Integrand size = 21, antiderivative size = 321 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {b c d^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))-\frac {i b d e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.59 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {272, 45, 5958, 12, 6874, 97, 92, 54, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+2 d e \log (x) (a+b \text {arccosh}(c x))+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {i b d e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {i b d e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b d e \sqrt {1-c^2 x^2} \log (x) \arcsin (c x)}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c d^2 \sqrt {c x-1} \sqrt {c x+1}}{2 x}-\frac {b e^2 x \sqrt {c x-1} \sqrt {c x+1}}{4 c} \]
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Rule 12
Rule 45
Rule 54
Rule 92
Rule 97
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}& = -\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))+2 d e (a+b \text {arccosh}(c x)) \log (x)-(b c) \int \frac {-\frac {d^2}{x^2}+e^2 x^2+4 d e \log (x)}{2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {1}{2} (b c) \int \frac {-\frac {d^2}{x^2}+e^2 x^2+4 d e \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {1}{2} (b c) \int \left (-\frac {d^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 d e \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))+2 d e (a+b \text {arccosh}(c x)) \log (x)+\frac {1}{2} \left (b c d^2\right ) \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx-(2 b c d e) \int \frac {\log (x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{2} \left (b c e^2\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {b c d^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {\left (b e^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 c}-\frac {\left (2 b c d 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^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b d 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^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b d 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^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))-\frac {i b d e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{\sqrt {-1+c x} \sqrt {1+c x}}+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 i b d 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^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))-\frac {i b d e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d 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}}+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b d 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^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))-\frac {i b d e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d 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}}+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b d 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 )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c d^2 \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b e^2 x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b e^2 \text {arccosh}(c x)}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \text {arccosh}(c x))-\frac {i b d e \sqrt {1-c^2 x^2} \arcsin (c x)^2}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d 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}}+2 d e (a+b \text {arccosh}(c x)) \log (x)-\frac {2 b d e \sqrt {1-c^2 x^2} \arcsin (c x) \log (x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d e \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.54 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {1}{4} \left (-\frac {2 a d^2}{x^2}+2 a e^2 x^2+\frac {2 b d^2 \left (c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)\right )}{x^2}+\frac {b e^2 \left (-c x \sqrt {-1+c x} \sqrt {1+c x}+2 c^2 x^2 \text {arccosh}(c x)-2 \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )}{c^2}+8 a d e \log (x)+4 b d e \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )\right ) \]
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Time = 1.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.61
method | result | size |
parts | \(a \left (\frac {e^{2} x^{2}}{2}-\frac {d^{2}}{2 x^{2}}+2 d e \ln \left (x \right )\right )-b \operatorname {arccosh}\left (c x \right )^{2} d e +\frac {b \,e^{2} \operatorname {arccosh}\left (c x \right ) x^{2}}{2}-\frac {b \,e^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {b \,e^{2} \operatorname {arccosh}\left (c x \right )}{4 c^{2}}+\frac {b c \,d^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{2 x}-\frac {b \,c^{2} d^{2}}{2}-\frac {b \,d^{2} \operatorname {arccosh}\left (c x \right )}{2 x^{2}}+2 b e d \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+b e d \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\) | \(196\) |
derivativedivides | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}-\frac {b d e \operatorname {arccosh}\left (c x \right )^{2}}{c^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x}{4 c^{3}}-\frac {b \,e^{2} \operatorname {arccosh}\left (c x \right )}{4 c^{4}}+\frac {b \,d^{2} \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b}{2}-\frac {b \,\operatorname {arccosh}\left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {2 b \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d e \,\operatorname {arccosh}\left (c x \right )}{c^{2}}+\frac {b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d e}{c^{2}}\right )\) | \(225\) |
default | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}-\frac {b d e \operatorname {arccosh}\left (c x \right )^{2}}{c^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2} x}{4 c^{3}}-\frac {b \,e^{2} \operatorname {arccosh}\left (c x \right )}{4 c^{4}}+\frac {b \,d^{2} \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b}{2}-\frac {b \,\operatorname {arccosh}\left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {2 b \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d e \,\operatorname {arccosh}\left (c x \right )}{c^{2}}+\frac {b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) d e}{c^{2}}\right )\) | \(225\) |
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\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \]
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\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^2 (a+b \text {arccosh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2}{x^3} \,d x \]
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