Integrand size = 24, antiderivative size = 61 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \sqrt {\pi } x}{3 c}-\frac {1}{9} b c \sqrt {\pi } x^3+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 \pi } \]
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Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5798} \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi c^2}-\frac {1}{9} \sqrt {\pi } b c x^3-\frac {\sqrt {\pi } b x}{3 c} \]
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Rule 5798
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^2 \pi }-\frac {\left (b \sqrt {\pi }\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 c} \\ & = -\frac {b \sqrt {\pi } x}{3 c}-\frac {1}{9} b c \sqrt {\pi } x^3+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2 \pi } \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.03 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi } \left (3 a \left (1+c^2 x^2\right )^{3/2}-b c x \left (3+c^2 x^2\right )+3 b \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )}{9 c^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(107\) vs. \(2(49)=98\).
Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.77
method | result | size |
default | \(\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,c^{2}}+\frac {b \sqrt {\pi }\, \left (3 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+3 \,\operatorname {arcsinh}\left (c x \right )-3 c x \sqrt {c^{2} x^{2}+1}\right )}{9 c^{2} \sqrt {c^{2} x^{2}+1}}\) | \(108\) |
parts | \(\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3 \pi \,c^{2}}+\frac {b \sqrt {\pi }\, \left (3 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+3 \,\operatorname {arcsinh}\left (c x \right )-3 c x \sqrt {c^{2} x^{2}+1}\right )}{9 c^{2} \sqrt {c^{2} x^{2}+1}}\) | \(108\) |
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Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (49) = 98\).
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.08 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 3 \, a\right )}}{9 \, {\left (c^{4} x^{2} + c^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (53) = 106\).
Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.31 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\sqrt {\pi } a x^{2} \sqrt {c^{2} x^{2} + 1}}{3} + \frac {\sqrt {\pi } a \sqrt {c^{2} x^{2} + 1}}{3 c^{2}} - \frac {\sqrt {\pi } b c x^{3}}{9} + \frac {\sqrt {\pi } b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {\sqrt {\pi } b x}{3 c} + \frac {\sqrt {\pi } b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{2}} & \text {for}\: c \neq 0 \\\frac {\sqrt {\pi } a x^{2}}{2} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.20 \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} b \operatorname {arsinh}\left (c x\right )}{3 \, \pi c^{2}} - \frac {{\left (\pi ^{\frac {3}{2}} c^{2} x^{3} + 3 \, \pi ^{\frac {3}{2}} x\right )} b}{9 \, \pi c} + \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} a}{3 \, \pi c^{2}} \]
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Exception generated. \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi } \,d x \]
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