Integrand size = 23, antiderivative size = 27 \[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a x))}{2 a^3}-\frac {\log (\text {arcsinh}(a x))}{2 a^3} \]
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Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5819, 3393, 3382} \[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a x))}{2 a^3}-\frac {\log (\text {arcsinh}(a x))}{2 a^3} \]
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Rule 3382
Rule 3393
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sinh ^2(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^3} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^3} \\ & = -\frac {\log (\text {arcsinh}(a x))}{2 a^3}+\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{2 a^3} \\ & = \frac {\text {Chi}(2 \text {arcsinh}(a x))}{2 a^3}-\frac {\log (\text {arcsinh}(a x))}{2 a^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a x))-\log (\text {arcsinh}(a x))}{2 a^3} \]
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Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\ln \left (\operatorname {arcsinh}\left (a x \right )\right )-\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{2 a^{3}}\) | \(21\) |
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\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {x^2}{\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \]
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