Optimal. Leaf size=39 \[ \frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{c^{3/2} \sqrt{d}}+\frac{a x}{c} \]
[Out]
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Rubi [A] time = 0.0718176, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{c^{3/2} \sqrt{d}}+\frac{a x}{c} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)/(c + d/x^2),x]
[Out]
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Rubi in Sympy [A] time = 11.336, size = 34, normalized size = 0.87 \[ \frac{a x}{c} - \frac{\left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{c^{\frac{3}{2}} \sqrt{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)/(c+d/x**2),x)
[Out]
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Mathematica [A] time = 0.0442703, size = 40, normalized size = 1.03 \[ \frac{a x}{c}-\frac{(a d-b c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )}{c^{3/2} \sqrt{d}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)/(c + d/x^2),x]
[Out]
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Maple [A] time = 0.007, size = 45, normalized size = 1.2 \[{\frac{ax}{c}}-{\frac{ad}{c}\arctan \left ({cx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{b\arctan \left ({cx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)/(c+d/x^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(c + d/x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234813, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \sqrt{-c d} a x -{\left (b c - a d\right )} \log \left (-\frac{2 \, c d x -{\left (c x^{2} - d\right )} \sqrt{-c d}}{c x^{2} + d}\right )}{2 \, \sqrt{-c d} c}, \frac{\sqrt{c d} a x +{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{c d} x}{d}\right )}{\sqrt{c d} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(c + d/x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.66471, size = 82, normalized size = 2.1 \[ \frac{a x}{c} + \frac{\sqrt{- \frac{1}{c^{3} d}} \left (a d - b c\right ) \log{\left (- c d \sqrt{- \frac{1}{c^{3} d}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{c^{3} d}} \left (a d - b c\right ) \log{\left (c d \sqrt{- \frac{1}{c^{3} d}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)/(c+d/x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.213993, size = 45, normalized size = 1.15 \[ \frac{a x}{c} + \frac{{\left (b c - a d\right )} \arctan \left (\frac{c x}{\sqrt{c d}}\right )}{\sqrt{c d} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/(c + d/x^2),x, algorithm="giac")
[Out]