Optimal. Leaf size=81 \[ \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {b \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e} \]
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Rubi [A] time = 0.11, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5866, 12, 5660, 3718, 2190, 2279, 2391} \[ \frac {b \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\log \left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{d e} \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5866
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c+d x)}{c e+d e x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {b \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c+d x)}\right )}{d e}+\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}(c+d x)}\right )}{2 d e}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 69, normalized size = 0.85 \[ \frac {2 a \log (c+d x)-b \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c+d x)}\right )+b \cosh ^{-1}(c+d x)^2+2 b \cosh ^{-1}(c+d x) \log \left (e^{-2 \cosh ^{-1}(c+d x)}+1\right )}{2 d e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{d e x + c e}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{d e x + c e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 111, normalized size = 1.37 \[ \frac {a \ln \left (d x +c \right )}{d e}-\frac {b \mathrm {arccosh}\left (d x +c \right )^{2}}{2 d e}+\frac {b \,\mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{d e}+\frac {b \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2 d e} \]
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 \[ b \int \frac {\log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{d e x + c e}\,{d x} + \frac {a \log \left (d e x + c e\right )}{d e} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{c\,e+d\,e\,x} \,d x \]
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 \[ \frac {\int \frac {a}{c + d x}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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