Integrand size = 10, antiderivative size = 73 \[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{8 a^5}+\frac {9 \text {Chi}(3 \text {arccosh}(a x))}{16 a^5}+\frac {5 \text {Chi}(5 \text {arccosh}(a x))}{16 a^5} \]
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Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5885, 3382} \[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=\frac {\text {Chi}(\text {arccosh}(a x))}{8 a^5}+\frac {9 \text {Chi}(3 \text {arccosh}(a x))}{16 a^5}+\frac {5 \text {Chi}(5 \text {arccosh}(a x))}{16 a^5}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)} \]
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Rule 3382
Rule 5885
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \left (-\frac {\cosh (x)}{8 x}-\frac {9 \cosh (3 x)}{16 x}-\frac {5 \cosh (5 x)}{16 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a^5} \\ & = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{8 a^5}+\frac {5 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{16 a^5}+\frac {9 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{16 a^5} \\ & = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{8 a^5}+\frac {9 \text {Chi}(3 \text {arccosh}(a x))}{16 a^5}+\frac {5 \text {Chi}(5 \text {arccosh}(a x))}{16 a^5} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.38 \[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=\frac {-16 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-16 a^5 x^5 \sqrt {\frac {-1+a x}{1+a x}}+2 \text {arccosh}(a x) \text {Chi}(\text {arccosh}(a x))+9 \text {arccosh}(a x) \text {Chi}(3 \text {arccosh}(a x))+5 \text {arccosh}(a x) \text {Chi}(5 \text {arccosh}(a x))}{16 a^5 \text {arccosh}(a x)} \]
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Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{8}-\frac {3 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \,\operatorname {arccosh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16}-\frac {\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \,\operatorname {arccosh}\left (a x \right )}+\frac {5 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{16}}{a^{5}}\) | \(83\) |
default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{8}-\frac {3 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \,\operatorname {arccosh}\left (a x \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16}-\frac {\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \,\operatorname {arccosh}\left (a x \right )}+\frac {5 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{16}}{a^{5}}\) | \(83\) |
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\[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^{4}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]
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