Optimal. Leaf size=133 \[ \frac{x \sec ^{-1}(x)^3}{8 \sqrt{x^2}}-\frac{3 \sqrt{x^2-1} \sec ^{-1}(x)^2}{8 x^2}+\frac{9 x \sec ^{-1}(x)}{64 \sqrt{x^2}}-\frac{3 \sec ^{-1}(x)}{8 x \sqrt{x^2}}+\frac{\sqrt{x^2-1} \left (17 x^2-2\right )}{64 x^4}-\frac{\left (x^2-1\right )^{3/2} \sec ^{-1}(x)^2}{4 x^4}+\frac{\left (x^2-1\right )^2 \sec ^{-1}(x)}{8 x^3 \sqrt{x^2}} \]
[Out]
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Rubi [A] time = 0.308679, antiderivative size = 172, normalized size of antiderivative = 1.29, number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{\left (1-\frac{1}{x^2}\right )^{3/2}}{32 \sqrt{x^2}}+\frac{15 \sqrt{1-\frac{1}{x^2}}}{64 \sqrt{x^2}}-\frac{9 \sqrt{x^2} \csc ^{-1}(x)}{64 x}+\frac{\sqrt{x^2} \sec ^{-1}(x)^3}{8 x}-\frac{\left (1-\frac{1}{x^2}\right )^{3/2} \sec ^{-1}(x)^2}{4 \sqrt{x^2}}-\frac{3 \sqrt{1-\frac{1}{x^2}} \sec ^{-1}(x)^2}{8 \sqrt{x^2}}+\frac{\left (1-\frac{1}{x^2}\right )^2 \sqrt{x^2} \sec ^{-1}(x)}{8 x}-\frac{3 \sqrt{x^2} \sec ^{-1}(x)}{8 x^3} \]
Antiderivative was successfully verified.
[In] Int[((-1 + x^2)^(3/2)*ArcSec[x]^2)/x^5,x]
[Out]
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Rubi in Sympy [A] time = 14.8492, size = 180, normalized size = 1.35 \[ \frac{\left (1 - \frac{1}{x^{2}}\right )^{2} \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{8 x} + \frac{\sqrt{x^{2}} \operatorname{acos}^{3}{\left (\frac{1}{x} \right )}}{8 x} - \frac{9 \sqrt{x^{2}} \operatorname{asin}{\left (\frac{1}{x} \right )}}{64 x} - \frac{\left (1 - \frac{1}{x^{2}}\right )^{\frac{3}{2}} \sqrt{x^{2}} \operatorname{acos}^{2}{\left (\frac{1}{x} \right )}}{4 x^{2}} + \frac{\left (1 - \frac{1}{x^{2}}\right )^{\frac{3}{2}} \sqrt{x^{2}}}{32 x^{2}} - \frac{3 \sqrt{1 - \frac{1}{x^{2}}} \sqrt{x^{2}} \operatorname{acos}^{2}{\left (\frac{1}{x} \right )}}{8 x^{2}} + \frac{15 \sqrt{1 - \frac{1}{x^{2}}} \sqrt{x^{2}}}{64 x^{2}} - \frac{3 \sqrt{x^{2}} \operatorname{acos}{\left (\frac{1}{x} \right )}}{8 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2-1)**(3/2)*asec(x)**2/x**5,x)
[Out]
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Mathematica [A] time = 0.324673, size = 84, normalized size = 0.63 \[ \frac{\sqrt{x^2-1} \left (32 \sec ^{-1}(x)^3+4 \sec ^{-1}(x) \left (\cos \left (4 \sec ^{-1}(x)\right )-16 \cos \left (2 \sec ^{-1}(x)\right )\right )+8 \sec ^{-1}(x)^2 \left (\sin \left (4 \sec ^{-1}(x)\right )-8 \sin \left (2 \sec ^{-1}(x)\right )\right )+32 \sin \left (2 \sec ^{-1}(x)\right )-\sin \left (4 \sec ^{-1}(x)\right )\right )}{256 \sqrt{1-\frac{1}{x^2}} x} \]
Antiderivative was successfully verified.
[In] Integrate[((-1 + x^2)^(3/2)*ArcSec[x]^2)/x^5,x]
[Out]
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Maple [C] time = 0.455, size = 327, normalized size = 2.5 \[{\frac{x \left ({\rm arcsec} \left (x\right ) \right ) ^{3}}{8}\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{\frac{1}{\sqrt{{x}^{2}-1}}}}-{\frac{4\,i{\rm arcsec} \left (x\right )+8\, \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-1}{512\,{x}^{4}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{5}-8\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}+4\,{x}^{4}+8\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-12\,{x}^{2}+8 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}-{\frac{2\, \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-1+2\,i{\rm arcsec} \left (x\right )}{16\,{x}^{2}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+2\,{x}^{2}-2 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{2\, \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-1-2\,i{\rm arcsec} \left (x\right )}{16\,{x}^{2}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-2\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x-2\,{x}^{2}+2 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}}+{\frac{-4\,i{\rm arcsec} \left (x\right )+8\, \left ({\rm arcsec} \left (x\right ) \right ) ^{2}-1}{512\,{x}^{4}} \left ( i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{5}-8\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}{x}^{3}-4\,{x}^{4}+8\,i\sqrt{{\frac{{x}^{2}-1}{{x}^{2}}}}x+12\,{x}^{2}-8 \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2-1)^(3/2)*arcsec(x)^2/x^5,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226313, size = 80, normalized size = 0.6 \[ \frac{8 \, x^{4} \operatorname{arcsec}\left (x\right )^{3} +{\left (17 \, x^{4} - 40 \, x^{2} + 8\right )} \operatorname{arcsec}\left (x\right ) -{\left (8 \,{\left (5 \, x^{2} - 2\right )} \operatorname{arcsec}\left (x\right )^{2} - 17 \, x^{2} + 2\right )} \sqrt{x^{2} - 1}}{64 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2-1)**(3/2)*asec(x)**2/x**5,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} - 1\right )}^{\frac{3}{2}} \operatorname{arcsec}\left (x\right )^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^2 - 1)^(3/2)*arcsec(x)^2/x^5,x, algorithm="giac")
[Out]