Integrand size = 10, antiderivative size = 87 \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {3 x^2}{2 a^2 \text {arccosh}(a x)}-\frac {2 x^4}{\text {arccosh}(a x)}+\frac {\text {Shi}(2 \text {arccosh}(a x))}{2 a^4}+\frac {\text {Shi}(4 \text {arccosh}(a x))}{a^4} \]
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Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5886, 5951, 5887, 5556, 3379, 12} \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\frac {\text {Shi}(2 \text {arccosh}(a x))}{2 a^4}+\frac {\text {Shi}(4 \text {arccosh}(a x))}{a^4}+\frac {3 x^2}{2 a^2 \text {arccosh}(a x)}-\frac {2 x^4}{\text {arccosh}(a x)}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2} \]
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Rule 12
Rule 3379
Rule 5556
Rule 5886
Rule 5887
Rule 5951
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}-\frac {3 \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2} \, dx}{2 a}+(2 a) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2} \, dx \\ & = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {3 x^2}{2 a^2 \text {arccosh}(a x)}-\frac {2 x^4}{\text {arccosh}(a x)}+8 \int \frac {x^3}{\text {arccosh}(a x)} \, dx-\frac {3 \int \frac {x}{\text {arccosh}(a x)} \, dx}{a^2} \\ & = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {3 x^2}{2 a^2 \text {arccosh}(a x)}-\frac {2 x^4}{\text {arccosh}(a x)}-\frac {3 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^4}+\frac {8 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^4} \\ & = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {3 x^2}{2 a^2 \text {arccosh}(a x)}-\frac {2 x^4}{\text {arccosh}(a x)}-\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\text {arccosh}(a x)\right )}{a^4}+\frac {8 \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a^4} \\ & = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {3 x^2}{2 a^2 \text {arccosh}(a x)}-\frac {2 x^4}{\text {arccosh}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^4}-\frac {3 \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{2 a^4}+\frac {2 \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^4} \\ & = -\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {3 x^2}{2 a^2 \text {arccosh}(a x)}-\frac {2 x^4}{\text {arccosh}(a x)}+\frac {\text {Shi}(2 \text {arccosh}(a x))}{2 a^4}+\frac {\text {Shi}(4 \text {arccosh}(a x))}{a^4} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\frac {-\frac {a^2 x^2 \left (a x \sqrt {-1+a x} \sqrt {1+a x}+\left (-3+4 a^2 x^2\right ) \text {arccosh}(a x)\right )}{\text {arccosh}(a x)^2}+\text {Shi}(2 \text {arccosh}(a x))+2 \text {Shi}(4 \text {arccosh}(a x))}{2 a^4} \]
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Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2}-\frac {\sinh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Shi}\left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{4}}\) | \(82\) |
default | \(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{8 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2}-\frac {\sinh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Shi}\left (4 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{4}}\) | \(82\) |
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\[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\int \frac {x^{3}}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x^{3}}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]
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Exception generated. \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{\text {arccosh}(a x)^3} \, dx=\int \frac {x^3}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]
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