Optimal. Leaf size=81 \[ \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \cosh ^{-1}(c+d x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )}{d e}-\frac {b \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e} \]
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Rubi [A]
time = 0.10, 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 = {5996, 12, 5882,
3799, 2221, 2317, 2438} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5996
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c+d x)}{c e+d e x} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\text {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 \text {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 \text {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 \text {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 = 0.04, size = 69, normalized size = 0.85 \begin {gather*} \frac {b \cosh ^{-1}(c+d x)^2+2 b \cosh ^{-1}(c+d x) \log \left (1+e^{-2 \cosh ^{-1}(c+d x)}\right )+2 a \log (c+d x)-b \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c+d x)}\right )}{2 d e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 8.79, size = 103, normalized size = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {a \ln \left (d x +c \right )}{e}-\frac {b \mathrm {arccosh}\left (d x +c \right )^{2}}{2 e}+\frac {b \,\mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {b \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2 e}}{d}\) | \(103\) |
default | \(\frac {\frac {a \ln \left (d x +c \right )}{e}-\frac {b \mathrm {arccosh}\left (d x +c \right )^{2}}{2 e}+\frac {b \,\mathrm {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{e}+\frac {b \polylog \left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2 e}}{d}\) | \(103\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {\int \frac {a}{c + d x}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{c\,e+d\,e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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