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

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

[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 = 0.0853401, 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 \[ \frac{625}{32} \sin ^{-1}\left (\frac{2 \sin (x)}{\sqrt{5}}\right )+\frac{1}{6} \sin (x) \left (5-4 \sin ^2(x)\right )^{5/2}+\frac{25}{24} \sin (x) \left (5-4 \sin ^2(x)\right )^{3/2}+\frac{125}{16} \sin (x) \sqrt{5-4 \sin ^2(x)} \]

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

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Rubi in Sympy [A]  time = 179.512, size = 70, normalized size = 1.01 \[ \frac{\left (- 4 \sin ^{2}{\left (x \right )} + 5\right )^{\frac{5}{2}} \sin{\left (x \right )}}{6} + \frac{25 \left (- 4 \sin ^{2}{\left (x \right )} + 5\right )^{\frac{3}{2}} \sin{\left (x \right )}}{24} + \frac{125 \sqrt{- 4 \sin ^{2}{\left (x \right )} + 5} \sin{\left (x \right )}}{16} + \frac{625 \operatorname{asin}{\left (\frac{2 \sqrt{5} \sin{\left (x \right )}}{5} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)*(5*cos(x)**2+sin(x)**2)**(5/2),x)

[Out]

(-4*sin(x)**2 + 5)**(5/2)*sin(x)/6 + 25*(-4*sin(x)**2 + 5)**(3/2)*sin(x)/24 + 12
5*sqrt(-4*sin(x)**2 + 5)*sin(x)/16 + 625*asin(2*sqrt(5)*sin(x)/5)/32

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Mathematica [A]  time = 0.126779, size = 55, normalized size = 0.8 \[ \frac{1}{48} (515 \sin (x)+90 \sin (3 x)+8 \sin (5 x)) \sqrt{2 \cos (2 x)+3}+\frac{625}{32} \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{2 \cos (2 x)+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(625*ArcTan[(2*Sin[x])/Sqrt[3 + 2*Cos[2*x]]])/32 + (Sqrt[3 + 2*Cos[2*x]]*(515*Si
n[x] + 90*Sin[3*x] + 8*Sin[5*x]))/48

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Maple [A]  time = 0.136, size = 103, normalized size = 1.5 \[ -{\frac{1}{192\,\sin \left ( x \right ) }\sqrt{ \left ( 4\, \left ( \cos \left ( x \right ) \right ) ^{2}+1 \right ) \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( -512\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}+5\, \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \sin \left ( x \right ) \right ) ^{4}+2080\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}+5\, \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \sin \left ( x \right ) \right ) ^{2}-3300\,\sqrt{-4\, \left ( \sin \left ( x \right ) \right ) ^{4}+5\, \left ( \sin \left ( x \right ) \right ) ^{2}}-1875\,\arcsin \left ( -1+8/5\, \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ){\frac{1}{\sqrt{4\, \left ( \cos \left ( x \right ) \right ) ^{2}+1}}}} \]

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 = 1.55701, size = 72, normalized size = 1.04 \[ \frac{1}{6} \,{\left (-4 \, \sin \left (x\right )^{2} + 5\right )}^{\frac{5}{2}} \sin \left (x\right ) + \frac{25}{24} \,{\left (-4 \, \sin \left (x\right )^{2} + 5\right )}^{\frac{3}{2}} \sin \left (x\right ) + \frac{125}{16} \, \sqrt{-4 \, \sin \left (x\right )^{2} + 5} \sin \left (x\right ) + \frac{625}{32} \, \arcsin \left (\frac{2}{5} \, \sqrt{5} \sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*cos(x)^2 + sin(x)^2)^(5/2)*cos(x),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 = 0.282593, size = 162, normalized size = 2.35 \[ \frac{1}{48} \,{\left (128 \, \cos \left (x\right )^{4} + 264 \, \cos \left (x\right )^{2} + 433\right )} \sqrt{4 \, \cos \left (x\right )^{2} + 1} \sin \left (x\right ) - \frac{625}{64} \, \arctan \left (-\frac{{\left (16 \, \cos \left (x\right )^{2} - 3\right )} \sqrt{4 \, \cos \left (x\right )^{2} + 1} \sin \left (x\right ) - 2 \,{\left (16 \, \cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right )}{32 \, \cos \left (x\right )^{4} - 18 \, \cos \left (x\right )^{2} -{\left (16 \, \cos \left (x\right )^{3} - 11 \, \cos \left (x\right )\right )} \sqrt{4 \, \cos \left (x\right )^{2} + 1} - 4}\right ) + \frac{625}{64} \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*cos(x)^2 + sin(x)^2)^(5/2)*cos(x),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*ar
ctan(-((16*cos(x)^2 - 3)*sqrt(4*cos(x)^2 + 1)*sin(x) - 2*(16*cos(x)^3 - cos(x))*
sin(x))/(32*cos(x)^4 - 18*cos(x)^2 - (16*cos(x)^3 - 11*cos(x))*sqrt(4*cos(x)^2 +
1) - 4)) + 625/64*arctan(sin(x)/cos(x))

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

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/XCAS [A]  time = 0.212135, size = 55, normalized size = 0.8 \[ \frac{1}{48} \,{\left (8 \,{\left (16 \, \sin \left (x\right )^{2} - 65\right )} \sin \left (x\right )^{2} + 825\right )} \sqrt{-4 \, \sin \left (x\right )^{2} + 5} \sin \left (x\right ) + \frac{625}{32} \, \arcsin \left (\frac{2}{5} \, \sqrt{5} \sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*cos(x)^2 + sin(x)^2)^(5/2)*cos(x),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))