Integrand size = 26, antiderivative size = 247 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {3 b c}{4 \pi ^{5/2} x}+\frac {b c}{4 \pi ^{5/2} x \left (1+c^2 x^2\right )}+\frac {5 b c^3 x}{12 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{2 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {13 b c^2 \arctan (c x)}{6 \pi ^{5/2}}+\frac {5 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {5 b c^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}}-\frac {5 b c^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}} \]
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Time = 0.33 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5809, 5811, 5816, 4267, 2317, 2438, 209, 205, 296, 331} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {5 c^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\pi ^{5/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{2 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {5 b c^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}}-\frac {5 b c^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}}+\frac {13 b c^2 \arctan (c x)}{6 \pi ^{5/2}}+\frac {b c}{4 \pi ^{5/2} x \left (c^2 x^2+1\right )}+\frac {5 b c^3 x}{12 \pi ^{5/2} \left (c^2 x^2+1\right )}-\frac {3 b c}{4 \pi ^{5/2} x} \]
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Rule 205
Rule 209
Rule 296
Rule 331
Rule 2317
Rule 2438
Rule 4267
Rule 5809
Rule 5811
Rule 5816
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {1}{2} \left (5 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )^2} \, dx}{2 \pi ^{5/2}} \\ & = \frac {b c}{4 \pi ^{5/2} x \left (1+c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {(3 b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{4 \pi ^{5/2}}+\frac {\left (5 b c^3\right ) \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{6 \pi ^{5/2}}-\frac {\left (5 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx}{2 \pi } \\ & = -\frac {3 b c}{4 \pi ^{5/2} x}+\frac {b c}{4 \pi ^{5/2} x \left (1+c^2 x^2\right )}+\frac {5 b c^3 x}{12 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{2 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (5 b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{12 \pi ^{5/2}}-\frac {\left (3 b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 \pi ^{5/2}}+\frac {\left (5 b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 \pi ^{5/2}}-\frac {\left (5 c^2\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {\pi +c^2 \pi x^2}} \, dx}{2 \pi ^2} \\ & = -\frac {3 b c}{4 \pi ^{5/2} x}+\frac {b c}{4 \pi ^{5/2} x \left (1+c^2 x^2\right )}+\frac {5 b c^3 x}{12 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{2 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {13 b c^2 \arctan (c x)}{6 \pi ^{5/2}}-\frac {\left (5 c^2\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{2 \pi ^{5/2}} \\ & = -\frac {3 b c}{4 \pi ^{5/2} x}+\frac {b c}{4 \pi ^{5/2} x \left (1+c^2 x^2\right )}+\frac {5 b c^3 x}{12 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{2 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {13 b c^2 \arctan (c x)}{6 \pi ^{5/2}}+\frac {5 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {\left (5 b c^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 \pi ^{5/2}}-\frac {\left (5 b c^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 \pi ^{5/2}} \\ & = -\frac {3 b c}{4 \pi ^{5/2} x}+\frac {b c}{4 \pi ^{5/2} x \left (1+c^2 x^2\right )}+\frac {5 b c^3 x}{12 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{2 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {13 b c^2 \arctan (c x)}{6 \pi ^{5/2}}+\frac {5 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {\left (5 b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}}-\frac {\left (5 b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}} \\ & = -\frac {3 b c}{4 \pi ^{5/2} x}+\frac {b c}{4 \pi ^{5/2} x \left (1+c^2 x^2\right )}+\frac {5 b c^3 x}{12 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{6 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{2 \pi x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 c^2 (a+b \text {arcsinh}(c x))}{2 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {13 b c^2 \arctan (c x)}{6 \pi ^{5/2}}+\frac {5 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\pi ^{5/2}}+\frac {5 b c^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}}-\frac {5 b c^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \pi ^{5/2}} \\ \end{align*}
Time = 4.11 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.34 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-\frac {8 a c^2}{\left (1+c^2 x^2\right )^{3/2}}+\frac {4 b c^3 x}{1+c^2 x^2}-\frac {48 a c^2}{\sqrt {1+c^2 x^2}}-\frac {12 a \sqrt {1+c^2 x^2}}{x^2}-\frac {56 b c^2 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}-\frac {48 b c^4 x^2 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+104 b c^2 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-6 b c^2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-3 b c^2 \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-60 b c^2 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+60 b c^2 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-60 a c^2 \log (x)+60 a c^2 \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )-60 b c^2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+60 b c^2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+6 b c^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\frac {6 b c \text {arcsinh}(c x) \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{x}}{24 \pi ^{5/2}} \]
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Time = 0.18 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.27
method | result | size |
default | \(a \left (-\frac {1}{2 \pi \,x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {5 c^{2} \left (\frac {1}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}}{\pi }\right )}{2}\right )-\frac {5 b \,x^{2} \operatorname {arcsinh}\left (c x \right ) c^{4}}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b \,c^{3} x}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )}-\frac {10 b \,\operatorname {arcsinh}\left (c x \right ) c^{2}}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b c}{2 \pi ^{\frac {5}{2}} x \left (c^{2} x^{2}+1\right )}-\frac {b \,\operatorname {arcsinh}\left (c x \right )}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} x^{2}}+\frac {13 b \,c^{2} \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}\) | \(314\) |
parts | \(a \left (-\frac {1}{2 \pi \,x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {5 c^{2} \left (\frac {1}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {\frac {1}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{\pi ^{\frac {3}{2}}}}{\pi }\right )}{2}\right )-\frac {5 b \,x^{2} \operatorname {arcsinh}\left (c x \right ) c^{4}}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b \,c^{3} x}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )}-\frac {10 b \,\operatorname {arcsinh}\left (c x \right ) c^{2}}{3 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {b c}{2 \pi ^{\frac {5}{2}} x \left (c^{2} x^{2}+1\right )}-\frac {b \,\operatorname {arcsinh}\left (c x \right )}{2 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} x^{2}}+\frac {13 b \,c^{2} \arctan \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {dilog}\left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}+\frac {5 b \,c^{2} \operatorname {dilog}\left (c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \pi ^{\frac {5}{2}}}\) | \(314\) |
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a}{c^{4} x^{7} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} + x^{3} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{7} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} + x^{3} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]
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