Optimal. Leaf size=15 \[ \frac{\tanh ^{-1}\left (\frac{b \sin (x)}{a}\right )}{a b} \]
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Rubi [A] time = 0.0508859, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tanh ^{-1}\left (\frac{b \sin (x)}{a}\right )}{a b} \]
Antiderivative was successfully verified.
[In] Int[Cos[x]/(a^2 - b^2*Sin[x]^2),x]
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Rubi in Sympy [A] time = 5.70068, size = 10, normalized size = 0.67 \[ \frac{\operatorname{atanh}{\left (\frac{b \sin{\left (x \right )}}{a} \right )}}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cos(x)/(a**2-b**2*sin(x)**2),x)
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Mathematica [A] time = 0.013909, size = 15, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{b \sin (x)}{a}\right )}{a b} \]
Antiderivative was successfully verified.
[In] Integrate[Cos[x]/(a^2 - b^2*Sin[x]^2),x]
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Maple [B] time = 0.017, size = 34, normalized size = 2.3 \[{\frac{\ln \left ( a+b\sin \left ( x \right ) \right ) }{2\,ab}}-{\frac{\ln \left ( b\sin \left ( x \right ) -a \right ) }{2\,ab}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cos(x)/(a^2-b^2*sin(x)^2),x)
[Out]
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Maxima [A] time = 1.38631, size = 45, normalized size = 3. \[ \frac{\log \left (b \sin \left (x\right ) + a\right )}{2 \, a b} - \frac{\log \left (b \sin \left (x\right ) - a\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-cos(x)/(b^2*sin(x)^2 - a^2),x, algorithm="maxima")
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Fricas [A] time = 0.250197, size = 35, normalized size = 2.33 \[ \frac{\log \left (b \sin \left (x\right ) + a\right ) - \log \left (-b \sin \left (x\right ) + a\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-cos(x)/(b^2*sin(x)^2 - a^2),x, algorithm="fricas")
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Sympy [A] time = 1.26538, size = 44, normalized size = 2.93 \[ \begin{cases} \frac{\tilde{\infty }}{\sin{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{1}{b^{2} \sin{\left (x \right )}} & \text{for}\: a = 0 \\\frac{\sin{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{\log{\left (- \frac{a}{b} + \sin{\left (x \right )} \right )}}{2 a b} + \frac{\log{\left (\frac{a}{b} + \sin{\left (x \right )} \right )}}{2 a b} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(x)/(a**2-b**2*sin(x)**2),x)
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GIAC/XCAS [A] time = 0.202103, size = 47, normalized size = 3.13 \[ \frac{{\rm ln}\left ({\left | b \sin \left (x\right ) + a \right |}\right )}{2 \, a b} - \frac{{\rm ln}\left ({\left | b \sin \left (x\right ) - a \right |}\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-cos(x)/(b^2*sin(x)^2 - a^2),x, algorithm="giac")
[Out]