Integrand size = 10, antiderivative size = 55 \[ \int \frac {x^6}{\text {arccosh}(a x)} \, dx=\frac {5 \text {Shi}(\text {arccosh}(a x))}{64 a^7}+\frac {9 \text {Shi}(3 \text {arccosh}(a x))}{64 a^7}+\frac {5 \text {Shi}(5 \text {arccosh}(a x))}{64 a^7}+\frac {\text {Shi}(7 \text {arccosh}(a x))}{64 a^7} \]
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Time = 0.08 (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 = {5887, 5556, 3379} \[ \int \frac {x^6}{\text {arccosh}(a x)} \, dx=\frac {5 \text {Shi}(\text {arccosh}(a x))}{64 a^7}+\frac {9 \text {Shi}(3 \text {arccosh}(a x))}{64 a^7}+\frac {5 \text {Shi}(5 \text {arccosh}(a x))}{64 a^7}+\frac {\text {Shi}(7 \text {arccosh}(a x))}{64 a^7} \]
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Rule 3379
Rule 5556
Rule 5887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cosh ^6(x) \sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^7} \\ & = \frac {\text {Subst}\left (\int \left (\frac {5 \sinh (x)}{64 x}+\frac {9 \sinh (3 x)}{64 x}+\frac {5 \sinh (5 x)}{64 x}+\frac {\sinh (7 x)}{64 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a^7} \\ & = \frac {\text {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a^7}+\frac {5 \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a^7}+\frac {5 \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a^7}+\frac {9 \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{64 a^7} \\ & = \frac {5 \text {Shi}(\text {arccosh}(a x))}{64 a^7}+\frac {9 \text {Shi}(3 \text {arccosh}(a x))}{64 a^7}+\frac {5 \text {Shi}(5 \text {arccosh}(a x))}{64 a^7}+\frac {\text {Shi}(7 \text {arccosh}(a x))}{64 a^7} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int \frac {x^6}{\text {arccosh}(a x)} \, dx=\frac {5 \text {Shi}(\text {arccosh}(a x))+9 \text {Shi}(3 \text {arccosh}(a x))+5 \text {Shi}(5 \text {arccosh}(a x))+\text {Shi}(7 \text {arccosh}(a x))}{64 a^7} \]
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Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {\frac {5 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )}{64}+\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{64}+\frac {5 \,\operatorname {Shi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{64}+\frac {\operatorname {Shi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right )}{64}}{a^{7}}\) | \(40\) |
default | \(\frac {\frac {5 \,\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )}{64}+\frac {9 \,\operatorname {Shi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{64}+\frac {5 \,\operatorname {Shi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{64}+\frac {\operatorname {Shi}\left (7 \,\operatorname {arccosh}\left (a x \right )\right )}{64}}{a^{7}}\) | \(40\) |
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\[ \int \frac {x^6}{\text {arccosh}(a x)} \, dx=\int { \frac {x^{6}}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^6}{\text {arccosh}(a x)} \, dx=\int \frac {x^{6}}{\operatorname {acosh}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^6}{\text {arccosh}(a x)} \, dx=\int { \frac {x^{6}}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^6}{\text {arccosh}(a x)} \, dx=\int { \frac {x^{6}}{\operatorname {arcosh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {x^6}{\text {arccosh}(a x)} \, dx=\int \frac {x^6}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
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