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

Optimal. Leaf size=48 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{\sin (x)}{2 a \left (a+b \sin ^2(x)\right )} \]

[Out]

ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sin[x]/(2*a*(a + b*Sin[x]^2))

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Rubi [A]  time = 0.0348803, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3190, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{\sin (x)}{2 a \left (a+b \sin ^2(x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/(a + b*Sin[x]^2)^2,x]

[Out]

ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sin[x]/(2*a*(a + b*Sin[x]^2))

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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{\cos (x)}{\left (a+b \sin ^2(x)\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2} \, dx,x,\sin (x)\right )\\ &=\frac{\sin (x)}{2 a \left (a+b \sin ^2(x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sin (x)\right )}{2 a}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{\sin (x)}{2 a \left (a+b \sin ^2(x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.0582955, size = 48, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{\sin (x)}{2 a \left (a+b \sin ^2(x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[b]*Sin[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sin[x]/(2*a*(a + b*Sin[x]^2))

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Maple [A]  time = 0.035, size = 39, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( x \right ) }{2\,a \left ( a+b \left ( \sin \left ( x \right ) \right ) ^{2} \right ) }}+{\frac{1}{2\,a}\arctan \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sin(x)/a/(a+b*sin(x)^2)+1/2/a/(a*b)^(1/2)*arctan(sin(x)*b/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.02593, size = 381, normalized size = 7.94 \begin{align*} \left [-\frac{2 \, a b \sin \left (x\right ) +{\left (b \cos \left (x\right )^{2} - a - b\right )} \sqrt{-a b} \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right )}{4 \,{\left (a^{2} b^{2} \cos \left (x\right )^{2} - a^{3} b - a^{2} b^{2}\right )}}, -\frac{a b \sin \left (x\right ) -{\left (b \cos \left (x\right )^{2} - a - b\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} \sin \left (x\right )}{a}\right )}{2 \,{\left (a^{2} b^{2} \cos \left (x\right )^{2} - a^{3} b - a^{2} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a*b*sin(x) + (b*cos(x)^2 - a - b)*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x
)^2 - a - b)))/(a^2*b^2*cos(x)^2 - a^3*b - a^2*b^2), -1/2*(a*b*sin(x) - (b*cos(x)^2 - a - b)*sqrt(a*b)*arctan(
sqrt(a*b)*sin(x)/a))/(a^2*b^2*cos(x)^2 - a^3*b - a^2*b^2)]

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Sympy [A]  time = 31.5219, size = 340, normalized size = 7.08 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sin ^{3}{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\sin{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{1}{3 b^{2} \sin ^{3}{\left (x \right )}} & \text{for}\: a = 0 \\\frac{2 i \sqrt{a} b \sqrt{\frac{1}{b}} \sin{\left (x \right )}}{4 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} \sin ^{2}{\left (x \right )}} + \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sin{\left (x \right )} \right )}}{4 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} \sin ^{2}{\left (x \right )}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sin{\left (x \right )} \right )}}{4 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} \sin ^{2}{\left (x \right )}} + \frac{b \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sin{\left (x \right )} \right )} \sin ^{2}{\left (x \right )}}{4 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} \sin ^{2}{\left (x \right )}} - \frac{b \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sin{\left (x \right )} \right )} \sin ^{2}{\left (x \right )}}{4 i a^{\frac{5}{2}} b \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} \sqrt{\frac{1}{b}} \sin ^{2}{\left (x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/sin(x)**3, Eq(a, 0) & Eq(b, 0)), (sin(x)/a**2, Eq(b, 0)), (-1/(3*b**2*sin(x)**3), Eq(a, 0)), (2
*I*sqrt(a)*b*sqrt(1/b)*sin(x)/(4*I*a**(5/2)*b*sqrt(1/b) + 4*I*a**(3/2)*b**2*sqrt(1/b)*sin(x)**2) + a*log(-I*sq
rt(a)*sqrt(1/b) + sin(x))/(4*I*a**(5/2)*b*sqrt(1/b) + 4*I*a**(3/2)*b**2*sqrt(1/b)*sin(x)**2) - a*log(I*sqrt(a)
*sqrt(1/b) + sin(x))/(4*I*a**(5/2)*b*sqrt(1/b) + 4*I*a**(3/2)*b**2*sqrt(1/b)*sin(x)**2) + b*log(-I*sqrt(a)*sqr
t(1/b) + sin(x))*sin(x)**2/(4*I*a**(5/2)*b*sqrt(1/b) + 4*I*a**(3/2)*b**2*sqrt(1/b)*sin(x)**2) - b*log(I*sqrt(a
)*sqrt(1/b) + sin(x))*sin(x)**2/(4*I*a**(5/2)*b*sqrt(1/b) + 4*I*a**(3/2)*b**2*sqrt(1/b)*sin(x)**2), True))

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Giac [A]  time = 1.15537, size = 51, normalized size = 1.06 \begin{align*} \frac{\arctan \left (\frac{b \sin \left (x\right )}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a} + \frac{\sin \left (x\right )}{2 \,{\left (b \sin \left (x\right )^{2} + a\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(b*sin(x)/sqrt(a*b))/(sqrt(a*b)*a) + 1/2*sin(x)/((b*sin(x)^2 + a)*a)