3.1.75 \(\int \cos ^3(x) \sqrt {\sin (x)} \, dx\) [75]

Optimal. Leaf size=21 \[ \frac {2}{3} \sin ^{\frac {3}{2}}(x)-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \]

[Out]

2/3*sin(x)^(3/2)-2/7*sin(x)^(7/2)

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2644, 14} \begin {gather*} \frac {2}{3} \sin ^{\frac {3}{2}}(x)-\frac {2}{7} \sin ^{\frac {7}{2}}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3*Sqrt[Sin[x]],x]

[Out]

(2*Sin[x]^(3/2))/3 - (2*Sin[x]^(7/2))/7

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rubi steps

\begin {align*} \int \cos ^3(x) \sqrt {\sin (x)} \, dx &=\text {Subst}\left (\int \sqrt {x} \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=\text {Subst}\left (\int \left (\sqrt {x}-x^{5/2}\right ) \, dx,x,\sin (x)\right )\\ &=\frac {2}{3} \sin ^{\frac {3}{2}}(x)-\frac {2}{7} \sin ^{\frac {7}{2}}(x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.86 \begin {gather*} \frac {1}{21} (11+3 \cos (2 x)) \sin ^{\frac {3}{2}}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3*Sqrt[Sin[x]],x]

[Out]

((11 + 3*Cos[2*x])*Sin[x]^(3/2))/21

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Mathics [A]
time = 8.35, size = 17, normalized size = 0.81 \begin {gather*} \frac {\left (6 \text {Sin}\left [3 x\right ]+38 \text {Sin}\left [x\right ]\right ) \sqrt {\text {Sin}\left [x\right ]}}{84} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Cos[x]^3*Sqrt[Sin[x]],x]')

[Out]

(6 Sin[3 x] + 38 Sin[x]) Sqrt[Sin[x]] / 84

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Maple [A]
time = 0.05, size = 14, normalized size = 0.67

method	result	size
default	\(\frac {2 \left (\sin ^{\frac {3}{2}}\left (x \right )\right )}{3}-\frac {2 \left (\sin ^{\frac {7}{2}}\left (x \right )\right )}{7}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sin(x)^(3/2)-2/7*sin(x)^(7/2)

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.62 \begin {gather*} -\frac {2}{7} \, \sin \left (x\right )^{\frac {7}{2}} + \frac {2}{3} \, \sin \left (x\right )^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*sin(x)^(7/2) + 2/3*sin(x)^(3/2)

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Fricas [A]
time = 0.34, size = 14, normalized size = 0.67 \begin {gather*} \frac {2}{21} \, {\left (3 \, \cos \left (x\right )^{2} + 4\right )} \sin \left (x\right )^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*cos(x)^2 + 4)*sin(x)^(3/2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (19) = 38\)
time = 4.87, size = 170, normalized size = 8.10 \begin {gather*} \frac {28 \sqrt {2} \sqrt {\frac {\tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1}} \tan ^{5}{\left (\frac {x}{2} \right )}}{21 \tan ^{6}{\left (\frac {x}{2} \right )} + 63 \tan ^{4}{\left (\frac {x}{2} \right )} + 63 \tan ^{2}{\left (\frac {x}{2} \right )} + 21} + \frac {8 \sqrt {2} \sqrt {\frac {\tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1}} \tan ^{3}{\left (\frac {x}{2} \right )}}{21 \tan ^{6}{\left (\frac {x}{2} \right )} + 63 \tan ^{4}{\left (\frac {x}{2} \right )} + 63 \tan ^{2}{\left (\frac {x}{2} \right )} + 21} + \frac {28 \sqrt {2} \sqrt {\frac {\tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} + 1}} \tan {\left (\frac {x}{2} \right )}}{21 \tan ^{6}{\left (\frac {x}{2} \right )} + 63 \tan ^{4}{\left (\frac {x}{2} \right )} + 63 \tan ^{2}{\left (\frac {x}{2} \right )} + 21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3*sin(x)**(1/2),x)

[Out]

28*sqrt(2)*sqrt(tan(x/2)/(tan(x/2)**2 + 1))*tan(x/2)**5/(21*tan(x/2)**6 + 63*tan(x/2)**4 + 63*tan(x/2)**2 + 21
) + 8*sqrt(2)*sqrt(tan(x/2)/(tan(x/2)**2 + 1))*tan(x/2)**3/(21*tan(x/2)**6 + 63*tan(x/2)**4 + 63*tan(x/2)**2 +
21) + 28*sqrt(2)*sqrt(tan(x/2)/(tan(x/2)**2 + 1))*tan(x/2)/(21*tan(x/2)**6 + 63*tan(x/2)**4 + 63*tan(x/2)**2
+ 21)

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Giac [A]
time = 0.00, size = 28, normalized size = 1.33 \begin {gather*} 2 \left (-\frac {1}{7} \sqrt {\sin x} \sin ^{3}x+\frac {1}{3} \sqrt {\sin x} \sin x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^(1/2),x)

[Out]

-2/7*sin(x)^(7/2) + 2/3*sin(x)^(3/2)

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Mupad [B]
time = 0.21, size = 25, normalized size = 1.19 \begin {gather*} -\frac {{\cos \left (x\right )}^4\,{\sin \left (x\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},2;\ 3;\ {\cos \left (x\right )}^2\right )}{4\,{\left ({\sin \left (x\right )}^2\right )}^{3/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*sin(x)^(1/2),x)

[Out]

-(cos(x)^4*sin(x)^(3/2)*hypergeom([1/4, 2], 3, cos(x)^2))/(4*(sin(x)^2)^(3/4))

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