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d}-\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d}-\frac {b \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )}{2 d} \]
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Rubi [A] time = 0.92, antiderivative size = 472, normalized size of antiderivative = 0.97, number of steps used = 25, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5792, 5660, 3718, 2190, 2279, 2391, 5800, 5562} \[ -\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 d}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 d}-\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 d}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 d}+\frac {b \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 d}+\frac {\log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d} \]
Warning: Unable to verify antiderivative.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5562
Rule 5660
Rule 5792
Rule 5800
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x \left (d+e x^2\right )} \, dx &=\int \left (\frac {a+b \cosh ^{-1}(c x)}{d x}-\frac {e x \left (a+b \cosh ^{-1}(c x)\right )}{d \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx}{d}-\frac {e \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d+e x^2} \, dx}{d}\\ &=\frac {\operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{d}-\frac {e \int \left (-\frac {a+b \cosh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b d}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac {\sqrt {e} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d}-\frac {\sqrt {e} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 b d}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d}-\frac {b \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac {\sqrt {e} \operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}-\frac {\sqrt {e} \operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d}+\frac {\sqrt {e} \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}+\frac {\sqrt {e} \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}-\frac {\sqrt {e} \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}-\frac {\sqrt {e} \operatorname {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d}+\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d}+\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}+\frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d}-\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d}+\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d}\\ \end {align*}
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Mathematica [C] time = 0.82, size = 418, normalized size = 0.85 \[ \frac {-2 a \log \left (d+e x^2\right )+4 a \log (x)+b \left (\text {Li}_2\left (-\frac {\left (2 d c^2+e-2 \sqrt {c^2 d \left (d c^2+e\right )}\right ) e^{-2 \cosh ^{-1}(c x)}}{e}\right )+\text {Li}_2\left (-\frac {\left (2 d c^2+e+2 \sqrt {c^2 d \left (d c^2+e\right )}\right ) e^{-2 \cosh ^{-1}(c x)}}{e}\right )-4 i \sin ^{-1}\left (\sqrt {\frac {c^2 d}{e}+1}\right ) \tanh ^{-1}\left (\frac {c d \sqrt {\frac {c x-1}{c x+1}} (c x+1)}{x \sqrt {c^2 d \left (c^2 d+e\right )}}\right )-2 \cosh ^{-1}(c x) \log \left (\frac {\left (-2 \sqrt {c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right )-2 \cosh ^{-1}(c x) \log \left (\frac {\left (2 \sqrt {c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right )+2 i \sin ^{-1}\left (\sqrt {\frac {c^2 d}{e}+1}\right ) \log \left (\frac {\left (-2 \sqrt {c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right )-2 i \sin ^{-1}\left (\sqrt {\frac {c^2 d}{e}+1}\right ) \log \left (\frac {\left (2 \sqrt {c^2 d \left (c^2 d+e\right )}+2 c^2 d+e\right ) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right )-2 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+4 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )}{4 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (c x\right ) + a}{e x^{3} + d x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.36, size = 393, normalized size = 0.80 \[ \frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c^{2} x^{2} e +c^{2} d \right )}{2 d}-\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right ) e}{4 d}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}+\frac {b \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}-\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 d} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {\log \left (e x^{2} + d\right )}{d} - \frac {2 \, \log \relax (x)}{d}\right )} + b \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e x^{3} + d x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,\left (e\,x^2+d\right )} \,d x \]
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 \[ \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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