3.395 \(\int (1-\sin (\frac {2 x}{3}))^{5/2} \, dx\)

Optimal. Leaf size=73 \[ \frac {3}{5} \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2} \cos \left (\frac {2 x}{3}\right )+\frac {8}{5} \sqrt {1-\sin \left (\frac {2 x}{3}\right )} \cos \left (\frac {2 x}{3}\right )+\frac {32 \cos \left (\frac {2 x}{3}\right )}{5 \sqrt {1-\sin \left (\frac {2 x}{3}\right )}} \]

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Rubi [A]  time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ \frac {3}{5} \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2} \cos \left (\frac {2 x}{3}\right )+\frac {8}{5} \sqrt {1-\sin \left (\frac {2 x}{3}\right )} \cos \left (\frac {2 x}{3}\right )+\frac {32 \cos \left (\frac {2 x}{3}\right )}{5 \sqrt {1-\sin \left (\frac {2 x}{3}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sin[(2*x)/3])^(5/2),x]

[Out]

(32*Cos[(2*x)/3])/(5*Sqrt[1 - Sin[(2*x)/3]]) + (8*Cos[(2*x)/3]*Sqrt[1 - Sin[(2*x)/3]])/5 + (3*Cos[(2*x)/3]*(1
- Sin[(2*x)/3])^(3/2))/5

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rubi steps

\begin {align*} \int \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{5/2} \, dx &=\frac {3}{5} \cos \left (\frac {2 x}{3}\right ) \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2}+\frac {8}{5} \int \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2} \, dx\\ &=\frac {8}{5} \cos \left (\frac {2 x}{3}\right ) \sqrt {1-\sin \left (\frac {2 x}{3}\right )}+\frac {3}{5} \cos \left (\frac {2 x}{3}\right ) \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2}+\frac {32}{15} \int \sqrt {1-\sin \left (\frac {2 x}{3}\right )} \, dx\\ &=\frac {32 \cos \left (\frac {2 x}{3}\right )}{5 \sqrt {1-\sin \left (\frac {2 x}{3}\right )}}+\frac {8}{5} \cos \left (\frac {2 x}{3}\right ) \sqrt {1-\sin \left (\frac {2 x}{3}\right )}+\frac {3}{5} \cos \left (\frac {2 x}{3}\right ) \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 76, normalized size = 1.04 \[ \frac {\left (1-\sin \left (\frac {2 x}{3}\right )\right )^{5/2} \left (150 \sin \left (\frac {x}{3}\right )-25 \sin (x)-3 \sin \left (\frac {5 x}{3}\right )+150 \cos \left (\frac {x}{3}\right )+25 \cos (x)-3 \cos \left (\frac {5 x}{3}\right )\right )}{20 \left (\cos \left (\frac {x}{3}\right )-\sin \left (\frac {x}{3}\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sin[(2*x)/3])^(5/2),x]

[Out]

((1 - Sin[(2*x)/3])^(5/2)*(150*Cos[x/3] + 25*Cos[x] - 3*Cos[(5*x)/3] + 150*Sin[x/3] - 25*Sin[x] - 3*Sin[(5*x)/
3]))/(20*(Cos[x/3] - Sin[x/3])^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1-\sin \left (\frac {2 x}{3}\right )\right )^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(1 - Sin[(2*x)/3])^(5/2),x]

[Out]

Could not integrate

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fricas [A]  time = 0.79, size = 71, normalized size = 0.97 \[ -\frac {{\left (3 \, \cos \left (\frac {2}{3} \, x\right )^{3} - 11 \, \cos \left (\frac {2}{3} \, x\right )^{2} + {\left (3 \, \cos \left (\frac {2}{3} \, x\right )^{2} + 14 \, \cos \left (\frac {2}{3} \, x\right ) - 32\right )} \sin \left (\frac {2}{3} \, x\right ) - 46 \, \cos \left (\frac {2}{3} \, x\right ) - 32\right )} \sqrt {-\sin \left (\frac {2}{3} \, x\right ) + 1}}{5 \, {\left (\cos \left (\frac {2}{3} \, x\right ) - \sin \left (\frac {2}{3} \, x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sin(2/3*x))^(5/2),x, algorithm="fricas")

[Out]

-1/5*(3*cos(2/3*x)^3 - 11*cos(2/3*x)^2 + (3*cos(2/3*x)^2 + 14*cos(2/3*x) - 32)*sin(2/3*x) - 46*cos(2/3*x) - 32
)*sqrt(-sin(2/3*x) + 1)/(cos(2/3*x) - sin(2/3*x) + 1)

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giac [B]  time = 0.75, size = 110, normalized size = 1.51 \[ -\frac {1}{20} \, \sqrt {2} {\left (5 \, \cos \left (\frac {1}{4} \, \pi + x\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right ) + 90 \, \cos \left (\frac {1}{4} \, \pi - \frac {1}{3} \, x\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right ) - 3 \, \cos \left (\frac {1}{4} \, \pi - \frac {5}{3} \, x\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right ) + 60 \, \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, x\right ) + 20 \, \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right ) \sin \left (\frac {1}{4} \, \pi - x\right ) - 128 \, \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sin(2/3*x))^(5/2),x, algorithm="giac")

[Out]

-1/20*sqrt(2)*(5*cos(1/4*pi + x)*sgn(sin(-1/4*pi + 1/3*x)) + 90*cos(1/4*pi - 1/3*x)*sgn(sin(-1/4*pi + 1/3*x))
- 3*cos(1/4*pi - 5/3*x)*sgn(sin(-1/4*pi + 1/3*x)) + 60*sgn(sin(-1/4*pi + 1/3*x))*sin(1/4*pi + 1/3*x) + 20*sgn(
sin(-1/4*pi + 1/3*x))*sin(1/4*pi - x) - 128*sgn(sin(-1/4*pi + 1/3*x)))

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maple [A]  time = 0.15, size = 47, normalized size = 0.64

method	result	size
default	\(-\frac {\left (-1+\sin \left (\frac {2 x}{3}\right )\right ) \left (\sin \left (\frac {2 x}{3}\right )+1\right ) \left (3 \left (\sin ^{2}\left (\frac {2 x}{3}\right )\right )-14 \sin \left (\frac {2 x}{3}\right )+43\right )}{5 \cos \left (\frac {2 x}{3}\right ) \sqrt {1-\sin \left (\frac {2 x}{3}\right )}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-sin(2/3*x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*(-1+sin(2/3*x))*(sin(2/3*x)+1)*(3*sin(2/3*x)^2-14*sin(2/3*x)+43)/cos(2/3*x)/(1-sin(2/3*x))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-\sin \left (\frac {2}{3} \, x\right ) + 1\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sin(2/3*x))^(5/2),x, algorithm="maxima")

[Out]

integrate((-sin(2/3*x) + 1)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (1-\sin \left (\frac {2\,x}{3}\right )\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - sin((2*x)/3))^(5/2),x)

[Out]

int((1 - sin((2*x)/3))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1 - \sin {\left (\frac {2 x}{3} \right )}\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-sin(2/3*x))**(5/2),x)

[Out]

Integral((1 - sin(2*x/3))**(5/2), x)

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