Integrand size = 24, antiderivative size = 65 \[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {-1+a x} \text {Chi}(2 \text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}}+\frac {\sqrt {-1+a x} \log (\text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}} \]
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Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5952, 3393, 3382} \[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {a x-1} \text {Chi}(2 \text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}}+\frac {\sqrt {a x-1} \log (\text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}} \]
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Rule 3382
Rule 3393
Rule 5952
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+a x} \text {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^3 \sqrt {1-a x}} \\ & = \frac {\sqrt {-1+a x} \text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a^3 \sqrt {1-a x}} \\ & = \frac {\sqrt {-1+a x} \log (\text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}}+\frac {\sqrt {-1+a x} \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{2 a^3 \sqrt {1-a x}} \\ & = \frac {\sqrt {-1+a x} \text {Chi}(2 \text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}}+\frac {\sqrt {-1+a x} \log (\text {arccosh}(a x))}{2 a^3 \sqrt {1-a x}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} (\text {Chi}(2 \text {arccosh}(a x))+\log (\text {arccosh}(a x)))}{2 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]
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Time = 0.59 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \left (\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )+\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (a x \right )\right )-2 \ln \left (\operatorname {arccosh}\left (a x \right )\right )\right )}{4 a^{3} \left (a^{2} x^{2}-1\right )}\) | \(67\) |
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\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x^{2}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {x^2}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
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