3.63 \(\int \cos ^3(x) \sin ^4(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{\sin ^5(x)}{5}-\frac{\sin ^7(x)}{7} \]

[Out]

Sin[x]^5/5 - Sin[x]^7/7

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Rubi [A]  time = 0.0222875, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2564, 14} \[ \frac{\sin ^5(x)}{5}-\frac{\sin ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3*Sin[x]^4,x]

[Out]

Sin[x]^5/5 - Sin[x]^7/7

Rule 2564

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])

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]]

Rubi steps

\begin{align*} \int \cos ^3(x) \sin ^4(x) \, dx &=\operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (x)\right )\\ &=\frac{\sin ^5(x)}{5}-\frac{\sin ^7(x)}{7}\\ \end{align*}

Mathematica [A]  time = 0.0093503, size = 31, normalized size = 1.82 \[ \frac{3 \sin (x)}{64}-\frac{1}{64} \sin (3 x)-\frac{1}{320} \sin (5 x)+\frac{1}{448} \sin (7 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^3*Sin[x]^4,x]

[Out]

(3*Sin[x])/64 - Sin[3*x]/64 - Sin[5*x]/320 + Sin[7*x]/448

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Maple [B]  time = 0.004, size = 30, normalized size = 1.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4} \left ( \sin \left ( x \right ) \right ) ^{3}}{7}}-{\frac{3\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{4}}{35}}+{\frac{ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{35}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3*sin(x)^4,x)

[Out]

-1/7*cos(x)^4*sin(x)^3-3/35*sin(x)*cos(x)^4+1/35*(2+cos(x)^2)*sin(x)

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Maxima [A]  time = 0.928444, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{7} \, \sin \left (x\right )^{7} + \frac{1}{5} \, \sin \left (x\right )^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^4,x, algorithm="maxima")

[Out]

-1/7*sin(x)^7 + 1/5*sin(x)^5

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Fricas [A]  time = 2.13228, size = 73, normalized size = 4.29 \begin{align*} \frac{1}{35} \,{\left (5 \, \cos \left (x\right )^{6} - 8 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^4,x, algorithm="fricas")

[Out]

1/35*(5*cos(x)^6 - 8*cos(x)^4 + cos(x)^2 + 2)*sin(x)

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Sympy [A]  time = 0.062694, size = 12, normalized size = 0.71 \begin{align*} - \frac{\sin ^{7}{\left (x \right )}}{7} + \frac{\sin ^{5}{\left (x \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**3*sin(x)**4,x)

[Out]

-sin(x)**7/7 + sin(x)**5/5

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Giac [A]  time = 1.04648, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{7} \, \sin \left (x\right )^{7} + \frac{1}{5} \, \sin \left (x\right )^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^4,x, algorithm="giac")

[Out]

-1/7*sin(x)^7 + 1/5*sin(x)^5