Integrand size = 10, antiderivative size = 170 \[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {2 x^3}{3 a^2 \text {arccosh}(a x)^2}-\frac {5 x^5}{6 \text {arccosh}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \text {arccosh}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{48 a^5}+\frac {27 \text {Chi}(3 \text {arccosh}(a x))}{32 a^5}+\frac {125 \text {Chi}(5 \text {arccosh}(a x))}{96 a^5} \]
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Time = 0.45 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5886, 5951, 5885, 3382} \[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\frac {\text {Chi}(\text {arccosh}(a x))}{48 a^5}+\frac {27 \text {Chi}(3 \text {arccosh}(a x))}{32 a^5}+\frac {125 \text {Chi}(5 \text {arccosh}(a x))}{96 a^5}+\frac {2 x^2 \sqrt {a x-1} \sqrt {a x+1}}{a^3 \text {arccosh}(a x)}+\frac {2 x^3}{3 a^2 \text {arccosh}(a x)^2}-\frac {5 x^5}{6 \text {arccosh}(a x)^2}-\frac {25 x^4 \sqrt {a x-1} \sqrt {a x+1}}{6 a \text {arccosh}(a x)}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{3 a \text {arccosh}(a x)^3} \]
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Rule 3382
Rule 5885
Rule 5886
Rule 5951
Rubi steps \begin{align*} \text {integral}& = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}-\frac {4 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3} \, dx}{3 a}+\frac {1}{3} (5 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3} \, dx \\ & = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {2 x^3}{3 a^2 \text {arccosh}(a x)^2}-\frac {5 x^5}{6 \text {arccosh}(a x)^2}+\frac {25}{6} \int \frac {x^4}{\text {arccosh}(a x)^2} \, dx-\frac {2 \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx}{a^2} \\ & = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {2 x^3}{3 a^2 \text {arccosh}(a x)^2}-\frac {5 x^5}{6 \text {arccosh}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \text {arccosh}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \text {arccosh}(a x)}+\frac {2 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}-\frac {3 \cosh (3 x)}{4 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a^5}-\frac {25 \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 )}{6 a^5} \\ & = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {2 x^3}{3 a^2 \text {arccosh}(a x)^2}-\frac {5 x^5}{6 \text {arccosh}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \text {arccosh}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{2 a^5}+\frac {25 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{48 a^5}+\frac {125 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{96 a^5}-\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{2 a^5}+\frac {75 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{32 a^5} \\ & = -\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \text {arccosh}(a x)^3}+\frac {2 x^3}{3 a^2 \text {arccosh}(a x)^2}-\frac {5 x^5}{6 \text {arccosh}(a x)^2}+\frac {2 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a^3 \text {arccosh}(a x)}-\frac {25 x^4 \sqrt {-1+a x} \sqrt {1+a x}}{6 a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{48 a^5}+\frac {27 \text {Chi}(3 \text {arccosh}(a x))}{32 a^5}+\frac {125 \text {Chi}(5 \text {arccosh}(a x))}{96 a^5} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(356\) vs. \(2(170)=340\).
Time = 0.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.09 \[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\frac {\sqrt {-1+a x} \left (32 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}}-32 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}}+64 a^3 x^3 \sqrt {-1+a x} \sqrt {\frac {-1+a x}{1+a x}} \sqrt {1+a x} \text {arccosh}(a x)-80 a^5 x^5 \sqrt {-1+a x} \sqrt {\frac {-1+a x}{1+a x}} \sqrt {1+a x} \text {arccosh}(a x)-192 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2+592 a^4 x^4 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2-400 a^6 x^6 \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^2+2 (-1+a x) \text {arccosh}(a x)^3 \text {Chi}(\text {arccosh}(a x))+81 (-1+a x) \text {arccosh}(a x)^3 \text {Chi}(3 \text {arccosh}(a x))-125 \text {arccosh}(a x)^3 \text {Chi}(5 \text {arccosh}(a x))+125 a x \text {arccosh}(a x)^3 \text {Chi}(5 \text {arccosh}(a x))\right )}{96 a^5 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^3} \]
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Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {a x}{48 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{48 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{48}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {3 \cosh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {9 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32 \,\operatorname {arccosh}\left (a x \right )}+\frac {27 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{48 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {5 \cosh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {25 \sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96 \,\operatorname {arccosh}\left (a x \right )}+\frac {125 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96}}{a^{5}}\) | \(175\) |
default | \(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{24 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {a x}{48 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{48 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{48}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{16 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {3 \cosh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {9 \sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32 \,\operatorname {arccosh}\left (a x \right )}+\frac {27 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{48 \operatorname {arccosh}\left (a x \right )^{3}}-\frac {5 \cosh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {25 \sinh \left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96 \,\operatorname {arccosh}\left (a x \right )}+\frac {125 \,\operatorname {Chi}\left (5 \,\operatorname {arccosh}\left (a x \right )\right )}{96}}{a^{5}}\) | \(175\) |
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\[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
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\[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int \frac {x^{4}}{\operatorname {acosh}^{4}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
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\[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int { \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\text {arccosh}(a x)^4} \, dx=\int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \]
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