Integrand size = 26, antiderivative size = 80 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^3 \pi ^{3/2}}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5810, 5783, 266} \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {(a+b \text {arcsinh}(c x))^2}{2 \pi ^{3/2} b c^3}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3} \]
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Rule 266
Rule 5783
Rule 5810
Rubi steps \begin{align*} \text {integral}& = -\frac {x (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b \int \frac {x}{1+c^2 x^2} \, dx}{c \pi ^{3/2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{c^2 \pi } \\ & = -\frac {x (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^3 \pi ^{3/2}}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-\frac {2 a c x}{\sqrt {1+c^2 x^2}}+\left (2 a-\frac {2 b c x}{\sqrt {1+c^2 x^2}}\right ) \text {arcsinh}(c x)+b \text {arcsinh}(c x)^2+b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(70)=140\).
Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.10
method | result | size |
default | \(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+b \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{3} \pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c^{3} \pi ^{\frac {3}{2}}}+\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{3} \pi ^{\frac {3}{2}}}\right )\) | \(168\) |
parts | \(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+b \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{3} \pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c^{3} \pi ^{\frac {3}{2}}}+\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{3} \pi ^{\frac {3}{2}}}\right )\) | \(168\) |
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a x^{2}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{2} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]
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