### 3.62 $$\int \cos ^4(x) \sin ^3(x) \, dx$$

Optimal. Leaf size=17 $\frac{\cos ^7(x)}{7}-\frac{\cos ^5(x)}{5}$

[Out]

-Cos[x]^5/5 + Cos[x]^7/7

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Rubi [A]  time = 0.022827, 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 = {2565, 14} $\frac{\cos ^7(x)}{7}-\frac{\cos ^5(x)}{5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cos[x]^5/5 + Cos[x]^7/7

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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 ^4(x) \sin ^3(x) \, dx &=-\operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{1}{5} \cos ^5(x)+\frac{\cos ^7(x)}{7}\\ \end{align*}

Mathematica [A]  time = 0.009907, size = 31, normalized size = 1.82 $-\frac{3 \cos (x)}{64}-\frac{1}{64} \cos (3 x)+\frac{1}{320} \cos (5 x)+\frac{1}{448} \cos (7 x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 18, normalized size = 1.1 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5} \left ( \sin \left ( x \right ) \right ) ^{2}}{7}}-{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{5}}{35}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*cos(x)^5*sin(x)^2-2/35*cos(x)^5

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Maxima [A]  time = 0.95728, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*cos(x)^7 - 1/5*cos(x)^5

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Fricas [A]  time = 2.21902, size = 39, normalized size = 2.29 \begin{align*} \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*cos(x)^7 - 1/5*cos(x)^5

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Sympy [A]  time = 0.061924, size = 12, normalized size = 0.71 \begin{align*} \frac{\cos ^{7}{\left (x \right )}}{7} - \frac{\cos ^{5}{\left (x \right )}}{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(x)**7/7 - cos(x)**5/5

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Giac [A]  time = 1.0605, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*cos(x)^7 - 1/5*cos(x)^5