Integrand size = 26, antiderivative size = 126 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {3 b x^2}{16 c^3 \sqrt {\pi }}-\frac {b x^4}{16 c \sqrt {\pi }}-\frac {3 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{8 c^4 \pi }+\frac {x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{4 c^2 \pi }+\frac {3 (a+b \text {arcsinh}(c x))^2}{16 b c^5 \sqrt {\pi }} \]
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Time = 0.17 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5812, 5783, 30} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {3 (a+b \text {arcsinh}(c x))^2}{16 \sqrt {\pi } b c^5}+\frac {x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{4 \pi c^2}-\frac {3 x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{8 \pi c^4}+\frac {3 b x^2}{16 \sqrt {\pi } c^3}-\frac {b x^4}{16 \sqrt {\pi } c} \]
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Rule 30
Rule 5783
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{4 c^2 \pi }-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{4 c^2}-\frac {b \int x^3 \, dx}{4 c \sqrt {\pi }} \\ & = -\frac {b x^4}{16 c \sqrt {\pi }}-\frac {3 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{8 c^4 \pi }+\frac {x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{4 c^2 \pi }+\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{8 c^4}+\frac {(3 b) \int x \, dx}{8 c^3 \sqrt {\pi }} \\ & = \frac {3 b x^2}{16 c^3 \sqrt {\pi }}-\frac {b x^4}{16 c \sqrt {\pi }}-\frac {3 x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{8 c^4 \pi }+\frac {x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{4 c^2 \pi }+\frac {3 (a+b \text {arcsinh}(c x))^2}{16 b c^5 \sqrt {\pi }} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {-48 a c x \sqrt {1+c^2 x^2}+32 a c^3 x^3 \sqrt {1+c^2 x^2}+24 b \text {arcsinh}(c x)^2+16 b \cosh (2 \text {arcsinh}(c x))-b \cosh (4 \text {arcsinh}(c x))+4 \text {arcsinh}(c x) (12 a-8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))}{128 c^5 \sqrt {\pi }} \]
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Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.31
method | result | size |
default | \(\frac {a \,x^{3} \sqrt {\pi \,c^{2} x^{2}+\pi }}{4 \pi \,c^{2}}-\frac {3 a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8 c^{4} \pi }+\frac {3 a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}+3\right )}{16 c^{5} \sqrt {\pi }}\) | \(165\) |
parts | \(\frac {a \,x^{3} \sqrt {\pi \,c^{2} x^{2}+\pi }}{4 \pi \,c^{2}}-\frac {3 a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8 c^{4} \pi }+\frac {3 a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}+3\right )}{16 c^{5} \sqrt {\pi }}\) | \(165\) |
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {\pi + \pi c^{2} x^{2}}} \,d x } \]
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Time = 2.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.47 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {a \left (\begin {cases} \frac {x^{3} \sqrt {c^{2} x^{2} + 1}}{4 c^{2}} - \frac {3 x \sqrt {c^{2} x^{2} + 1}}{8 c^{4}} + \frac {3 \log {\left (2 c^{2} x + 2 \sqrt {c^{2} x^{2} + 1} \sqrt {c^{2}} \right )}}{8 c^{4} \sqrt {c^{2}}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{5}}{5} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x^{4}}{16 c} + \frac {x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4 c^{2}} + \frac {3 x^{2}}{16 c^{3}} - \frac {3 x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8 c^{4}} + \frac {3 \operatorname {asinh}^{2}{\left (c x \right )}}{16 c^{5}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \]
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Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{\sqrt {\pi + \pi c^{2} x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \]
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