Integrand size = 10, antiderivative size = 97 \[ \int \frac {x^4}{\text {arcsinh}(a x)^3} \, dx=-\frac {x^4 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {2 x^3}{a^2 \text {arcsinh}(a x)}-\frac {5 x^5}{2 \text {arcsinh}(a x)}+\frac {\text {Chi}(\text {arcsinh}(a x))}{16 a^5}-\frac {27 \text {Chi}(3 \text {arcsinh}(a x))}{32 a^5}+\frac {25 \text {Chi}(5 \text {arcsinh}(a x))}{32 a^5} \]
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Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5779, 5818, 5780, 5556, 3382} \[ \int \frac {x^4}{\text {arcsinh}(a x)^3} \, dx=\frac {\text {Chi}(\text {arcsinh}(a x))}{16 a^5}-\frac {27 \text {Chi}(3 \text {arcsinh}(a x))}{32 a^5}+\frac {25 \text {Chi}(5 \text {arcsinh}(a x))}{32 a^5}-\frac {2 x^3}{a^2 \text {arcsinh}(a x)}-\frac {x^4 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {5 x^5}{2 \text {arcsinh}(a x)} \]
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Rule 3382
Rule 5556
Rule 5779
Rule 5780
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}+\frac {2 \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2} \, dx}{a}+\frac {1}{2} (5 a) \int \frac {x^5}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2} \, dx \\ & = -\frac {x^4 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {2 x^3}{a^2 \text {arcsinh}(a x)}-\frac {5 x^5}{2 \text {arcsinh}(a x)}+\frac {25}{2} \int \frac {x^4}{\text {arcsinh}(a x)} \, dx+\frac {6 \int \frac {x^2}{\text {arcsinh}(a x)} \, dx}{a^2} \\ & = -\frac {x^4 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {2 x^3}{a^2 \text {arcsinh}(a x)}-\frac {5 x^5}{2 \text {arcsinh}(a x)}+\frac {6 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^5} \\ & = -\frac {x^4 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {2 x^3}{a^2 \text {arcsinh}(a x)}-\frac {5 x^5}{2 \text {arcsinh}(a x)}+\frac {6 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^5}+\frac {25 \text {Subst}\left (\int \left (\frac {\cosh (x)}{8 x}-\frac {3 \cosh (3 x)}{16 x}+\frac {\cosh (5 x)}{16 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{2 a^5} \\ & = -\frac {x^4 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {2 x^3}{a^2 \text {arcsinh}(a x)}-\frac {5 x^5}{2 \text {arcsinh}(a x)}+\frac {25 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^5}+\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^5}+\frac {25 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{16 a^5}-\frac {75 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^5} \\ & = -\frac {x^4 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {2 x^3}{a^2 \text {arcsinh}(a x)}-\frac {5 x^5}{2 \text {arcsinh}(a x)}+\frac {\text {Chi}(\text {arcsinh}(a x))}{16 a^5}-\frac {27 \text {Chi}(3 \text {arcsinh}(a x))}{32 a^5}+\frac {25 \text {Chi}(5 \text {arcsinh}(a x))}{32 a^5} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.05 \[ \int \frac {x^4}{\text {arcsinh}(a x)^3} \, dx=-\frac {16 a^4 x^4 \sqrt {1+a^2 x^2}+64 a^3 x^3 \text {arcsinh}(a x)+80 a^5 x^5 \text {arcsinh}(a x)-2 \text {arcsinh}(a x)^2 \text {Chi}(\text {arcsinh}(a x))+27 \text {arcsinh}(a x)^2 \text {Chi}(3 \text {arcsinh}(a x))-25 \text {arcsinh}(a x)^2 \text {Chi}(5 \text {arcsinh}(a x))}{32 a^5 \text {arcsinh}(a x)^2} \]
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Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{16 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {a x}{16 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{16}+\frac {3 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {9 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}-\frac {27 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}-\frac {\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {5 \sinh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}+\frac {25 \,\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}}{a^{5}}\) | \(120\) |
default | \(\frac {-\frac {\sqrt {a^{2} x^{2}+1}}{16 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {a x}{16 \,\operatorname {arcsinh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{16}+\frac {3 \cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {9 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}-\frac {27 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}-\frac {\cosh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {5 \sinh \left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{32 \,\operatorname {arcsinh}\left (a x \right )}+\frac {25 \,\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}}{a^{5}}\) | \(120\) |
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\[ \int \frac {x^4}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x^4}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x^{4}}{\operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x^4}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]
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