3.146 \(\int \frac{1}{(b \cos (x)+a \sin (x))^2} \, dx\)

Optimal. Leaf size=17 \[ \frac{\sin (x)}{b (a \sin (x)+b \cos (x))} \]

[Out]

Sin[x]/(b*(b*Cos[x] + a*Sin[x]))

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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]

Sin[x]/(b*(b*Cos[x] + a*Sin[x]))

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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]

sin(x)/(b*(a*sin(x) + b*cos(x)))

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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]

Sin[x]/(b*(b*Cos[x] + a*Sin[x]))

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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]

-1/a/(a*tan(x)+b)

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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]

-1/(a^2*tan(x) + a*b)

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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]

-(a*cos(x) - b*sin(x))/((a^2*b + b^3)*cos(x) + (a^3 + a*b^2)*sin(x))

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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]

Timed 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]

-1/((a*tan(x) + b)*a)