Optimal. Leaf size=111 \[ -\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \sinh ^{-1}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} b d \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5801, 5775,
3797, 2221, 2317, 2438, 201, 221} \begin {gather*} \frac {1}{2} d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{4} b c d x \sqrt {c^2 x^2+1}-\frac {1}{2} b d \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )-\frac {1}{4} b d \sinh ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5801
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+d \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx-\frac {1}{2} (b c d) \int \sqrt {1+c^2 x^2} \, dx\\ &=-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )+d \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )-\frac {1}{4} (b c d) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \sinh ^{-1}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}-(2 d) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \sinh ^{-1}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-(b d) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \sinh ^{-1}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\frac {1}{2} (b d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \sinh ^{-1}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b}+d \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+\frac {1}{2} b d \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 113, normalized size = 1.02 \begin {gather*} \frac {1}{2} a c^2 d x^2-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}+\frac {1}{4} b d \sinh ^{-1}(c x)+\frac {1}{2} b c^2 d x^2 \sinh ^{-1}(c x)-\frac {1}{2} b d \sinh ^{-1}(c x)^2+b d \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+a d \log (x)+\frac {1}{2} b d \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.98, size = 162, normalized size = 1.46
method | result | size |
derivativedivides | \(\frac {a \,c^{2} d \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {b d \arcsinh \left (c x \right )^{2}}{2}+\frac {b d \arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {b c d x \sqrt {c^{2} x^{2}+1}}{4}+\frac {b d \arcsinh \left (c x \right )}{4}+b d \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b d \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+b d \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+b d \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(162\) |
default | \(\frac {a \,c^{2} d \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {b d \arcsinh \left (c x \right )^{2}}{2}+\frac {b d \arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {b c d x \sqrt {c^{2} x^{2}+1}}{4}+\frac {b d \arcsinh \left (c x \right )}{4}+b d \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b d \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+b d \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+b d \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(162\) |
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*} d \left (\int \frac {a}{x}\, dx + \int a c^{2} x\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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