Integrand size = 21, antiderivative size = 370 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx=-\frac {b e \left (6 c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{6 c^3}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x^3}{12 c}+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \]
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Time = 0.85 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6438, 272, 45, 5958, 6874, 465, 97, 2365, 2363, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx=-d^2 \log \left (\frac {1}{x}\right ) \left (a+b \text {sech}^{-1}(c x)\right )+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x)}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {b e x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \left (6 c^2 d+e\right )}{6 c^3}-\frac {b e^2 x^3 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{12 c} \]
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Rule 45
Rule 97
Rule 272
Rule 465
Rule 2221
Rule 2317
Rule 2363
Rule 2365
Rule 2438
Rule 3798
Rule 4721
Rule 5958
Rule 6438
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (e+d x^2\right )^2 \left (a+b \text {arccosh}\left (\frac {x}{c}\right )\right )}{x^5} \, dx,x,\frac {1}{x}\right ) \\ & = d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {-\frac {e \left (e+4 d x^2\right )}{4 x^4}+d^2 \log (x)}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \left (-\frac {e \left (e+4 d x^2\right )}{4 x^4 \sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}}+\frac {d^2 \log (x)}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}}\right ) \, dx,x,\frac {1}{x}\right )}{c} \\ & = d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {(b e) \text {Subst}\left (\int \frac {e+4 d x^2}{x^4 \sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{4 c} \\ & = -\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x^3}{12 c}+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (b e \left (6 c^2 d+e\right )\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{6 c^3}+\frac {\left (b d^2 \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = -\frac {b e \left (6 c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{6 c^3}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x^3}{12 c}+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (b d^2 \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {\arcsin \left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = -\frac {b e \left (6 c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{6 c^3}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x^3}{12 c}+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (b d^2 \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = -\frac {b e \left (6 c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{6 c^3}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x^3}{12 c}+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (2 i b d^2 \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = -\frac {b e \left (6 c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{6 c^3}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x^3}{12 c}+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {\left (b d^2 \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = -\frac {b e \left (6 c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{6 c^3}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x^3}{12 c}+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {\left (i b d^2 \sqrt {1-\frac {1}{c^2 x^2}}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ & = -\frac {b e \left (6 c^2 d+e\right ) \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{6 c^3}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x^3}{12 c}+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-d^2 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.48 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx=a d e x^2+\frac {1}{4} a e^2 x^4-\frac {b d e \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c^2}-\frac {b e^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (2+c^2 x^2\right )}{12 c^4}+b d e x^2 \text {sech}^{-1}(c x)+\frac {1}{4} b e^2 x^4 \text {sech}^{-1}(c x)-\frac {1}{2} b d^2 \text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+a d^2 \log (x)+\frac {1}{2} b d^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right ) \]
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Time = 1.23 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.75
method | result | size |
parts | \(a \left (\frac {e^{2} x^{4}}{4}+d e \,x^{2}+d^{2} \ln \left (x \right )\right )+b \left (\frac {\operatorname {arcsech}\left (c x \right )^{2} d^{2}}{2}+\frac {e \left (12 \,\operatorname {arcsech}\left (c x \right ) c^{4} d \,x^{2}+3 e \,\operatorname {arcsech}\left (c x \right ) c^{4} x^{4}-12 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} d x -\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e \,c^{3} x^{3}-2 \sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, e c x +12 c^{2} d +2 e \right )}{12 c^{4}}-d^{2} \operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {d^{2} \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\right )\) | \(276\) |
derivativedivides | \(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {b \left (\frac {c^{4} d^{2} \operatorname {arcsech}\left (c x \right )^{2}}{2}+\frac {e \left (12 \,\operatorname {arcsech}\left (c x \right ) c^{4} d \,x^{2}+3 e \,\operatorname {arcsech}\left (c x \right ) c^{4} x^{4}-12 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} d x -\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e \,c^{3} x^{3}-2 \sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, e c x +12 c^{2} d +2 e \right )}{12}-\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right ) c^{4} d^{2} \operatorname {arcsech}\left (c x \right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right ) c^{4} d^{2}}{2}\right )}{c^{4}}\) | \(287\) |
default | \(a d e \,x^{2}+\frac {a \,e^{2} x^{4}}{4}+a \,d^{2} \ln \left (c x \right )+\frac {b \left (\frac {c^{4} d^{2} \operatorname {arcsech}\left (c x \right )^{2}}{2}+\frac {e \left (12 \,\operatorname {arcsech}\left (c x \right ) c^{4} d \,x^{2}+3 e \,\operatorname {arcsech}\left (c x \right ) c^{4} x^{4}-12 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{3} d x -\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e \,c^{3} x^{3}-2 \sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, e c x +12 c^{2} d +2 e \right )}{12}-\ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right ) c^{4} d^{2} \operatorname {arcsech}\left (c x \right )-\frac {\operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right ) c^{4} d^{2}}{2}\right )}{c^{4}}\) | \(287\) |
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\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \]
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\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \]
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