Optimal. Leaf size=370 \[ -d^2 \log \left (\frac {1}{x}\right ) \left (a+b \text {sech}^{-1}(c x)\right )+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x)}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {b e x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \left (6 c^2 d+e\right )}{6 c^3}-\frac {b e^2 x^3 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{12 c} \]
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Rubi [A] time = 1.10, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 14, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6303, 266, 43, 5790, 6742, 454, 95, 2328, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-d^2 \log \left (\frac {1}{x}\right ) \left (a+b \text {sech}^{-1}(c x)\right )+d e x^2 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {i b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {b d^2 \sqrt {1-\frac {1}{c^2 x^2}} \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x)}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {b e x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \left (6 c^2 d+e\right )}{6 c^3}-\frac {b e^2 x^3 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{12 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 95
Rule 266
Rule 454
Rule 2190
Rule 2279
Rule 2326
Rule 2328
Rule 2391
Rule 3717
Rule 4625
Rule 5790
Rule 6303
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (e+d x^2\right )^2 \left (a+b \cosh ^{-1}\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 \operatorname {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 \operatorname {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 ) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\sin ^{-1}\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 176, normalized size = 0.48 \[ a d^2 \log (x)+a d e x^2+\frac {1}{4} a e^2 x^4-\frac {b d e \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c^2}-\frac {b e^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^2 x^2+2\right )}{12 c^4}+\frac {1}{2} b d^2 \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right )-\frac {1}{2} b d^2 \text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right )\right )+b d e x^2 \text {sech}^{-1}(c x)+\frac {1}{4} b e^2 x^4 \text {sech}^{-1}(c x) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname {arsech}\left (c x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.82, size = 286, normalized size = 0.77 \[ \frac {a \,e^{2} x^{4}}{4}+a d e \,x^{2}+a \,d^{2} \ln \left (c x \right )+\frac {b \mathrm {arcsech}\left (c x \right )^{2} d^{2}}{2}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x^{3} e^{2}}{12 c}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x d e}{c}+\frac {b \,\mathrm {arcsech}\left (c x \right ) e^{2} x^{4}}{4}+b \,\mathrm {arcsech}\left (c x \right ) d e \,x^{2}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x \,e^{2}}{6 c^{3}}+\frac {b d e}{c^{2}}+\frac {b \,e^{2}}{6 c^{4}}-b \,d^{2} \mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {b \,d^{2} \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, a e^{2} x^{4} + a d e x^{2} + a d^{2} \log \relax (x) + \int b e^{2} x^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right ) + 2 \, b d e x \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right ) + \frac {b d^{2} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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