Optimal. Leaf size=19 \[ -\frac{\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \]
[Out]
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Rubi [A] time = 0.066265, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Int[Sin[2*x]/(a^2 - b^2*Sin[x]^2),x]
[Out]
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Rubi in Sympy [A] time = 4.00925, size = 17, normalized size = 0.89 \[ - \frac{\log{\left (a^{2} - b^{2} \sin ^{2}{\left (x \right )} \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(sin(2*x)/(a**2-b**2*sin(x)**2),x)
[Out]
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Mathematica [A] time = 0.0216296, size = 25, normalized size = 1.32 \[ -\frac{\log \left (2 a^2+b^2 \cos (2 x)-b^2\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sin[2*x]/(a^2 - b^2*Sin[x]^2),x]
[Out]
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Maple [A] time = 0.032, size = 20, normalized size = 1.1 \[ -{\frac{\ln \left ({a}^{2}-{b}^{2} \left ( \sin \left ( x \right ) \right ) ^{2} \right ) }{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(sin(2*x)/(a^2-b^2*sin(x)^2),x)
[Out]
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Maxima [A] time = 1.42873, size = 27, normalized size = 1.42 \[ -\frac{\log \left (b^{2} \sin \left (x\right )^{2} - a^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sin(2*x)/(b^2*sin(x)^2 - a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247466, size = 31, normalized size = 1.63 \[ -\frac{\log \left (b^{2} \cos \left (x\right )^{2} + a^{2} - b^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sin(2*x)/(b^2*sin(x)^2 - a^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.74202, size = 34, normalized size = 1.79 \[ 2 \left (\begin{cases} - \frac{\cos ^{2}{\left (x \right )}}{2 a^{2}} & \text{for}\: b^{2} = 0 \\- \frac{\log{\left (a^{2} - b^{2} \sin ^{2}{\left (x \right )} \right )}}{2 b^{2}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sin(2*x)/(a**2-b**2*sin(x)**2),x)
[Out]
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GIAC/XCAS [A] time = 0.20716, size = 104, normalized size = 5.47 \[ -\frac{{\rm ln}\left (a^{2} - \frac{2 \, a^{2}{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{4 \, b^{2}{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{a^{2}{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{b^{2}} + \frac{2 \,{\rm ln}\left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sin(2*x)/(b^2*sin(x)^2 - a^2),x, algorithm="giac")
[Out]