Integrand size = 24, antiderivative size = 75 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b x}{6 c \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {b \arctan (c x)}{6 c^2 \pi ^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5798, 205, 209} \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {a+b \text {arcsinh}(c x)}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {b \arctan (c x)}{6 \pi ^{5/2} c^2}+\frac {b x}{6 \pi ^{5/2} c \left (c^2 x^2+1\right )} \]
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Rule 205
Rule 209
Rule 5798
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {b \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{3 c \pi ^{5/2}} \\ & = \frac {b x}{6 c \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{6 c \pi ^{5/2}} \\ & = \frac {b x}{6 c \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {b \arctan (c x)}{6 c^2 \pi ^{5/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-2 a+b c x \sqrt {1+c^2 x^2}-2 b \text {arcsinh}(c x)+b \left (1+c^2 x^2\right )^{3/2} \arctan (c x)}{6 c^2 \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}} \]
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Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.61
method | result | size |
default | \(-\frac {a}{3 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+b \left (-\frac {-c x \sqrt {c^{2} x^{2}+1}+2 \,\operatorname {arcsinh}\left (c x \right )}{6 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}+\frac {i \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{6 c^{2} \pi ^{\frac {5}{2}}}-\frac {i \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{6 c^{2} \pi ^{\frac {5}{2}}}\right )\) | \(121\) |
parts | \(-\frac {a}{3 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+b \left (-\frac {-c x \sqrt {c^{2} x^{2}+1}+2 \,\operatorname {arcsinh}\left (c x \right )}{6 \pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}+\frac {i \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{6 c^{2} \pi ^{\frac {5}{2}}}-\frac {i \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{6 c^{2} \pi ^{\frac {5}{2}}}\right )\) | \(121\) |
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (63) = 126\).
Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.20 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {\sqrt {\pi } {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) + 4 \, \sqrt {\pi + \pi c^{2} x^{2}} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (\sqrt {c^{2} x^{2} + 1} b c x - 2 \, a\right )}}{12 \, {\left (\pi ^{3} c^{6} x^{4} + 2 \, \pi ^{3} c^{4} x^{2} + \pi ^{3} c^{2}\right )}} \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a x}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]
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