3.3.55 \(\int \cos ^3(x) \sin ^{\frac {5}{2}}(x) \, dx\) [255]

Optimal. Leaf size=21 \[ \frac {2}{7} \sin ^{\frac {7}{2}}(x)-\frac {2}{11} \sin ^{\frac {11}{2}}(x) \]

[Out]

2/7*sin(x)^(7/2)-2/11*sin(x)^(11/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, 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}{7} \sin ^{\frac {7}{2}}(x)-\frac {2}{11} \sin ^{\frac {11}{2}}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3*Sin[x]^(5/2),x]

[Out]

(2*Sin[x]^(7/2))/7 - (2*Sin[x]^(11/2))/11

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) \sin ^{\frac {5}{2}}(x) \, dx &=\text {Subst}\left (\int x^{5/2} \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=\text {Subst}\left (\int \left (x^{5/2}-x^{9/2}\right ) \, dx,x,\sin (x)\right )\\ &=\frac {2}{7} \sin ^{\frac {7}{2}}(x)-\frac {2}{11} \sin ^{\frac {11}{2}}(x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 18, normalized size = 0.86 \begin {gather*} \frac {1}{77} (15+7 \cos (2 x)) \sin ^{\frac {7}{2}}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3*Sin[x]^(5/2),x]

[Out]

((15 + 7*Cos[2*x])*Sin[x]^(7/2))/77

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 14, normalized size = 0.67

method	result	size
default	\(\frac {2 \left (\sin ^{\frac {7}{2}}\left (x \right )\right )}{7}-\frac {2 \left (\sin ^{\frac {11}{2}}\left (x \right )\right )}{11}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*sin(x)^(7/2)-2/11*sin(x)^(11/2)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 13, normalized size = 0.62 \begin {gather*} -\frac {2}{11} \, \sin \left (x\right )^{\frac {11}{2}} + \frac {2}{7} \, \sin \left (x\right )^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*sin(x)^(11/2) + 2/7*sin(x)^(7/2)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 20, normalized size = 0.95 \begin {gather*} -\frac {2}{77} \, {\left (7 \, \cos \left (x\right )^{4} - 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)^(5/2),x, algorithm="fricas")

[Out]

-2/77*(7*cos(x)^4 - 3*cos(x)^2 - 4)*sin(x)^(3/2)

________________________________________________________________________________________

Sympy [A]
time = 44.49, size = 24, normalized size = 1.14 \begin {gather*} \frac {8 \sin ^{\frac {11}{2}}{\left (x \right )}}{77} + \frac {2 \sin ^{\frac {7}{2}}{\left (x \right )} \cos ^{2}{\left (x \right )}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3*sin(x)**(5/2),x)

[Out]

8*sin(x)**(11/2)/77 + 2*sin(x)**(7/2)*cos(x)**2/7

________________________________________________________________________________________

Giac [A]
time = 6.48, size = 13, normalized size = 0.62 \begin {gather*} -\frac {2}{11} \, \sin \left (x\right )^{\frac {11}{2}} + \frac {2}{7} \, \sin \left (x\right )^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*sin(x)^(11/2) + 2/7*sin(x)^(7/2)

________________________________________________________________________________________

Mupad [B]
time = 0.45, size = 25, normalized size = 1.19 \begin {gather*} -\frac {{\cos \left (x\right )}^4\,{\sin \left (x\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},2;\ 3;\ {\cos \left (x\right )}^2\right )}{4\,{\left ({\sin \left (x\right )}^2\right )}^{7/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*sin(x)^(5/2),x)

[Out]

-(cos(x)^4*sin(x)^(7/2)*hypergeom([-3/4, 2], 3, cos(x)^2))/(4*(sin(x)^2)^(7/4))

________________________________________________________________________________________