Integrand size = 19, antiderivative size = 264 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \text {arccosh}(c x)}{4 c^2}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))-\frac {i b d \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d (a+b \text {arccosh}(c x)) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.48 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {14, 5958, 12, 6874, 92, 54, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=d \log (x) (a+b \text {arccosh}(c x))+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))-\frac {b e \text {arccosh}(c x)}{4 c^2}-\frac {i b d \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {i b d \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b e x \sqrt {c x-1} \sqrt {c x+1}}{4 c} \]
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Rule 12
Rule 14
Rule 54
Rule 92
Rule 2221
Rule 2317
Rule 2363
Rule 2365
Rule 2438
Rule 3798
Rule 4721
Rule 5958
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} e x^2 (a+b \text {arccosh}(c x))+d (a+b \text {arccosh}(c x)) \log (x)-(b c) \int \frac {e x^2+2 d \log (x)}{2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{2} e x^2 (a+b \text {arccosh}(c x))+d (a+b \text {arccosh}(c x)) \log (x)-\frac {1}{2} (b c) \int \frac {e x^2+2 d \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {1}{2} e x^2 (a+b \text {arccosh}(c x))+d (a+b \text {arccosh}(c x)) \log (x)-\frac {1}{2} (b c) \int \left (\frac {e x^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 d \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx \\ & = \frac {1}{2} e x^2 (a+b \text {arccosh}(c x))+d (a+b \text {arccosh}(c x)) \log (x)-(b c d) \int \frac {\log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{2} (b c e) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))+d (a+b \text {arccosh}(c x)) \log (x)-\frac {(b e) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c}-\frac {\left (b c d \sqrt {1-c^2 x^2}\right ) \int \frac {\log (x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \text {arccosh}(c x)}{4 c^2}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))+d (a+b \text {arccosh}(c x)) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d \sqrt {1-c^2 x^2}\right ) \int \frac {\arcsin (c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \text {arccosh}(c x)}{4 c^2}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))+d (a+b \text {arccosh}(c x)) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int x \cot (x) \, dx,x,\arcsin (c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \text {arccosh}(c x)}{4 c^2}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))-\frac {i b d \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+d (a+b \text {arccosh}(c x)) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\arcsin (c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \text {arccosh}(c x)}{4 c^2}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))-\frac {i b d \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d (a+b \text {arccosh}(c x)) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \text {arccosh}(c x)}{4 c^2}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))-\frac {i b d \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d (a+b \text {arccosh}(c x)) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e \text {arccosh}(c x)}{4 c^2}+\frac {1}{2} e x^2 (a+b \text {arccosh}(c x))-\frac {i b d \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d (a+b \text {arccosh}(c x)) \log (x)-\frac {b d \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.49 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{2} a e x^2-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}+\frac {1}{2} b e x^2 \text {arccosh}(c x)-\frac {b e \text {arctanh}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{2 c^2}+a d \log (x)+\frac {1}{2} b d \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]
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Time = 1.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.48
method | result | size |
parts | \(\frac {a e \,x^{2}}{2}+a d \ln \left (x \right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {b e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {b e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(128\) |
derivativedivides | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {b e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {b e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(130\) |
default | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {b e x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}+\frac {b \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {b e \,\operatorname {arccosh}\left (c x \right )}{4 c^{2}}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(130\) |
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x} \,d x \]
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