Integrand size = 26, antiderivative size = 148 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {b c x}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {a+b \text {arcsinh}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {7 b \arctan (c x)}{6 \pi ^{5/2}}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}} \]
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Time = 0.23 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5811, 5816, 4267, 2317, 2438, 209, 205} \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\pi ^{5/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi ^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {a+b \text {arcsinh}(c x)}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}-\frac {7 b \arctan (c x)}{6 \pi ^{5/2}}-\frac {b c x}{6 \pi ^{5/2} \left (c^2 x^2+1\right )} \]
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Rule 205
Rule 209
Rule 2317
Rule 2438
Rule 4267
Rule 5811
Rule 5816
Rubi steps \begin{align*} \text {integral}& = \frac {a+b \text {arcsinh}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{3 \pi ^{5/2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx}{\pi } \\ & = -\frac {b c x}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {a+b \text {arcsinh}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {(b c) \int \frac {1}{1+c^2 x^2} \, dx}{6 \pi ^{5/2}}-\frac {(b c) \int \frac {1}{1+c^2 x^2} \, dx}{\pi ^{5/2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx}{\pi ^2} \\ & = -\frac {b c x}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {a+b \text {arcsinh}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {7 b \arctan (c x)}{6 \pi ^{5/2}}+\frac {\text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{\pi ^{5/2}} \\ & = -\frac {b c x}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {a+b \text {arcsinh}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {7 b \arctan (c x)}{6 \pi ^{5/2}}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}-\frac {b \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\pi ^{5/2}}+\frac {b \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\pi ^{5/2}} \\ & = -\frac {b c x}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {a+b \text {arcsinh}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {7 b \arctan (c x)}{6 \pi ^{5/2}}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}-\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}} \\ & = -\frac {b c x}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {a+b \text {arcsinh}(c x)}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {a+b \text {arcsinh}(c x)}{\pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {7 b \arctan (c x)}{6 \pi ^{5/2}}-\frac {2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}-\frac {b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\frac {2 a}{\left (1+c^2 x^2\right )^{3/2}}-\frac {b c x}{1+c^2 x^2}+\frac {6 a}{\sqrt {1+c^2 x^2}}+\frac {8 b \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+\frac {6 b c^2 x^2 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}-14 b \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+6 b \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-6 b \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+6 a \log (x)-6 a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+6 b \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-6 b \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )}{6 \pi ^{5/2}} \]
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Time = 0.23 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.41
method | result | size |
default | \(a \left (\frac {1}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}}{\pi }\right )+b \left (\frac {6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+8 \,\operatorname {arcsinh}\left (c x \right )}{6 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {7 \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3 \pi ^{\frac {5}{2}}}-\frac {\operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {5}{2}}}-\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {5}{2}}}-\frac {\operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {5}{2}}}\right )\) | \(208\) |
parts | \(a \left (\frac {1}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}}{\pi }\right )+b \left (\frac {6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+8 \,\operatorname {arcsinh}\left (c x \right )}{6 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {7 \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3 \pi ^{\frac {5}{2}}}-\frac {\operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {5}{2}}}-\frac {\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {5}{2}}}-\frac {\operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )}{\pi ^{\frac {5}{2}}}\right )\) | \(208\) |
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a}{c^{4} x^{5} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} + x \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{5} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} + x \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]
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