Optimal. Leaf size=17 \[ \frac{\sin (x)}{b (a \sin (x)+b \cos (x))} \]
[Out]
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Rubi [A] time = 0.0217313, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\sin (x)}{b (a \sin (x)+b \cos (x))} \]
Antiderivative was successfully verified.
[In] Int[(b*Cos[x] + a*Sin[x])^(-2),x]
[Out]
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Rubi in Sympy [A] time = 0.675075, size = 14, normalized size = 0.82 \[ \frac{\sin{\left (x \right )}}{b \left (a \sin{\left (x \right )} + b \cos{\left (x \right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*cos(x)+a*sin(x))**2,x)
[Out]
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Mathematica [A] time = 0.0377791, size = 17, normalized size = 1. \[ \frac{\sin (x)}{b (a \sin (x)+b \cos (x))} \]
Antiderivative was successfully verified.
[In] Integrate[(b*Cos[x] + a*Sin[x])^(-2),x]
[Out]
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Maple [A] time = 0.234, size = 14, normalized size = 0.8 \[ -{\frac{1}{a \left ( a\tan \left ( x \right ) +b \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*cos(x)+a*sin(x))^2,x)
[Out]
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Maxima [A] time = 1.34446, size = 19, normalized size = 1.12 \[ -\frac{1}{a^{2} \tan \left (x\right ) + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*cos(x) + a*sin(x))^(-2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218437, size = 53, normalized size = 3.12 \[ -\frac{a \cos \left (x\right ) - b \sin \left (x\right )}{{\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) +{\left (a^{3} + a b^{2}\right )} \sin \left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*cos(x) + a*sin(x))^(-2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*cos(x)+a*sin(x))**2,x)
[Out]
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GIAC/XCAS [A] time = 0.216294, size = 18, normalized size = 1.06 \[ -\frac{1}{{\left (a \tan \left (x\right ) + b\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*cos(x) + a*sin(x))^(-2),x, algorithm="giac")
[Out]