3.420 \(\int \cos (x) (5 \cos ^2(x)+\sin ^2(x))^{5/2} \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac{25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac{125}{16} \sin (x) \sqrt{5-4 \sin ^2(x)}+\frac{625}{32} \sin ^{-1}\left (\frac{2 \sin (x)}{\sqrt{5}}\right ) \]

[Out]

(625*ArcSin[(2*Sin[x])/Sqrt[5]])/32 + (125*Sin[x]*Sqrt[5 - 4*Sin[x]^2])/16 + (25*Sin[x]*(5 - 4*Sin[x]^2)^(3/2)
)/24 + (Sin[x]*(5 - 4*Sin[x]^2)^(5/2))/6

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Rubi [A]  time = 0.0533013, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4356, 195, 216} \[ \frac{1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac{25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac{125}{16} \sin (x) \sqrt{5-4 \sin ^2(x)}+\frac{625}{32} \sin ^{-1}\left (\frac{2 \sin (x)}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(625*ArcSin[(2*Sin[x])/Sqrt[5]])/32 + (125*Sin[x]*Sqrt[5 - 4*Sin[x]^2])/16 + (25*Sin[x]*(5 - 4*Sin[x]^2)^(3/2)
)/24 + (Sin[x]*(5 - 4*Sin[x]^2)^(5/2))/6

Rule 4356

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

Rule 195

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

Rule 216

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

Rubi steps

\begin{align*} \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx &=\operatorname{Subst}\left (\int \left (5-4 x^2\right )^{5/2} \, dx,x,\sin (x)\right )\\ &=\frac{1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac{25}{6} \operatorname{Subst}\left (\int \left (5-4 x^2\right )^{3/2} \, dx,x,\sin (x)\right )\\ &=\frac{25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac{1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac{125}{8} \operatorname{Subst}\left (\int \sqrt{5-4 x^2} \, dx,x,\sin (x)\right )\\ &=\frac{125}{16} \sin (x) \sqrt{5-4 \sin ^2(x)}+\frac{25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac{1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac{625}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5-4 x^2}} \, dx,x,\sin (x)\right )\\ &=\frac{625}{32} \sin ^{-1}\left (\frac{2 \sin (x)}{\sqrt{5}}\right )+\frac{125}{16} \sin (x) \sqrt{5-4 \sin ^2(x)}+\frac{25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac{1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0734648, size = 48, normalized size = 0.7 \[ \frac{1}{96} \left (1875 \sin ^{-1}\left (\frac{2 \sin (x)}{\sqrt{5}}\right )+2 (515 \sin (x)+90 \sin (3 x)+8 \sin (5 x)) \sqrt{2 \cos (2 x)+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1875*ArcSin[(2*Sin[x])/Sqrt[5]] + 2*Sqrt[3 + 2*Cos[2*x]]*(515*Sin[x] + 90*Sin[3*x] + 8*Sin[5*x]))/96

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Maple [A]  time = 0.092, size = 103, normalized size = 1.5 \begin{align*}{\frac{1}{192\,\sin \left ( x \right ) }\sqrt{ \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}+1 \right ) \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 512\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}+5\, \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \sin \left ( x \right ) \right ) ^{4}-2080\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}+5\, \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \sin \left ( x \right ) \right ) ^{2}+3300\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}+5\, \left ( \sin \left ( x \right ) \right ) ^{2}}+1875\,\arcsin \left ( -1+8/5\, \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ){\frac{1}{\sqrt{4\, \left ( \cos \left ( x \right ) \right ) ^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*((4*cos(x)^2+1)*sin(x)^2)^(1/2)*(512*(-4*sin(x)^4+5*sin(x)^2)^(1/2)*sin(x)^4-2080*(-4*sin(x)^4+5*sin(x)^
2)^(1/2)*sin(x)^2+3300*(-4*sin(x)^4+5*sin(x)^2)^(1/2)+1875*arcsin(-1+8/5*sin(x)^2))/sin(x)/(4*cos(x)^2+1)^(1/2
)

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Maxima [A]  time = 1.43121, size = 72, normalized size = 1.04 \begin{align*} \frac{1}{6} \,{\left (-4 \, \sin \left (x\right )^{2} + 5\right )}^{\frac{5}{2}} \sin \left (x\right ) + \frac{25}{24} \,{\left (-4 \, \sin \left (x\right )^{2} + 5\right )}^{\frac{3}{2}} \sin \left (x\right ) + \frac{125}{16} \, \sqrt{-4 \, \sin \left (x\right )^{2} + 5} \sin \left (x\right ) + \frac{625}{32} \, \arcsin \left (\frac{2}{5} \, \sqrt{5} \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-4*sin(x)^2 + 5)^(5/2)*sin(x) + 25/24*(-4*sin(x)^2 + 5)^(3/2)*sin(x) + 125/16*sqrt(-4*sin(x)^2 + 5)*sin(x
) + 625/32*arcsin(2/5*sqrt(5)*sin(x))

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Fricas [A]  time = 3.0801, size = 296, normalized size = 4.29 \begin{align*} \frac{1}{48} \,{\left (128 \, \cos \left (x\right )^{4} + 264 \, \cos \left (x\right )^{2} + 433\right )} \sqrt{4 \, \cos \left (x\right )^{2} + 1} \sin \left (x\right ) + \frac{625}{64} \, \arctan \left (\frac{4 \,{\left (8 \, \cos \left (x\right )^{2} - 3\right )} \sqrt{4 \, \cos \left (x\right )^{2} + 1} \sin \left (x\right ) - 25 \, \cos \left (x\right ) \sin \left (x\right )}{64 \, \cos \left (x\right )^{4} - 23 \, \cos \left (x\right )^{2} - 16}\right ) + \frac{625}{64} \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(128*cos(x)^4 + 264*cos(x)^2 + 433)*sqrt(4*cos(x)^2 + 1)*sin(x) + 625/64*arctan((4*(8*cos(x)^2 - 3)*sqrt(
4*cos(x)^2 + 1)*sin(x) - 25*cos(x)*sin(x))/(64*cos(x)^4 - 23*cos(x)^2 - 16)) + 625/64*arctan(sin(x)/cos(x))

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

[Out]

Timed out

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Giac [A]  time = 1.08751, size = 55, normalized size = 0.8 \begin{align*} \frac{1}{48} \,{\left (8 \,{\left (16 \, \sin \left (x\right )^{2} - 65\right )} \sin \left (x\right )^{2} + 825\right )} \sqrt{-4 \, \sin \left (x\right )^{2} + 5} \sin \left (x\right ) + \frac{625}{32} \, \arcsin \left (\frac{2}{5} \, \sqrt{5} \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(8*(16*sin(x)^2 - 65)*sin(x)^2 + 825)*sqrt(-4*sin(x)^2 + 5)*sin(x) + 625/32*arcsin(2/5*sqrt(5)*sin(x))