3.814 \(\int (x+60 \cos ^5(x) \sin ^4(x)) \, dx\)

Optimal. Leaf size=30 \[ \frac{x^2}{2}+\frac{20 \sin ^9(x)}{3}-\frac{120 \sin ^7(x)}{7}+12 \sin ^5(x) \]

[Out]

x^2/2 + 12*Sin[x]^5 - (120*Sin[x]^7)/7 + (20*Sin[x]^9)/3

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Rubi [A]  time = 0.0303572, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2564, 270} \[ \frac{x^2}{2}+\frac{20 \sin ^9(x)}{3}-\frac{120 \sin ^7(x)}{7}+12 \sin ^5(x) \]

Antiderivative was successfully verified.

[In]

Int[x + 60*Cos[x]^5*Sin[x]^4,x]

[Out]

x^2/2 + 12*Sin[x]^5 - (120*Sin[x]^7)/7 + (20*Sin[x]^9)/3

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (x+60 \cos ^5(x) \sin ^4(x)\right ) \, dx &=\frac{x^2}{2}+60 \int \cos ^5(x) \sin ^4(x) \, dx\\ &=\frac{x^2}{2}+60 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (x)\right )\\ &=\frac{x^2}{2}+60 \operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (x)\right )\\ &=\frac{x^2}{2}+12 \sin ^5(x)-\frac{120 \sin ^7(x)}{7}+\frac{20 \sin ^9(x)}{3}\\ \end{align*}

Mathematica [A]  time = 0.0175568, size = 46, normalized size = 1.53 \[ \frac{x^2}{2}+\frac{45 \sin (x)}{32}-\frac{5}{16} \sin (3 x)-\frac{3}{16} \sin (5 x)+\frac{15}{448} \sin (7 x)+\frac{5}{192} \sin (9 x) \]

Antiderivative was successfully verified.

[In]

Integrate[x + 60*Cos[x]^5*Sin[x]^4,x]

[Out]

x^2/2 + (45*Sin[x])/32 - (5*Sin[3*x])/16 - (3*Sin[5*x])/16 + (15*Sin[7*x])/448 + (5*Sin[9*x])/192

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Maple [A]  time = 0.007, size = 41, normalized size = 1.4 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{20\, \left ( \cos \left ( x \right ) \right ) ^{6} \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}-{\frac{20\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{6}}{7}}+{\frac{4\,\sin \left ( x \right ) }{7} \left ({\frac{8}{3}}+ \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x+60*cos(x)^5*sin(x)^4,x)

[Out]

1/2*x^2-20/3*cos(x)^6*sin(x)^3-20/7*sin(x)*cos(x)^6+4/7*(8/3+cos(x)^4+4/3*cos(x)^2)*sin(x)

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Maxima [A]  time = 0.962846, size = 32, normalized size = 1.07 \begin{align*} \frac{20}{3} \, \sin \left (x\right )^{9} - \frac{120}{7} \, \sin \left (x\right )^{7} + 12 \, \sin \left (x\right )^{5} + \frac{1}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x+60*cos(x)^5*sin(x)^4,x, algorithm="maxima")

[Out]

20/3*sin(x)^9 - 120/7*sin(x)^7 + 12*sin(x)^5 + 1/2*x^2

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Fricas [A]  time = 2.33896, size = 109, normalized size = 3.63 \begin{align*} \frac{1}{2} \, x^{2} + \frac{4}{21} \,{\left (35 \, \cos \left (x\right )^{8} - 50 \, \cos \left (x\right )^{6} + 3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x+60*cos(x)^5*sin(x)^4,x, algorithm="fricas")

[Out]

1/2*x^2 + 4/21*(35*cos(x)^8 - 50*cos(x)^6 + 3*cos(x)^4 + 4*cos(x)^2 + 8)*sin(x)

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Sympy [A]  time = 0.068034, size = 27, normalized size = 0.9 \begin{align*} \frac{x^{2}}{2} + \frac{20 \sin ^{9}{\left (x \right )}}{3} - \frac{120 \sin ^{7}{\left (x \right )}}{7} + 12 \sin ^{5}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x+60*cos(x)**5*sin(x)**4,x)

[Out]

x**2/2 + 20*sin(x)**9/3 - 120*sin(x)**7/7 + 12*sin(x)**5

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Giac [A]  time = 1.10109, size = 32, normalized size = 1.07 \begin{align*} \frac{20}{3} \, \sin \left (x\right )^{9} - \frac{120}{7} \, \sin \left (x\right )^{7} + 12 \, \sin \left (x\right )^{5} + \frac{1}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x+60*cos(x)^5*sin(x)^4,x, algorithm="giac")

[Out]

20/3*sin(x)^9 - 120/7*sin(x)^7 + 12*sin(x)^5 + 1/2*x^2