Integrand size = 23, antiderivative size = 27 \[ \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=-\frac {3 \text {Shi}(\text {arcsinh}(a x))}{4 a^4}+\frac {\text {Shi}(3 \text {arcsinh}(a x))}{4 a^4} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5819, 3393, 3379} \[ \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {\text {Shi}(3 \text {arcsinh}(a x))}{4 a^4}-\frac {3 \text {Shi}(\text {arcsinh}(a x))}{4 a^4} \]
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Rule 3379
Rule 3393
Rule 5819
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sinh ^3(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^4} \\ & = \frac {i \text {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 x}-\frac {i \sinh (3 x)}{4 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^4} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^4}-\frac {3 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{4 a^4} \\ & = -\frac {3 \text {Shi}(\text {arcsinh}(a x))}{4 a^4}+\frac {\text {Shi}(3 \text {arcsinh}(a x))}{4 a^4} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\frac {-3 \text {Shi}(\text {arcsinh}(a x))+\text {Shi}(3 \text {arcsinh}(a x))}{4 a^4} \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {3 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (a x \right )\right )-\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{4 a^{4}}\) | \(23\) |
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\[ \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {x^{3}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {x^{3}}{\sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {x^{3}}{\sqrt {a^{2} x^{2} + 1} \operatorname {arsinh}\left (a x\right )} \,d x } \]
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Exception generated. \[ \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {x^3}{\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \]
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