3.5.20 \(\int \cos (x) (5 \cos ^2(x)+\sin ^2(x))^{5/2} \, dx\) [420]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {4441, 201, 222} \begin {gather*} \frac {625}{32} \text {ArcSin}\left (\frac {2 \sin (x)}{\sqrt {5}}\right )+\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)} \end {gather*}

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 201

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 222

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]

Rule 4441

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])

Rubi steps

\begin {align*} \int \cos (x) \left (5 \cos ^2(x)+\sin ^2(x)\right )^{5/2} \, dx &=\text {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} \text {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} \text {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} \text {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*}

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Mathematica [A]
time = 0.05, size = 48, normalized size = 0.70 \begin {gather*} \frac {1}{96} \left (1875 \sin ^{-1}\left (\frac {2 \sin (x)}{\sqrt {5}}\right )+2 \sqrt {3+2 \cos (2 x)} (515 \sin (x)+90 \sin (3 x)+8 \sin (5 x))\right ) \end {gather*}

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 = 0.16, size = 103, normalized size = 1.49

method	result	size
default	\(\frac {\sqrt {\left (4 \left (\cos ^{2}\left (x \right )\right )+1\right ) \left (\sin ^{2}\left (x \right )\right )}\, \left (512 \sqrt {-4 \left (\sin ^{4}\left (x \right )\right )+5 \left (\sin ^{2}\left (x \right )\right )}\, \left (\sin ^{4}\left (x \right )\right )-2080 \sqrt {-4 \left (\sin ^{4}\left (x \right )\right )+5 \left (\sin ^{2}\left (x \right )\right )}\, \left (\sin ^{2}\left (x \right )\right )+3300 \sqrt {-4 \left (\sin ^{4}\left (x \right )\right )+5 \left (\sin ^{2}\left (x \right )\right )}+1875 \arcsin \left (-1+\frac {8 \left (\sin ^{2}\left (x \right )\right )}{5}\right )\right )}{192 \sin \left (x \right ) \sqrt {4 \left (\cos ^{2}\left (x \right )\right )+1}}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 2.21, size = 53, normalized size = 0.77 \begin {gather*} \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 {gather*}

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 = 1.42, size = 88, normalized size = 1.28 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 4846 deep

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Giac [A]
time = 1.74, size = 41, normalized size = 0.59 \begin {gather*} \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 {gather*}

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \cos \left (x\right )\,{\left (5\,{\cos \left (x\right )}^2+{\sin \left (x\right )}^2\right )}^{5/2} \,d x \end {gather*}

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]

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

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