3.30 \(\int \frac{\cos ^3(x)}{a+b \cos ^2(x)} \, dx\)

Optimal. Leaf size=38 \[ \frac{\sin (x)}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]

[Out]

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

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Rubi [A]  time = 0.0550803, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3186, 388, 208} \[ \frac{\sin (x)}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3/(a + b*Cos[x]^2),x]

[Out]

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

Rule 3186

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

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.0294949, size = 38, normalized size = 1. \[ \frac{\sin (x)}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right )}{b^{3/2} \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3/(a + b*Cos[x]^2),x]

[Out]

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

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Maple [A]  time = 0.013, size = 33, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( x \right ) }{b}}-{\frac{a}{b}{\it Artanh} \left ({\sin \left ( x \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3/(a+b*cos(x)^2),x)

[Out]

sin(x)/b-1/b*a/((a+b)*b)^(1/2)*arctanh(b*sin(x)/((a+b)*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)^3/(a+b*cos(x)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.87761, size = 319, normalized size = 8.39 \begin{align*} \left [\frac{\sqrt{a b + b^{2}} a \log \left (-\frac{b \cos \left (x\right )^{2} + 2 \, \sqrt{a b + b^{2}} \sin \left (x\right ) - a - 2 \, b}{b \cos \left (x\right )^{2} + a}\right ) + 2 \,{\left (a b + b^{2}\right )} \sin \left (x\right )}{2 \,{\left (a b^{2} + b^{3}\right )}}, \frac{\sqrt{-a b - b^{2}} a \arctan \left (\frac{\sqrt{-a b - b^{2}} \sin \left (x\right )}{a + b}\right ) +{\left (a b + b^{2}\right )} \sin \left (x\right )}{a b^{2} + b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3/(a+b*cos(x)^2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3/(a+b*cos(x)**2),x)

[Out]

Timed out

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Giac [A]  time = 1.15699, size = 55, normalized size = 1.45 \begin{align*} \frac{a \arctan \left (\frac{b \sin \left (x\right )}{\sqrt{-a b - b^{2}}}\right )}{\sqrt{-a b - b^{2}} b} + \frac{\sin \left (x\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3/(a+b*cos(x)^2),x, algorithm="giac")

[Out]

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