Integrand size = 10, antiderivative size = 43 \[ \int \frac {x^5}{\text {arcsinh}(a x)} \, dx=\frac {5 \text {Shi}(2 \text {arcsinh}(a x))}{32 a^6}-\frac {\text {Shi}(4 \text {arcsinh}(a x))}{8 a^6}+\frac {\text {Shi}(6 \text {arcsinh}(a x))}{32 a^6} \]
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Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5780, 5556, 3379} \[ \int \frac {x^5}{\text {arcsinh}(a x)} \, dx=\frac {5 \text {Shi}(2 \text {arcsinh}(a x))}{32 a^6}-\frac {\text {Shi}(4 \text {arcsinh}(a x))}{8 a^6}+\frac {\text {Shi}(6 \text {arcsinh}(a x))}{32 a^6} \]
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Rule 3379
Rule 5556
Rule 5780
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^5(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^6} \\ & = \frac {\text {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 x}-\frac {\sinh (4 x)}{8 x}+\frac {\sinh (6 x)}{32 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^6} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (6 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^6}-\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{8 a^6}+\frac {5 \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{32 a^6} \\ & = \frac {5 \text {Shi}(2 \text {arcsinh}(a x))}{32 a^6}-\frac {\text {Shi}(4 \text {arcsinh}(a x))}{8 a^6}+\frac {\text {Shi}(6 \text {arcsinh}(a x))}{32 a^6} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \frac {x^5}{\text {arcsinh}(a x)} \, dx=\frac {5 \text {Shi}(2 \text {arcsinh}(a x))-4 \text {Shi}(4 \text {arcsinh}(a x))+\text {Shi}(6 \text {arcsinh}(a x))}{32 a^6} \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {5 \,\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}-\frac {\operatorname {Shi}\left (4 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}+\frac {\operatorname {Shi}\left (6 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}}{a^{6}}\) | \(33\) |
default | \(\frac {\frac {5 \,\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}-\frac {\operatorname {Shi}\left (4 \,\operatorname {arcsinh}\left (a x \right )\right )}{8}+\frac {\operatorname {Shi}\left (6 \,\operatorname {arcsinh}\left (a x \right )\right )}{32}}{a^{6}}\) | \(33\) |
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\[ \int \frac {x^5}{\text {arcsinh}(a x)} \, dx=\int { \frac {x^{5}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^5}{\text {arcsinh}(a x)} \, dx=\int \frac {x^{5}}{\operatorname {asinh}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^5}{\text {arcsinh}(a x)} \, dx=\int { \frac {x^{5}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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Exception generated. \[ \int \frac {x^5}{\text {arcsinh}(a x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5}{\text {arcsinh}(a x)} \, dx=\int \frac {x^5}{\mathrm {asinh}\left (a\,x\right )} \,d x \]
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