Optimal. Leaf size=186 \[ -d^2 \log \left (\frac {1}{x}\right ) \left (a+b \sec ^{-1}(c x)\right )+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e^2 x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}-\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (6 c^2 d+e\right )}{6 c^3}-\frac {1}{2} i b d^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \]
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Rubi [A] time = 0.41, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5240, 266, 43, 4732, 6742, 453, 264, 2326, 4625, 3717, 2190, 2279, 2391} \[ -\frac {1}{2} i b d^2 \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-d^2 \log \left (\frac {1}{x}\right ) \left (a+b \sec ^{-1}(c x)\right )+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (6 c^2 d+e\right )}{6 c^3}-\frac {b e^2 x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}-\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 264
Rule 266
Rule 453
Rule 2190
Rule 2279
Rule 2326
Rule 2391
Rule 3717
Rule 4625
Rule 4732
Rule 5240
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \sec ^{-1}(c x)\right )}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (e+d x^2\right )^2 \left (a+b \cos ^{-1}\left (\frac {x}{c}\right )\right )}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-d^2 \left (a+b \sec ^{-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^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-d^2 \left (a+b \sec ^{-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^2}{c^2}}}+\frac {d^2 \log (x)}{\sqrt {1-\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{c}\\ &=d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-d^2 \left (a+b \sec ^{-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^2}{c^2}}} \, 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^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{4 c}\\ &=-\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\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^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{6 c^3}\\ &=-\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\left (b d^2\right ) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\left (2 i b d^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\left (b d^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} \left (i b d^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=-\frac {b e \left (6 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}-\frac {b e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {1}{2} i b d^2 \csc ^{-1}(c x)^2+d e x^2 \left (a+b \sec ^{-1}(c x)\right )+\frac {1}{4} e^2 x^4 \left (a+b \sec ^{-1}(c x)\right )+b d^2 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )-b d^2 \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d^2 \left (a+b \sec ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} i b d^2 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.39, size = 160, normalized size = 0.86 \[ a d^2 \log (x)+a d e x^2+\frac {1}{4} a e^2 x^4+\frac {b d e x \left (c x \sec ^{-1}(c x)-\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}-\frac {b e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 x^2+2\right )}{12 c^3}+\frac {1}{2} i b d^2 \left (\text {Li}_2\left (-e^{2 i \sec ^{-1}(c x)}\right )+\sec ^{-1}(c x) \left (\sec ^{-1}(c x)+2 i \log \left (1+e^{2 i \sec ^{-1}(c x)}\right )\right )\right )+\frac {1}{4} b e^2 x^4 \sec ^{-1}(c x) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} + {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \operatorname {arcsec}\left (c x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.52, size = 242, normalized size = 1.30 \[ \frac {a \,x^{4} e^{2}}{4}+a e d \,x^{2}+a \,d^{2} \ln \left (c x \right )+\frac {i b \,d^{2} \mathrm {arcsec}\left (c x \right )^{2}}{2}+\frac {b \,\mathrm {arcsec}\left (c x \right ) x^{4} e^{2}}{4}+b e \,\mathrm {arcsec}\left (c x \right ) x^{2} d -\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3} e^{2}}{12 c}-\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d e}{c}-\frac {i b d e}{c^{2}}-\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \,e^{2}}{6 c^{3}}-\frac {i b \,e^{2}}{6 c^{4}}-b \,d^{2} \mathrm {arcsec}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )+\frac {i b \,d^{2} \polylog \left (2, -\left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )^{2}\right )}{2} \]
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 \[ \frac {1}{4} \, a e^{2} x^{4} + a d e x^{2} + a d^{2} \log \relax (x) - \frac {-2 i \, b c^{4} e^{2} x^{4} \log \relax (c) - 4 i \, b c^{4} d^{2} \log \left (-c x + 1\right ) \log \relax (x) - 4 i \, b c^{4} d^{2} \log \relax (x)^{2} - 4 i \, b c^{4} d^{2} {\rm Li}_2\left (c x\right ) - 4 i \, b c^{4} d^{2} {\rm Li}_2\left (-c x\right ) + i \, {\left (4 \, {\left ({\left (\log \left (c x + 1\right ) + \log \left (c x - 1\right ) - 2 \, \log \relax (x)\right )} \log \relax (x) - \log \left (c x - 1\right ) \log \relax (x) + \log \left (-c x + 1\right ) \log \relax (x) + \log \relax (x)^{2} + {\rm Li}_2\left (c x\right ) + {\rm Li}_2\left (-c x\right )\right )} b d^{2} + b e^{2} {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c x + 1\right )}{c^{4}} + \frac {\log \left (c x - 1\right )}{c^{4}}\right )} + 4 \, b d e {\left (\frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )}\right )} c^{4} + \frac {2}{3} \, {\left (12 \, b d^{2} \int \frac {\sqrt {c x + 1} \sqrt {c x - 1} \log \relax (x)}{c^{2} x^{3} - x}\,{d x} + \frac {12 \, \sqrt {c x + 1} \sqrt {c x - 1} b d e}{c^{2}} + \frac {{\left (c^{2} x^{2} + 2\right )} \sqrt {c x + 1} \sqrt {c x - 1} b e^{2}}{c^{4}}\right )} c^{4} + {\left (-8 i \, b c^{4} d e \log \relax (c) - i \, b c^{2} e^{2}\right )} x^{2} - 2 \, {\left (b c^{4} e^{2} x^{4} + 4 \, b c^{4} d e x^{2} + 4 \, b c^{4} d^{2} \log \relax (x)\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (i \, b c^{4} e^{2} x^{4} + 4 i \, b c^{4} d e x^{2} + 4 i \, b c^{4} d^{2} \log \relax (x)\right )} \log \left (c^{2} x^{2}\right ) + {\left (-4 i \, b c^{4} d^{2} \log \relax (x) - 4 i \, b c^{2} d e - i \, b e^{2}\right )} \log \left (c x + 1\right ) + {\left (-4 i \, b c^{2} d e - i \, b e^{2}\right )} \log \left (c x - 1\right ) + {\left (-2 i \, b c^{4} e^{2} x^{4} - 8 i \, b c^{4} d e x^{2} - 8 i \, b c^{4} d^{2} \log \relax (c)\right )} \log \relax (x)}{8 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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