Integrand size = 26, antiderivative size = 56 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \]
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Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5816, 4267, 2317, 2438} \[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {\pi }}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \]
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Rule 2317
Rule 2438
Rule 4267
Rule 5816
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{\sqrt {\pi }} \\ & = -\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {\pi }}+\frac {b \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {\pi }} \\ & = -\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}+\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \\ & = -\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.71 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {b \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-b \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+a \log (x)-a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+b \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-b \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )}{\sqrt {\pi }} \]
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Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.11
method | result | size |
default | \(-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\sqrt {\pi }}+\frac {b \left (-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{\sqrt {\pi }}\) | \(118\) |
parts | \(-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\sqrt {\pi }}+\frac {b \left (-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{\sqrt {\pi }}\) | \(118\) |
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {\pi + \pi c^{2} x^{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {\int \frac {a}{x \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x \sqrt {c^{2} x^{2} + 1}}\, dx}{\sqrt {\pi }} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {\pi + \pi c^{2} x^{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {\pi + \pi c^{2} x^{2}} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \]
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