3.807 \(\int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx\)

Optimal. Leaf size=20 \[ \sqrt {5} \tan ^{-1}\left (\frac {\cos (x)}{\sqrt {5}}\right )-\cos (x) \]

[Out]

-cos(x)+arctan(1/5*cos(x)*5^(1/2))*5^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4335, 321, 203} \[ \sqrt {5} \tan ^{-1}\left (\frac {\cos (x)}{\sqrt {5}}\right )-\cos (x) \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]^2*Sin[x])/(5 + Cos[x]^2),x]

[Out]

Sqrt[5]*ArcTan[Cos[x]/Sqrt[5]] - Cos[x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 321

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

Rule 4335

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rubi steps

\begin {align*} \int \frac {\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2}{5+x^2} \, dx,x,\cos (x)\right )\\ &=-\cos (x)+5 \operatorname {Subst}\left (\int \frac {1}{5+x^2} \, dx,x,\cos (x)\right )\\ &=\sqrt {5} \tan ^{-1}\left (\frac {\cos (x)}{\sqrt {5}}\right )-\cos (x)\\ \end {align*}

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Mathematica [B]  time = 0.17, size = 82, normalized size = 4.10 \[ \frac {1}{20} \left (-20 \cos (x)+21 \sqrt {5} \tan ^{-1}\left (\frac {1}{\sqrt {5}}-\sqrt {\frac {6}{5}} \tan \left (\frac {x}{2}\right )\right )+21 \sqrt {5} \tan ^{-1}\left (\sqrt {\frac {6}{5}} \tan \left (\frac {x}{2}\right )+\frac {1}{\sqrt {5}}\right )-\sqrt {5} \tan ^{-1}\left (\frac {\cos (x)}{\sqrt {5}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]^2*Sin[x])/(5 + Cos[x]^2),x]

[Out]

(-(Sqrt[5]*ArcTan[Cos[x]/Sqrt[5]]) + 21*Sqrt[5]*ArcTan[1/Sqrt[5] - Sqrt[6/5]*Tan[x/2]] + 21*Sqrt[5]*ArcTan[1/S
qrt[5] + Sqrt[6/5]*Tan[x/2]] - 20*Cos[x])/20

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fricas [A]  time = 0.62, size = 17, normalized size = 0.85 \[ \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} \cos \relax (x)\right ) - \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)/(5+cos(x)^2),x, algorithm="fricas")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*cos(x)) - cos(x)

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giac [A]  time = 0.12, size = 17, normalized size = 0.85 \[ \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} \cos \relax (x)\right ) - \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)/(5+cos(x)^2),x, algorithm="giac")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*cos(x)) - cos(x)

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maple [A]  time = 0.04, size = 18, normalized size = 0.90 \[ -\cos \relax (x )+\arctan \left (\frac {\cos \relax (x ) \sqrt {5}}{5}\right ) \sqrt {5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2*sin(x)/(5+cos(x)^2),x)

[Out]

-cos(x)+arctan(1/5*cos(x)*5^(1/2))*5^(1/2)

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maxima [A]  time = 0.43, size = 17, normalized size = 0.85 \[ \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} \cos \relax (x)\right ) - \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)/(5+cos(x)^2),x, algorithm="maxima")

[Out]

sqrt(5)*arctan(1/5*sqrt(5)*cos(x)) - cos(x)

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mupad [B]  time = 2.90, size = 17, normalized size = 0.85 \[ \sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\cos \relax (x)}{5}\right )-\cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)^2*sin(x))/(cos(x)^2 + 5),x)

[Out]

5^(1/2)*atan((5^(1/2)*cos(x))/5) - cos(x)

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sympy [A]  time = 0.43, size = 19, normalized size = 0.95 \[ - \cos {\relax (x )} + \sqrt {5} \operatorname {atan}{\left (\frac {\sqrt {5} \cos {\relax (x )}}{5} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**2*sin(x)/(5+cos(x)**2),x)

[Out]

-cos(x) + sqrt(5)*atan(sqrt(5)*cos(x)/5)

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