Integrand size = 23, antiderivative size = 187 \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {45 x^2}{128 a^3}-\frac {3 x^4}{128 a}-\frac {45 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{64 a^4}+\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{32 a^2}+\frac {45 \text {arcsinh}(a x)^2}{128 a^5}+\frac {9 x^2 \text {arcsinh}(a x)^2}{16 a^3}-\frac {3 x^4 \text {arcsinh}(a x)^2}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{4 a^2}+\frac {3 \text {arcsinh}(a x)^4}{32 a^5} \]
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Time = 0.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5812, 5783, 5776, 30} \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {3 \text {arcsinh}(a x)^4}{32 a^5}+\frac {45 \text {arcsinh}(a x)^2}{128 a^5}+\frac {9 x^2 \text {arcsinh}(a x)^2}{16 a^3}+\frac {45 x^2}{128 a^3}+\frac {x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{4 a^2}+\frac {3 x^3 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{32 a^2}-\frac {3 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{8 a^4}-\frac {45 x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{64 a^4}-\frac {3 x^4 \text {arcsinh}(a x)^2}{16 a}-\frac {3 x^4}{128 a} \]
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Rule 30
Rule 5776
Rule 5783
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{4 a^2}-\frac {3 \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2}-\frac {3 \int x^3 \text {arcsinh}(a x)^2 \, dx}{4 a} \\ & = -\frac {3 x^4 \text {arcsinh}(a x)^2}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{4 a^2}+\frac {3}{8} \int \frac {x^4 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx+\frac {3 \int \frac {\text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{8 a^4}+\frac {9 \int x \text {arcsinh}(a x)^2 \, dx}{8 a^3} \\ & = \frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{32 a^2}+\frac {9 x^2 \text {arcsinh}(a x)^2}{16 a^3}-\frac {3 x^4 \text {arcsinh}(a x)^2}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{4 a^2}+\frac {3 \text {arcsinh}(a x)^4}{32 a^5}-\frac {9 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{32 a^2}-\frac {9 \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2}-\frac {3 \int x^3 \, dx}{32 a} \\ & = -\frac {3 x^4}{128 a}-\frac {45 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{64 a^4}+\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{32 a^2}+\frac {9 x^2 \text {arcsinh}(a x)^2}{16 a^3}-\frac {3 x^4 \text {arcsinh}(a x)^2}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{4 a^2}+\frac {3 \text {arcsinh}(a x)^4}{32 a^5}+\frac {9 \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{64 a^4}+\frac {9 \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 a^4}+\frac {9 \int x \, dx}{64 a^3}+\frac {9 \int x \, dx}{16 a^3} \\ & = \frac {45 x^2}{128 a^3}-\frac {3 x^4}{128 a}-\frac {45 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{64 a^4}+\frac {3 x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{32 a^2}+\frac {45 \text {arcsinh}(a x)^2}{128 a^5}+\frac {9 x^2 \text {arcsinh}(a x)^2}{16 a^3}-\frac {3 x^4 \text {arcsinh}(a x)^2}{16 a}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{8 a^4}+\frac {x^3 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{4 a^2}+\frac {3 \text {arcsinh}(a x)^4}{32 a^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.65 \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\frac {45 a^2 x^2-3 a^4 x^4+6 a x \sqrt {1+a^2 x^2} \left (-15+2 a^2 x^2\right ) \text {arcsinh}(a x)+\left (45+72 a^2 x^2-24 a^4 x^4\right ) \text {arcsinh}(a x)^2+16 a x \sqrt {1+a^2 x^2} \left (-3+2 a^2 x^2\right ) \text {arcsinh}(a x)^3+12 \text {arcsinh}(a x)^4}{128 a^5} \]
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Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {32 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}-24 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2}+12 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}-3 a^{4} x^{4}-48 \operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x +72 \operatorname {arcsinh}\left (a x \right )^{2} a^{2} x^{2}+12 \operatorname {arcsinh}\left (a x \right )^{4}-90 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +45 a^{2} x^{2}+45 \operatorname {arcsinh}\left (a x \right )^{2}+45}{128 a^{5}}\) | \(156\) |
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Time = 0.26 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=-\frac {3 \, a^{4} x^{4} - 16 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 45 \, a^{2} x^{2} - 12 \, \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} + 3 \, {\left (8 \, a^{4} x^{4} - 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 6 \, {\left (2 \, a^{3} x^{3} - 15 \, a x\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{128 \, a^{5}} \]
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Time = 0.73 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.99 \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\begin {cases} - \frac {3 x^{4} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a} - \frac {3 x^{4}}{128 a} + \frac {x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 x^{3} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{32 a^{2}} + \frac {9 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{16 a^{3}} + \frac {45 x^{2}}{128 a^{3}} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{8 a^{4}} - \frac {45 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{64 a^{4}} + \frac {3 \operatorname {asinh}^{4}{\left (a x \right )}}{32 a^{5}} + \frac {45 \operatorname {asinh}^{2}{\left (a x \right )}}{128 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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\[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]
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\[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{4} \operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]
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Timed out. \[ \int \frac {x^4 \text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^4\,{\mathrm {asinh}\left (a\,x\right )}^3}{\sqrt {a^2\,x^2+1}} \,d x \]
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