3.823 \(\int \cos (\frac {1}{2} (1+3 x)) \sin ^3(\frac {1}{2} (1+3 x)) \, dx\)

Optimal. Leaf size=16 \[ \frac {1}{6} \sin ^4\left (\frac {3 x}{2}+\frac {1}{2}\right ) \]

[Out]

1/6*sin(1/2+3/2*x)^4

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2564, 30} \[ \frac {1}{6} \sin ^4\left (\frac {3 x}{2}+\frac {1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[(1 + 3*x)/2]*Sin[(1 + 3*x)/2]^3,x]

[Out]

Sin[1/2 + (3*x)/2]^4/6

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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])

Rubi steps

\begin {align*} \int \cos \left (\frac {1}{2} (1+3 x)\right ) \sin ^3\left (\frac {1}{2} (1+3 x)\right ) \, dx &=\frac {2}{3} \operatorname {Subst}\left (\int x^3 \, dx,x,\sin \left (\frac {1}{2}+\frac {3 x}{2}\right )\right )\\ &=\frac {1}{6} \sin ^4\left (\frac {1}{2}+\frac {3 x}{2}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 25, normalized size = 1.56 \[ \frac {1}{2} \left (\frac {1}{24} \cos (6 x+2)-\frac {1}{6} \cos (3 x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[(1 + 3*x)/2]*Sin[(1 + 3*x)/2]^3,x]

[Out]

(-1/6*Cos[1 + 3*x] + Cos[2 + 6*x]/24)/2

________________________________________________________________________________________

fricas [B]  time = 0.75, size = 21, normalized size = 1.31 \[ \frac {1}{6} \, \cos \left (\frac {3}{2} \, x + \frac {1}{2}\right )^{4} - \frac {1}{3} \, \cos \left (\frac {3}{2} \, x + \frac {1}{2}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2+3/2*x)*sin(1/2+3/2*x)^3,x, algorithm="fricas")

[Out]

1/6*cos(3/2*x + 1/2)^4 - 1/3*cos(3/2*x + 1/2)^2

________________________________________________________________________________________

giac [A]  time = 0.20, size = 10, normalized size = 0.62 \[ \frac {1}{6} \, \sin \left (\frac {3}{2} \, x + \frac {1}{2}\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2+3/2*x)*sin(1/2+3/2*x)^3,x, algorithm="giac")

[Out]

1/6*sin(3/2*x + 1/2)^4

________________________________________________________________________________________

maple [A]  time = 0.04, size = 11, normalized size = 0.69 \[ \frac {\left (\sin ^{4}\left (\frac {1}{2}+\frac {3 x}{2}\right )\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2+3/2*x)*sin(1/2+3/2*x)^3,x)

[Out]

1/6*sin(1/2+3/2*x)^4

________________________________________________________________________________________

maxima [A]  time = 0.33, size = 10, normalized size = 0.62 \[ \frac {1}{6} \, \sin \left (\frac {3}{2} \, x + \frac {1}{2}\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2+3/2*x)*sin(1/2+3/2*x)^3,x, algorithm="maxima")

[Out]

1/6*sin(3/2*x + 1/2)^4

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 14, normalized size = 0.88 \[ \frac {{\left (\frac {\cos \left (3\,x+1\right )}{2}-\frac {1}{2}\right )}^2}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos((3*x)/2 + 1/2)*sin((3*x)/2 + 1/2)^3,x)

[Out]

(cos(3*x + 1)/2 - 1/2)^2/6

________________________________________________________________________________________

sympy [A]  time = 0.46, size = 12, normalized size = 0.75 \[ \frac {\sin ^{4}{\left (\frac {3 x}{2} + \frac {1}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2+3/2*x)*sin(1/2+3/2*x)**3,x)

[Out]

sin(3*x/2 + 1/2)**4/6

________________________________________________________________________________________