3.1.63 \(\int \cos ^3(x) \sin ^4(x) \, dx\) [63]

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

[Out]

1/5*sin(x)^5-1/7*sin(x)^7

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Rubi [A]
time = 0.01, 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 = {2644, 14} \begin {gather*} \frac {\sin ^5(x)}{5}-\frac {\sin ^7(x)}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]^5/5 - Sin[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 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 ^4(x) \, dx &=\text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=\text {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*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.82 \begin {gather*} \frac {3 \sin (x)}{64}-\frac {1}{64} \sin (3 x)-\frac {1}{320} \sin (5 x)+\frac {1}{448} \sin (7 x) \end {gather*}

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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Mathics [A]
time = 1.81, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\text {Sin}\left [x\right ]^7}{7}+\frac {\text {Sin}\left [x\right ]^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sin[x] ^ 7 / 7 + Sin[x] ^ 5 / 5

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs. \(2(13)=26\).
time = 0.03, size = 30, normalized size = 1.76

method	result	size
risch	\(\frac {3 \sin \left (x \right )}{64}+\frac {\sin \left (7 x \right )}{448}-\frac {\sin \left (5 x \right )}{320}-\frac {\sin \left (3 x \right )}{64}\)	\(24\)
default	\(-\frac {\left (\cos ^{4}\left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{7}-\frac {3 \sin \left (x \right ) \left (\cos ^{4}\left (x \right )\right )}{35}+\frac {\left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{35}\)	\(30\)
norman	\(\frac {\frac {32 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {192 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{35}+\frac {32 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{5}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.26, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{7} \, \sin \left (x\right )^{7} + \frac {1}{5} \, \sin \left (x\right )^{5} \end {gather*}

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 = 0.34, size = 22, normalized size = 1.29 \begin {gather*} \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 {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*sin(x)^7 + 1/5*sin(x)^5

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Mupad [B]
time = 0.15, size = 14, normalized size = 0.82 \begin {gather*} -\frac {{\sin \left (x\right )}^5\,\left (5\,{\sin \left (x\right )}^2-7\right )}{35} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sin(x)^5*(5*sin(x)^2 - 7))/35

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