Integrand size = 22, antiderivative size = 111 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \text {arcsinh}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {d (a+b \text {arcsinh}(c x))^2}{2 b}+d (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5801, 5775, 3797, 2221, 2317, 2438, 201, 221} \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {1}{2} d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))+\frac {d (a+b \text {arcsinh}(c x))^2}{2 b}+d \log \left (1-e^{-2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b d \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{4} b d \text {arcsinh}(c x)-\frac {1}{4} b c d x \sqrt {c^2 x^2+1} \]
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Rule 201
Rule 221
Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5775
Rule 5801
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+d \int \frac {a+b \text {arcsinh}(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 ) (a+b \text {arcsinh}(c x))-\frac {d \text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b}-\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 \text {arcsinh}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {d (a+b \text {arcsinh}(c x))^2}{2 b}+\frac {(2 d) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b} \\ & = -\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \text {arcsinh}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {d (a+b \text {arcsinh}(c x))^2}{2 b}+d (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-d \text {Subst}\left (\int \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right ) \\ & = -\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \text {arcsinh}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {d (a+b \text {arcsinh}(c x))^2}{2 b}+d (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+\frac {1}{2} (b d) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ & = -\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \text {arcsinh}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {d (a+b \text {arcsinh}(c x))^2}{2 b}+d (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {1}{2} a c^2 d x^2-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}+\frac {1}{4} b d \text {arcsinh}(c x)+\frac {1}{2} b c^2 d x^2 \text {arcsinh}(c x)-\frac {1}{2} b d \text {arcsinh}(c x)^2+b d \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+a d \log (x)+\frac {1}{2} b d \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.43
method | result | size |
parts | \(d a \left (\frac {c^{2} x^{2}}{2}+\ln \left (x \right )\right )-\frac {d b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {d b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b c d x \sqrt {c^{2} x^{2}+1}}{4}+\frac {b d \,\operatorname {arcsinh}\left (c x \right )}{4}+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(159\) |
derivativedivides | \(d a \left (\frac {c^{2} x^{2}}{2}+\ln \left (c x \right )\right )-\frac {d b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {d b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b c d x \sqrt {c^{2} x^{2}+1}}{4}+\frac {b d \,\operatorname {arcsinh}\left (c x \right )}{4}+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(161\) |
default | \(d a \left (\frac {c^{2} x^{2}}{2}+\ln \left (c x \right )\right )-\frac {d b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {d b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b c d x \sqrt {c^{2} x^{2}+1}}{4}+\frac {b d \,\operatorname {arcsinh}\left (c x \right )}{4}+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(161\) |
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=d \left (\int \frac {a}{x}\, dx + \int a c^{2} x\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right )}{x} \,d x \]
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