Integrand size = 19, antiderivative size = 251 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.42 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {14, 5958, 6874, 97, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e \log (x) (a+b \text {arccosh}(c x))-\frac {i b e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {i b e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d \sqrt {c x-1} \sqrt {c x+1}}{2 x} \]
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Rule 14
Rule 97
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 {d (a+b \text {arccosh}(c x))}{2 x^2}+e (a+b \text {arccosh}(c x)) \log (x)-(b c) \int \frac {-\frac {d}{2 x^2}+e \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e (a+b \text {arccosh}(c x)) \log (x)-(b c) \int \left (-\frac {d}{2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {e \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx \\ & = -\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e (a+b \text {arccosh}(c x)) \log (x)+\frac {1}{2} (b c d) \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx-(b c e) \int \frac {\log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {\left (b c e \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 c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b e \sqrt {1-c^2 x^2}\right ) \int \frac {\arcsin (c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b e \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 c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b e \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 c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \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}}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b e \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 c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \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}}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b e \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 c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {d (a+b \text {arccosh}(c x))}{2 x^2}-\frac {i b e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \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}}+e (a+b \text {arccosh}(c x)) \log (x)-\frac {b e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b e \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 = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.40 \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {a d}{2 x^2}+\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b d \text {arccosh}(c x)}{2 x^2}+a e \log (x)+\frac {1}{2} b e \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]
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Time = 0.60 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.54
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a e \ln \left (c x \right )}{c^{2}}-\frac {a d}{2 c^{2} x^{2}}+\frac {b \left (-\frac {e \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {d \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )\right )}{2 x^{2}}+\ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) e \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) e}{2}\right )}{c^{2}}\right )\) | \(135\) |
default | \(c^{2} \left (\frac {a e \ln \left (c x \right )}{c^{2}}-\frac {a d}{2 c^{2} x^{2}}+\frac {b \left (-\frac {e \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {d \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )\right )}{2 x^{2}}+\ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) e \,\operatorname {arccosh}\left (c x \right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) e}{2}\right )}{c^{2}}\right )\) | \(135\) |
parts | \(-\frac {a d}{2 x^{2}}+a e \ln \left (x \right )+b \,c^{2} \left (-\frac {e \operatorname {arccosh}\left (c x \right )^{2}}{2 c^{2}}-\frac {d \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\frac {e \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{c^{2}}+\frac {e \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2 c^{2}}\right )\) | \(135\) |
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{3}}\, dx \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^3} \,d x \]
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