Integrand size = 10, antiderivative size = 55 \[ \int \frac {x^6}{\text {arcsinh}(a x)} \, dx=-\frac {5 \text {Chi}(\text {arcsinh}(a x))}{64 a^7}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{64 a^7}-\frac {5 \text {Chi}(5 \text {arcsinh}(a x))}{64 a^7}+\frac {\text {Chi}(7 \text {arcsinh}(a x))}{64 a^7} \]
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Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5780, 5556, 3382} \[ \int \frac {x^6}{\text {arcsinh}(a x)} \, dx=-\frac {5 \text {Chi}(\text {arcsinh}(a x))}{64 a^7}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{64 a^7}-\frac {5 \text {Chi}(5 \text {arcsinh}(a x))}{64 a^7}+\frac {\text {Chi}(7 \text {arcsinh}(a x))}{64 a^7} \]
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Rule 3382
Rule 5556
Rule 5780
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^6(x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{a^7} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {5 \cosh (x)}{64 x}+\frac {9 \cosh (3 x)}{64 x}-\frac {5 \cosh (5 x)}{64 x}+\frac {\cosh (7 x)}{64 x}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{a^7} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (7 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^7}-\frac {5 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^7}-\frac {5 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^7}+\frac {9 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a x)\right )}{64 a^7} \\ & = -\frac {5 \text {Chi}(\text {arcsinh}(a x))}{64 a^7}+\frac {9 \text {Chi}(3 \text {arcsinh}(a x))}{64 a^7}-\frac {5 \text {Chi}(5 \text {arcsinh}(a x))}{64 a^7}+\frac {\text {Chi}(7 \text {arcsinh}(a x))}{64 a^7} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int \frac {x^6}{\text {arcsinh}(a x)} \, dx=\frac {-5 \text {Chi}(\text {arcsinh}(a x))+9 \text {Chi}(3 \text {arcsinh}(a x))-5 \text {Chi}(5 \text {arcsinh}(a x))+\text {Chi}(7 \text {arcsinh}(a x))}{64 a^7} \]
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Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {-\frac {5 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{64}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{64}-\frac {5 \,\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{64}+\frac {\operatorname {Chi}\left (7 \,\operatorname {arcsinh}\left (a x \right )\right )}{64}}{a^{7}}\) | \(40\) |
default | \(\frac {-\frac {5 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (a x \right )\right )}{64}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (a x \right )\right )}{64}-\frac {5 \,\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (a x \right )\right )}{64}+\frac {\operatorname {Chi}\left (7 \,\operatorname {arcsinh}\left (a x \right )\right )}{64}}{a^{7}}\) | \(40\) |
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\[ \int \frac {x^6}{\text {arcsinh}(a x)} \, dx=\int { \frac {x^{6}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^6}{\text {arcsinh}(a x)} \, dx=\int \frac {x^{6}}{\operatorname {asinh}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^6}{\text {arcsinh}(a x)} \, dx=\int { \frac {x^{6}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^6}{\text {arcsinh}(a x)} \, dx=\int { \frac {x^{6}}{\operatorname {arsinh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {x^6}{\text {arcsinh}(a x)} \, dx=\int \frac {x^6}{\mathrm {asinh}\left (a\,x\right )} \,d x \]
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