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

Optimal. Leaf size=26 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

-(ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]))

________________________________________________________________________________________

Rubi [A]  time = 0.0272752, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3190, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(a + b*Cos[x]^2),x]

[Out]

-(ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]))

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0192716, size = 26, normalized size = 1. \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(a + b*Cos[x]^2),x]

[Out]

-(ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]))

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 18, normalized size = 0.7 \begin{align*} -{\arctan \left ({b\cos \left ( x \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a*b)^(1/2)*arctan(b*cos(x)/(a*b)^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.67127, size = 181, normalized size = 6.96 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{-a b} \cos \left (x\right ) - a}{b \cos \left (x\right )^{2} + a}\right )}{2 \, a b}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} \cos \left (x\right )}{a}\right )}{a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*cos(x) - a)/(b*cos(x)^2 + a))/(a*b), -sqrt(a*b)*arctan(sqrt(a
*b)*cos(x)/a)/(a*b)]

________________________________________________________________________________________

Sympy [A]  time = 1.94691, size = 87, normalized size = 3.35 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\cos{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{1}{b \cos{\left (x \right )}} & \text{for}\: a = 0 \\- \frac{\cos{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \cos{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} - \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \cos{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/cos(x), Eq(a, 0) & Eq(b, 0)), (1/(b*cos(x)), Eq(a, 0)), (-cos(x)/a, Eq(b, 0)), (I*log(-I*sqrt(a
)*sqrt(1/b) + cos(x))/(2*sqrt(a)*b*sqrt(1/b)) - I*log(I*sqrt(a)*sqrt(1/b) + cos(x))/(2*sqrt(a)*b*sqrt(1/b)), T
rue))

________________________________________________________________________________________

Giac [A]  time = 1.12251, size = 23, normalized size = 0.88 \begin{align*} -\frac{\arctan \left (\frac{b \cos \left (x\right )}{\sqrt{a b}}\right )}{\sqrt{a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(b*cos(x)/sqrt(a*b))/sqrt(a*b)