Integrand size = 21, antiderivative size = 342 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {b e \left (16 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b e \left (16 c^2 d+3 e\right ) \text {arccosh}(c x)}{32 c^4}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))-\frac {i b d^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+d^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.57 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {272, 45, 5958, 6874, 92, 54, 102, 12, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=d^2 \log (x) (a+b \text {arccosh}(c x))+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))-\frac {3 b e^2 \text {arccosh}(c x)}{32 c^4}-\frac {b d e \text {arccosh}(c x)}{2 c^2}-\frac {i b d^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {i b d^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^2 \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b e^2 x \sqrt {c x-1} \sqrt {c x+1}}{32 c^3}-\frac {b d e x \sqrt {c x-1} \sqrt {c x+1}}{2 c}-\frac {b e^2 x^3 \sqrt {c x-1} \sqrt {c x+1}}{16 c} \]
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Rule 12
Rule 45
Rule 54
Rule 92
Rule 102
Rule 272
Rule 2221
Rule 2317
Rule 2363
Rule 2365
Rule 2438
Rule 3798
Rule 4721
Rule 5958
Rule 6874
Rubi steps \begin{align*} \text {integral}& = d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))+d^2 (a+b \text {arccosh}(c x)) \log (x)-(b c) \int \frac {d e x^2+\frac {e^2 x^4}{4}+d^2 \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))+d^2 (a+b \text {arccosh}(c x)) \log (x)-(b c) \int \left (\frac {d e x^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^2 x^4}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx \\ & = d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))+d^2 (a+b \text {arccosh}(c x)) \log (x)-\left (b c d^2\right ) \int \frac {\log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-(b c d e) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{4} \left (b c e^2\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {b d e x \sqrt {-1+c x} \sqrt {1+c x}}{2 c}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))+d^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {(b d e) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c}-\frac {\left (b e^2\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c}-\frac {\left (b c d^2 \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 d e x \sqrt {-1+c x} \sqrt {1+c x}}{2 c}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b d e \text {arccosh}(c x)}{2 c^2}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))+d^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b e^2\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 c}+\frac {\left (b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\arcsin (c x)}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b d e x \sqrt {-1+c x} \sqrt {1+c x}}{2 c}-\frac {3 b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b d e \text {arccosh}(c x)}{2 c^2}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))+d^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b e^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^3}+\frac {\left (b d^2 \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 d e x \sqrt {-1+c x} \sqrt {1+c x}}{2 c}-\frac {3 b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b d e \text {arccosh}(c x)}{2 c^2}-\frac {3 b e^2 \text {arccosh}(c x)}{32 c^4}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))-\frac {i b d^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^2 \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^2 \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 d e x \sqrt {-1+c x} \sqrt {1+c x}}{2 c}-\frac {3 b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b d e \text {arccosh}(c x)}{2 c^2}-\frac {3 b e^2 \text {arccosh}(c x)}{32 c^4}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))-\frac {i b d^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \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^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b d^2 \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 d e x \sqrt {-1+c x} \sqrt {1+c x}}{2 c}-\frac {3 b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b d e \text {arccosh}(c x)}{2 c^2}-\frac {3 b e^2 \text {arccosh}(c x)}{32 c^4}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))-\frac {i b d^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \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^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b d^2 \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 d e x \sqrt {-1+c x} \sqrt {1+c x}}{2 c}-\frac {3 b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e^2 x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b d e \text {arccosh}(c x)}{2 c^2}-\frac {3 b e^2 \text {arccosh}(c x)}{32 c^4}+d e x^2 (a+b \text {arccosh}(c x))+\frac {1}{4} e^2 x^4 (a+b \text {arccosh}(c x))-\frac {i b d^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \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^2 (a+b \text {arccosh}(c x)) \log (x)-\frac {b d^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d^2 \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 = 0.39 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.67 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=a d e x^2+\frac {1}{4} a e^2 x^4+b d e x^2 \text {arccosh}(c x)+\frac {1}{4} b e^2 x^4 \text {arccosh}(c x)-\frac {b d e \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{2 c^2}-\frac {b e^2 \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )+6 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{32 c^4}+\frac {1}{2} b d^2 \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )+a d^2 \log (x)-\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]
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Time = 0.99 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.65
method | result | size |
parts | \(a \left (\frac {e^{2} x^{4}}{4}+d e \,x^{2}+d^{2} \ln \left (x \right )\right )+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{2} x^{4}}{4}-\frac {b d e \,\operatorname {arccosh}\left (c x \right )}{2 c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d e x}{2 c}+b \,d^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {3 b \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x}{32 c^{3}}+b \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}-\frac {b \,d^{2} \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(223\) |
derivativedivides | \(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{2} x^{4}}{4}-\frac {3 b \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d e x}{2 c}+b \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}-\frac {b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}+b \,d^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {b d e \,\operatorname {arccosh}\left (c x \right )}{2 c^{2}}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x}{32 c^{3}}-\frac {b \,d^{2} \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(225\) |
default | \(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {b \,\operatorname {arccosh}\left (c x \right ) e^{2} x^{4}}{4}-\frac {3 b \,e^{2} \operatorname {arccosh}\left (c x \right )}{32 c^{4}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, d e x}{2 c}+b \,\operatorname {arccosh}\left (c x \right ) d e \,x^{2}-\frac {b \,e^{2} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}+b \,d^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {b d e \,\operatorname {arccosh}\left (c x \right )}{2 c^{2}}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x}{32 c^{3}}-\frac {b \,d^{2} \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b \,d^{2} \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\) | \(225\) |
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\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \]
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\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2}{x} \,d x \]
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