3.1.62 \(\int \cos ^4(x) \sin ^3(x) \, dx\) [62]

Optimal. Leaf size=17 \[ -\frac {1}{5} \cos ^5(x)+\frac {\cos ^7(x)}{7} \]

[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 = {2645, 14} \begin {gather*} \frac {\cos ^7(x)}{7}-\frac {\cos ^5(x)}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*Cos[x]^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 2645

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 &=-\text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (x)\right )\\ &=-\text {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 \begin {gather*} -\frac {3 \cos (x)}{64}-\frac {1}{64} \cos (3 x)+\frac {1}{320} \cos (5 x)+\frac {1}{448} \cos (7 x) \end {gather*}

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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Mathics [A]
time = 1.79, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\text {Cos}\left [x\right ]^5}{5}+\frac {\text {Cos}\left [x\right ]^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sin[x]^3*Cos[x]^4,x]')

[Out]

-Cos[x] ^ 5 / 5 + Cos[x] ^ 7 / 7

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Maple [A]
time = 0.03, size = 18, normalized size = 1.06

method	result	size
default	\(-\frac {\left (\cos ^{5}\left (x \right )\right ) \left (\sin ^{2}\left (x \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (x \right )\right )}{35}\)	\(18\)
risch	\(-\frac {3 \cos \left (x \right )}{64}+\frac {\cos \left (7 x \right )}{448}+\frac {\cos \left (5 x \right )}{320}-\frac {\cos \left (3 x \right )}{64}\)	\(24\)
norman	\(\frac {-12 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )-\frac {32 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{5}-\frac {8 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{5}-\frac {4 \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{35}-\frac {8}{35}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^4*sin(x)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, \cos \left (x\right )^{7} - \frac {1}{5} \, \cos \left (x\right )^{5} \end {gather*}

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 = 0.32, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, \cos \left (x\right )^{7} - \frac {1}{5} \, \cos \left (x\right )^{5} \end {gather*}

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 = 0.03, size = 12, normalized size = 0.71 \begin {gather*} \frac {\cos ^{7}{\left (x \right )}}{7} - \frac {\cos ^{5}{\left (x \right )}}{5} \end {gather*}

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 = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {\cos ^{7}x}{7}-\frac {\cos ^{5}x}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \frac {{\cos \left (x\right )}^5\,\left (5\,{\cos \left (x\right )}^2-7\right )}{35} \end {gather*}

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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