Optimal. Leaf size=60 \[ -\frac {\sinh ^{-1}(a+b x)^2}{2 d}+\frac {\sinh ^{-1}(a+b x) \log \left (1-e^{2 \sinh ^{-1}(a+b x)}\right )}{d}+\frac {\text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a+b x)}\right )}{2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5859, 12, 5775,
3797, 2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (e^{2 \sinh ^{-1}(a+b x)}\right )}{2 d}-\frac {\sinh ^{-1}(a+b x)^2}{2 d}+\frac {\sinh ^{-1}(a+b x) \log \left (1-e^{2 \sinh ^{-1}(a+b x)}\right )}{d} \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 {\sinh ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx &=\frac {\text {Subst}\left (\int \frac {b \sinh ^{-1}(x)}{d x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\sinh ^{-1}(x)}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac {\text {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{d}\\ &=-\frac {\sinh ^{-1}(a+b x)^2}{2 d}-\frac {2 \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{d}\\ &=-\frac {\sinh ^{-1}(a+b x)^2}{2 d}+\frac {\sinh ^{-1}(a+b x) \log \left (1-e^{2 \sinh ^{-1}(a+b x)}\right )}{d}-\frac {\text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{d}\\ &=-\frac {\sinh ^{-1}(a+b x)^2}{2 d}+\frac {\sinh ^{-1}(a+b x) \log \left (1-e^{2 \sinh ^{-1}(a+b x)}\right )}{d}-\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a+b x)}\right )}{2 d}\\ &=-\frac {\sinh ^{-1}(a+b x)^2}{2 d}+\frac {\sinh ^{-1}(a+b x) \log \left (1-e^{2 \sinh ^{-1}(a+b x)}\right )}{d}+\frac {\text {Li}_2\left (e^{2 \sinh ^{-1}(a+b x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 52, normalized size = 0.87 \begin {gather*} \frac {-\sinh ^{-1}(a+b x) \left (\sinh ^{-1}(a+b x)-2 \log \left (1-e^{2 \sinh ^{-1}(a+b x)}\right )\right )+\text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a+b x)}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.26, size = 134, normalized size = 2.23
method | result | size |
derivativedivides | \(\frac {-\frac {b \arcsinh \left (b x +a \right )^{2}}{2 d}+\frac {b \arcsinh \left (b x +a \right ) \ln \left (1+b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{d}+\frac {b \polylog \left (2, -b x -a -\sqrt {1+\left (b x +a \right )^{2}}\right )}{d}+\frac {b \arcsinh \left (b x +a \right ) \ln \left (1-b x -a -\sqrt {1+\left (b x +a \right )^{2}}\right )}{d}+\frac {b \polylog \left (2, b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{d}}{b}\) | \(134\) |
default | \(\frac {-\frac {b \arcsinh \left (b x +a \right )^{2}}{2 d}+\frac {b \arcsinh \left (b x +a \right ) \ln \left (1+b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{d}+\frac {b \polylog \left (2, -b x -a -\sqrt {1+\left (b x +a \right )^{2}}\right )}{d}+\frac {b \arcsinh \left (b x +a \right ) \ln \left (1-b x -a -\sqrt {1+\left (b x +a \right )^{2}}\right )}{d}+\frac {b \polylog \left (2, b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{d}}{b}\) | \(134\) |
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 {b \int \frac {\operatorname {asinh}{\left (a + b x \right )}}{a + b x}\, dx}{d} \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 = -0.02 \begin {gather*} \int \frac {\mathrm {asinh}\left (a+b\,x\right )}{d\,x+\frac {a\,d}{b}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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