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

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

[Out]

sin(x)/a/(a*cos(x)+b*sin(x))

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3075} \[ \frac {\sin (x)}{a (a \cos (x)+b \sin (x))} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[x] + b*Sin[x])^(-2),x]

[Out]

Sin[x]/(a*(a*Cos[x] + b*Sin[x]))

Rule 3075

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-2), x_Symbol] :> Simp[Sin[c + d*x]/(a*d*
(a*Cos[c + d*x] + b*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac {\sin (x)}{a (a \cos (x)+b \sin (x))}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 1.00 \[ \frac {\sin (x)}{a (a \cos (x)+b \sin (x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[x] + b*Sin[x])^(-2),x]

[Out]

Sin[x]/(a*(a*Cos[x] + b*Sin[x]))

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fricas [B]  time = 0.44, size = 39, normalized size = 2.29 \[ -\frac {b \cos \relax (x) - a \sin \relax (x)}{{\left (a^{3} + a b^{2}\right )} \cos \relax (x) + {\left (a^{2} b + b^{3}\right )} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(x)+b*sin(x))^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 13, normalized size = 0.76 \[ -\frac {1}{{\left (b \tan \relax (x) + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(x)+b*sin(x))^2,x, algorithm="giac")

[Out]

-1/((b*tan(x) + a)*b)

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maple [A]  time = 0.53, size = 14, normalized size = 0.82 \[ -\frac {1}{b \left (a +b \tan \relax (x )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cos(x)+b*sin(x))^2,x)

[Out]

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

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maxima [A]  time = 0.31, size = 14, normalized size = 0.82 \[ -\frac {1}{b^{2} \tan \relax (x) + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(x)+b*sin(x))^2,x, algorithm="maxima")

[Out]

-1/(b^2*tan(x) + a*b)

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mupad [B]  time = 0.43, size = 29, normalized size = 1.71 \[ \frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )}{a\,\left (-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cos(x) + b*sin(x))^2,x)

[Out]

(2*tan(x/2))/(a*(a + 2*b*tan(x/2) - a*tan(x/2)^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(x)+b*sin(x))**2,x)

[Out]

Timed out

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