Optimal. Leaf size=81 \[ \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c+d x)}\right )}{d e}-\frac {b \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c+d x)}\right )}{2 d e} \]
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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 = {5859, 12, 5775,
3797, 2221, 2317, 2438} \begin {gather*} \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\log \left (1-e^{-2 \sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e}-\frac {b \text {Li}_2\left (e^{-2 \sinh ^{-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 3797
Rule 5775
Rule 5859
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c+d x)}{c e+d e x} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-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,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}-\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 70, normalized size = 0.86 \begin {gather*} \frac {-\left (\left (a+b \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)-2 b \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )\right )\right )+b^2 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c+d x)}\right )}{2 b d e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.39, size = 145, normalized size = 1.79
method | result | size |
derivativedivides | \(\frac {\frac {a \ln \left (d x +c \right )}{e}-\frac {b \arcsinh \left (d x +c \right )^{2}}{2 e}+\frac {b \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {b \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {b \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {b \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}}{d}\) | \(145\) |
default | \(\frac {\frac {a \ln \left (d x +c \right )}{e}-\frac {b \arcsinh \left (d x +c \right )^{2}}{2 e}+\frac {b \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {b \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {b \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}+\frac {b \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{e}}{d}\) | \(145\) |
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 {asinh}{\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 {asinh}\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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