Integrand size = 26, antiderivative size = 109 \[ \int x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b \sqrt {\pi } x}{15 c^3}-\frac {b \sqrt {\pi } x^3}{45 c}-\frac {1}{25} b c \sqrt {\pi } x^5-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 \pi ^2} \]
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Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45, 5804, 12} \[ \int x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi ^2 c^4}-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi c^4}+\frac {2 \sqrt {\pi } b x}{15 c^3}-\frac {1}{25} \sqrt {\pi } b c x^5-\frac {\sqrt {\pi } b x^3}{45 c} \]
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Rule 12
Rule 45
Rule 272
Rule 5804
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 \pi ^2}-\left (b c \sqrt {\pi }\right ) \int \frac {-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx \\ & = -\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 \pi ^2}-\frac {\left (b \sqrt {\pi }\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{15 c^3} \\ & = \frac {2 b \sqrt {\pi } x}{15 c^3}-\frac {b \sqrt {\pi } x^3}{45 c}-\frac {1}{25} b c \sqrt {\pi } x^5-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 \pi ^2} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97 \[ \int x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi } \left (15 a \sqrt {1+c^2 x^2} \left (-2+c^2 x^2+3 c^4 x^4\right )+b \left (30 c x-5 c^3 x^3-9 c^5 x^5\right )+15 b \sqrt {1+c^2 x^2} \left (-2+c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)\right )}{225 c^4} \]
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Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.50
method | result | size |
default | \(a \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{5 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{15 \pi \,c^{4}}\right )+\frac {b \sqrt {\pi }\, \left (45 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}+60 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-9 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-15 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-5 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-30 \,\operatorname {arcsinh}\left (c x \right )+30 c x \sqrt {c^{2} x^{2}+1}\right )}{225 c^{4} \sqrt {c^{2} x^{2}+1}}\) | \(164\) |
parts | \(a \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{5 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{15 \pi \,c^{4}}\right )+\frac {b \sqrt {\pi }\, \left (45 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}+60 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-9 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-15 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-5 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-30 \,\operatorname {arcsinh}\left (c x \right )+30 c x \sqrt {c^{2} x^{2}+1}\right )}{225 c^{4} \sqrt {c^{2} x^{2}+1}}\) | \(164\) |
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Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.45 \[ \int x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {15 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, b c^{6} x^{6} + 4 \, b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (45 \, a c^{6} x^{6} + 60 \, a c^{4} x^{4} - 15 \, a c^{2} x^{2} - {\left (9 \, b c^{5} x^{5} + 5 \, b c^{3} x^{3} - 30 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 30 \, a\right )}}{225 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (100) = 200\).
Time = 0.90 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.03 \[ \int x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\sqrt {\pi } a x^{4} \sqrt {c^{2} x^{2} + 1}}{5} + \frac {\sqrt {\pi } a x^{2} \sqrt {c^{2} x^{2} + 1}}{15 c^{2}} - \frac {2 \sqrt {\pi } a \sqrt {c^{2} x^{2} + 1}}{15 c^{4}} - \frac {\sqrt {\pi } b c x^{5}}{25} + \frac {\sqrt {\pi } b x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {\sqrt {\pi } b x^{3}}{45 c} + \frac {\sqrt {\pi } b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{15 c^{2}} + \frac {2 \sqrt {\pi } b x}{15 c^{3}} - \frac {2 \sqrt {\pi } b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{15 c^{4}} & \text {for}\: c \neq 0 \\\frac {\sqrt {\pi } a x^{4}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.23 \[ \int x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{15} \, b {\left (\frac {3 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{2}}{\pi c^{2}} - \frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}{\pi c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, a {\left (\frac {3 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{2}}{\pi c^{2}} - \frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}{\pi c^{4}}\right )} - \frac {{\left (9 \, \sqrt {\pi } c^{4} x^{5} + 5 \, \sqrt {\pi } c^{2} x^{3} - 30 \, \sqrt {\pi } x\right )} b}{225 \, c^{3}} \]
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Exception generated. \[ \int x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi } \,d x \]
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