Integrand size = 19, antiderivative size = 75 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))+b c d \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 75, 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 = {5956, 471, 94, 211} \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))+b c d \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )-\frac {b e \sqrt {c x-1} \sqrt {c x+1}}{c} \]
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Rule 94
Rule 211
Rule 471
Rule 5956
Rubi steps \begin{align*} \text {integral}& = -\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))+(b c) \int \frac {d-e x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))+(b c d) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))+\left (b c^2 d\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right ) \\ & = -\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {d (a+b \text {arccosh}(c x))}{x}+e x (a+b \text {arccosh}(c x))+b c d \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.40 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-\frac {a d}{x}+a e x-\frac {b e \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {b d \text {arccosh}(c x)}{x}+b e x \text {arccosh}(c x)+\frac {b c d \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.36
method | result | size |
parts | \(a \left (e x -\frac {d}{x}\right )+b c \left (\frac {\operatorname {arccosh}\left (c x \right ) e x}{c}-\frac {\operatorname {arccosh}\left (c x \right ) d}{c x}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (d \,c^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {c^{2} x^{2}-1}}\right )\) | \(102\) |
derivativedivides | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) e c x -\frac {\operatorname {arccosh}\left (c x \right ) d c}{x}+\frac {\left (-d \,c^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-e \sqrt {c^{2} x^{2}-1}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{\sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\right )\) | \(105\) |
default | \(c \left (\frac {a \left (e c x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) e c x -\frac {\operatorname {arccosh}\left (c x \right ) d c}{x}+\frac {\left (-d \,c^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-e \sqrt {c^{2} x^{2}-1}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{\sqrt {c^{2} x^{2}-1}}\right )}{c^{2}}\right )\) | \(105\) |
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Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.76 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\frac {2 \, b c^{2} d x \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + a c e x^{2} - \sqrt {c^{2} x^{2} - 1} b e x - a c d + {\left (b c d - b c e\right )} x \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{c x} \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{2}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=-{\left (c \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arcosh}\left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b e}{c} - \frac {a d}{x} \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^2} \,d x \]
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