Integrand size = 19, antiderivative size = 113 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {arccosh}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 \sqrt {d} e \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5957, 533, 385, 214} \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {b c \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 \sqrt {d} e \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}}-\frac {a+b \text {arccosh}(c x)}{2 e \left (d+e x^2\right )} \]
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Rule 214
Rule 385
Rule 533
Rule 5957
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{2 e \left (d+e x^2\right )}+\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )} \, dx}{2 e} \\ & = -\frac {a+b \text {arccosh}(c x)}{2 e \left (d+e x^2\right )}+\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{2 e \left (d+e x^2\right )}+\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{2 e \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a+b \text {arccosh}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 \sqrt {d} e \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=-\frac {\frac {a}{d+e x^2}+\frac {b \text {arccosh}(c x)}{d+e x^2}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\frac {\sqrt {-c^2 d-e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{\sqrt {d} \sqrt {-c^2 d-e} \sqrt {-1+c^2 x^2}}}{2 e} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(398\) vs. \(2(96)=192\).
Time = 7.56 (sec) , antiderivative size = 399, normalized size of antiderivative = 3.53
method | result | size |
parts | \(-\frac {a}{2 e \left (e \,x^{2}+d \right )}+\frac {b \left (-\frac {c^{4} \operatorname {arccosh}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, \left (\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) c^{2} d -\ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{e c x +\sqrt {-c^{2} d e}}\right ) c^{2} d +\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) e -\ln \left (-\frac {2 \left (-\sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{e c x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (-\sqrt {-c^{2} d e}+e \right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(399\) |
derivativedivides | \(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) c^{2} d -\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x +\sqrt {-c^{2} d e}}\right ) c^{2} d +\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) e -\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (-\sqrt {-c^{2} d e}+e \right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(411\) |
default | \(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) c^{2} d -\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x +\sqrt {-c^{2} d e}}\right ) c^{2} d +\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) e -\ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x +\sqrt {-c^{2} d e}}\right ) e \right )}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (-\sqrt {-c^{2} d e}+e \right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(411\) |
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (93) = 186\).
Time = 0.28 (sec) , antiderivative size = 537, normalized size of antiderivative = 4.75 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {2 \, a c^{2} d^{2} - 2 \, {\left (b c^{2} d e + b e^{2}\right )} x^{2} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, a d e - {\left (b c e x^{2} + b c d\right )} \sqrt {c^{2} d^{2} + d e} \log \left (-\frac {2 \, c^{2} d^{2} - {\left (4 \, c^{4} d^{2} + 4 \, c^{2} d e + e^{2}\right )} x^{2} + d e - 2 \, \sqrt {c^{2} d^{2} + d e} {\left ({\left (2 \, c^{3} d + c e\right )} x^{2} - c d\right )} - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (\sqrt {c^{2} d^{2} + d e} {\left (2 \, c^{2} d + e\right )} x + 2 \, {\left (c^{3} d^{2} + c d e\right )} x\right )}}{e x^{2} + d}\right ) - 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} - {\left (b c^{2} d e + b e^{2}\right )} x^{2} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a d e - {\left (b c e x^{2} + b c d\right )} \sqrt {-c^{2} d^{2} - d e} \arctan \left (\frac {\sqrt {-c^{2} d^{2} - d e} \sqrt {c^{2} x^{2} - 1} e x - \sqrt {-c^{2} d^{2} - d e} {\left (c e x^{2} + c d\right )}}{c^{2} d^{2} + d e}\right ) - {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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