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]

-1/5*cos(x)^5+1/7*cos(x)^7

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 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 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])

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.42, size = 13, normalized size = 0.76 \[ \frac {1}{7} \, \cos \relax (x)^{7} - \frac {1}{5} \, \cos \relax (x)^{5} \]

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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giac [A]  time = 0.96, size = 13, normalized size = 0.76 \[ \frac {1}{7} \, \cos \relax (x)^{7} - \frac {1}{5} \, \cos \relax (x)^{5} \]

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

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maple [A]  time = 0.01, size = 18, normalized size = 1.06 \[ -\frac {\left (\cos ^{5}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )}{7}-\frac {2 \left (\cos ^{5}\relax (x )\right )}{35} \]

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 = 0.44, size = 13, normalized size = 0.76 \[ \frac {1}{7} \, \cos \relax (x)^{7} - \frac {1}{5} \, \cos \relax (x)^{5} \]

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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mupad [B]  time = 0.04, size = 14, normalized size = 0.82 \[ \frac {{\cos \relax (x)}^5\,\left (5\,{\cos \relax (x)}^2-7\right )}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cos(x)^5*(5*cos(x)^2 - 7))/35

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sympy [A]  time = 0.07, size = 12, normalized size = 0.71 \[ \frac {\cos ^{7}{\relax (x )}}{7} - \frac {\cos ^{5}{\relax (x )}}{5} \]

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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