3.834 \(\int 35 \cos ^3(x) \sin ^4(x) \, dx\)

Optimal. Leaf size=13 \[ 7 \sin ^5(x)-5 \sin ^7(x) \]

[Out]

7*Sin[x]^5 - 5*Sin[x]^7

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Rubi [A]  time = 0.0241216, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {12, 2564, 14} \[ 7 \sin ^5(x)-5 \sin ^7(x) \]

Antiderivative was successfully verified.

[In]

Int[35*Cos[x]^3*Sin[x]^4,x]

[Out]

7*Sin[x]^5 - 5*Sin[x]^7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 35 \cos ^3(x) \sin ^4(x) \, dx &=35 \int \cos ^3(x) \sin ^4(x) \, dx\\ &=35 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=35 \operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\sin (x)\right )\\ &=7 \sin ^5(x)-5 \sin ^7(x)\\ \end{align*}

Mathematica [B]  time = 0.0094767, size = 33, normalized size = 2.54 \[ 35 \left (\frac{3 \sin (x)}{64}-\frac{1}{64} \sin (3 x)-\frac{1}{320} \sin (5 x)+\frac{1}{448} \sin (7 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[35*Cos[x]^3*Sin[x]^4,x]

[Out]

35*((3*Sin[x])/64 - Sin[3*x]/64 - Sin[5*x]/320 + Sin[7*x]/448)

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Maple [B]  time = 0.005, size = 29, normalized size = 2.2 \begin{align*} -5\, \left ( \cos \left ( x \right ) \right ) ^{4} \left ( \sin \left ( x \right ) \right ) ^{3}-3\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{4}+ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(35*cos(x)^3*sin(x)^4,x)

[Out]

-5*cos(x)^4*sin(x)^3-3*sin(x)*cos(x)^4+(2+cos(x)^2)*sin(x)

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Maxima [A]  time = 0.979526, size = 18, normalized size = 1.38 \begin{align*} -5 \, \sin \left (x\right )^{7} + 7 \, \sin \left (x\right )^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(35*cos(x)^3*sin(x)^4,x, algorithm="maxima")

[Out]

-5*sin(x)^7 + 7*sin(x)^5

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Fricas [A]  time = 2.29107, size = 66, normalized size = 5.08 \begin{align*}{\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(35*cos(x)^3*sin(x)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.061939, size = 12, normalized size = 0.92 \begin{align*} - 5 \sin ^{7}{\left (x \right )} + 7 \sin ^{5}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(35*cos(x)**3*sin(x)**4,x)

[Out]

-5*sin(x)**7 + 7*sin(x)**5

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Giac [A]  time = 1.07909, size = 18, normalized size = 1.38 \begin{align*} -5 \, \sin \left (x\right )^{7} + 7 \, \sin \left (x\right )^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(35*cos(x)^3*sin(x)^4,x, algorithm="giac")

[Out]

-5*sin(x)^7 + 7*sin(x)^5