Optimal. Leaf size=70 \[ \frac{1}{2} \sqrt{x^4-1} \sec ^{-1}(x)-\frac{\sqrt{x^4-1}}{2 \sqrt{1-\frac{1}{x^2}} x}+\frac{1}{2} \tanh ^{-1}\left (\frac{\sqrt{1-\frac{1}{x^2}} x}{\sqrt{x^4-1}}\right ) \]
[Out]
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Rubi [A] time = 0.333074, antiderivative size = 94, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ \frac{1}{2} \sqrt{x^4-1} \sec ^{-1}(x)-\frac{\sqrt{x^4-1}}{2 \sqrt{1-\frac{1}{x^2}} x}+\frac{\sqrt{1-x^2} \tan ^{-1}\left (\frac{\sqrt{x^4-1}}{\sqrt{1-x^2}}\right )}{2 \sqrt{1-\frac{1}{x^2}} x} \]
Antiderivative was successfully verified.
[In] Int[(x^3*ArcSec[x])/Sqrt[-1 + x^4],x]
[Out]
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Rubi in Sympy [A] time = 10.8354, size = 68, normalized size = 0.97 \[ \frac{x \operatorname{atanh}{\left (\frac{\sqrt{x^{4} - 1}}{\sqrt{x^{2} - 1}} \right )}}{2 \sqrt{x^{2}}} - \frac{x \sqrt{x^{4} - 1}}{2 \sqrt{x^{2} - 1} \sqrt{x^{2}}} + \frac{\sqrt{x^{4} - 1} \operatorname{asec}{\left (x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*asec(x)/(x**4-1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.187791, size = 88, normalized size = 1.26 \[ \frac{1}{2} \left (\sqrt{x^4-1} \sec ^{-1}(x)-\log \left (x-x^3\right )-\frac{\sqrt{1-\frac{1}{x^2}} \sqrt{x^4-1} x}{x^2-1}+\log \left (-x^2-\sqrt{1-\frac{1}{x^2}} \sqrt{x^4-1} x+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*ArcSec[x])/Sqrt[-1 + x^4],x]
[Out]
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Maple [F] time = 0.667, size = 0, normalized size = 0. \[ \int{{x}^{3}{\rm arcsec} \left (x\right ){\frac{1}{\sqrt{{x}^{4}-1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*arcsec(x)/(x^4-1)^(1/2),x)
[Out]
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Maxima [A] time = 1.6877, size = 38, normalized size = 0.54 \[ \frac{1}{2} \, \sqrt{x^{4} - 1} \operatorname{arcsec}\left (x\right ) - \frac{1}{2} \, \sqrt{x^{2} + 1} + \frac{1}{2} \, \operatorname{arsinh}\left (\frac{1}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3*arcsec(x)/sqrt(x^4 - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240511, size = 149, normalized size = 2.13 \[ \frac{{\left (x^{2} - 1\right )} \log \left (\frac{x^{2} + \sqrt{x^{4} - 1} \sqrt{x^{2} - 1} - 1}{x^{2} - 1}\right ) -{\left (x^{2} - 1\right )} \log \left (-\frac{x^{2} - \sqrt{x^{4} - 1} \sqrt{x^{2} - 1} - 1}{x^{2} - 1}\right ) + 2 \, \sqrt{x^{4} - 1}{\left ({\left (x^{2} - 1\right )} \operatorname{arcsec}\left (x\right ) - \sqrt{x^{2} - 1}\right )}}{4 \,{\left (x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3*arcsec(x)/sqrt(x^4 - 1),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**3*asec(x)/(x**4-1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215627, size = 70, normalized size = 1. \[ \frac{1}{2} \, \sqrt{x^{4} - 1} \arccos \left (\frac{1}{x}\right ) - \frac{2 \, \sqrt{x^{2} + 1} -{\rm ln}\left (\sqrt{x^{2} + 1} + 1\right ) +{\rm ln}\left (\sqrt{x^{2} + 1} - 1\right )}{4 \,{\rm sign}\left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3*arcsec(x)/sqrt(x^4 - 1),x, algorithm="giac")
[Out]