Optimal. Leaf size=17 \[ \frac{1}{7} \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
[Out]
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Rubi [A] time = 0.109498, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{1}{7} \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((7 + 5*x^2)*Sqrt[2 + x^2 - x^4]),x]
[Out]
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Rubi in Sympy [A] time = 17.7466, size = 17, normalized size = 1. \[ \frac{\Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(5*x**2+7)/(-x**4+x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0443375, size = 24, normalized size = 1.41 \[ -\frac{i \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )}{7 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((7 + 5*x^2)*Sqrt[2 + x^2 - x^4]),x]
[Out]
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Maple [B] time = 0.018, size = 48, normalized size = 2.8 \[{\frac{\sqrt{2}}{7}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(5*x^2+7)/(-x^4+x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(5*x**2+7)/(-x**4+x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)),x, algorithm="giac")
[Out]