Integrand size = 10, antiderivative size = 81 \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=-\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}-\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3} \]
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Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5779, 5818, 5780, 5556, 3382, 5774} \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=-\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3}-\frac {x^2 \sqrt {a^2 x^2+1}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)} \]
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Rule 3382
Rule 5556
Rule 5774
Rule 5779
Rule 5780
Rule 5818
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}+\frac {\int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2} \, dx}{a}+\frac {1}{2} (3 a) \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2} \, dx \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}+\frac {9}{2} \int \frac {x^2}{\text {arcsinh}(a x)} \, dx+\frac {\int \frac {1}{\text {arcsinh}(a x)} \, dx}{a^2} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}+\frac {\text {Chi}(\text {arcsinh}(a x))}{a^3}+\frac {9 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}+\frac {\text {Chi}(\text {arcsinh}(a x))}{a^3}-\frac {9 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^3} \\ & = -\frac {x^2 \sqrt {1+a^2 x^2}}{2 a \text {arcsinh}(a x)^2}-\frac {x}{a^2 \text {arcsinh}(a x)}-\frac {3 x^3}{2 \text {arcsinh}(a x)}-\frac {\text {Chi}(\text {arcsinh}(a x))}{8 a^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=-\frac {\frac {4 a x \left (a x \sqrt {1+a^2 x^2}+\left (2+3 a^2 x^2\right ) \text {arcsinh}(a x)\right )}{\text {arcsinh}(a x)^2}+\text {Chi}(\text {arcsinh}(a x))-9 \text {Chi}(3 \text {arcsinh}(a x))}{8 a^3} \]
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Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{8 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {a x}{8 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(81\) |
default | \(\frac {\frac {\sqrt {a^{2} x^{2}+1}}{8 \operatorname {arcsinh}\left (a x \right )^{2}}+\frac {a x}{8 \,\operatorname {arcsinh}\left (a x \right )}-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{8}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \operatorname {arcsinh}\left (a x \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8 \,\operatorname {arcsinh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}}{a^{3}}\) | \(81\) |
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^3} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^3} \,d x \]
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