Optimal. Leaf size=17 \[ \frac{\log \left (a^2+b^2 \sin ^2(x)\right )}{b^2} \]
[Out]
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Rubi [A] time = 0.0632104, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\log \left (a^2+b^2 \sin ^2(x)\right )}{b^2} \]
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 = 3.76368, size = 15, normalized size = 0.88 \[ \frac{\log{\left (a^{2} + b^{2} \sin ^{2}{\left (x \right )} \right )}}{b^{2}} \]
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 = 0.0223703, size = 23, normalized size = 1.35 \[ \frac{\log \left (2 a^2-b^2 \cos (2 x)+b^2\right )}{b^2} \]
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 = 0.031, size = 18, 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 = 1.3524, size = 23, normalized size = 1.35 \[ \frac{\log \left (b^{2} \sin \left (x\right )^{2} + a^{2}\right )}{b^{2}} \]
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 = 0.260401, size = 28, normalized size = 1.65 \[ \frac{\log \left (-b^{2} \cos \left (x\right )^{2} + a^{2} + b^{2}\right )}{b^{2}} \]
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 = 3.99181, size = 32, normalized size = 1.88 \[ 2 \left (\begin{cases} - \frac{\cos ^{2}{\left (x \right )}}{2 a^{2}} & \text{for}\: b^{2} = 0 \\\frac{\log{\left (a^{2} + b^{2} \sin ^{2}{\left (x \right )} \right )}}{2 b^{2}} & \text{otherwise} \end{cases}\right ) \]
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 = 0.238326, size = 104, normalized size = 6.12 \[ -\frac{2 \,{\rm ln}\left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}{b^{2}} + \frac{{\rm ln}\left ({\left | a^{2} - \frac{2 \, a^{2}{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac{4 \, b^{2}{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{a^{2}{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} \right |}\right )}{b^{2}} \]
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]