3.29 \(\int \sin (\frac{x}{4}) \sin (x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{3} \sin \left (\frac{3 x}{4}\right )-\frac{2}{5} \sin \left (\frac{5 x}{4}\right ) \]

[Out]

(2*Sin[(3*x)/4])/3 - (2*Sin[(5*x)/4])/5

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Rubi [A]  time = 0.0080012, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4282} \[ \frac{2}{3} \sin \left (\frac{3 x}{4}\right )-\frac{2}{5} \sin \left (\frac{5 x}{4}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x/4]*Sin[x],x]

[Out]

(2*Sin[(3*x)/4])/3 - (2*Sin[(5*x)/4])/5

Rule 4282

Int[sin[(a_.) + (b_.)*(x_)]*sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin{align*} \int \sin \left (\frac{x}{4}\right ) \sin (x) \, dx &=\frac{2}{3} \sin \left (\frac{3 x}{4}\right )-\frac{2}{5} \sin \left (\frac{5 x}{4}\right )\\ \end{align*}

Mathematica [A]  time = 0.0108375, size = 21, normalized size = 1. \[ \frac{2}{3} \sin \left (\frac{3 x}{4}\right )-\frac{2}{5} \sin \left (\frac{5 x}{4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x/4]*Sin[x],x]

[Out]

(2*Sin[(3*x)/4])/3 - (2*Sin[(5*x)/4])/5

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Maple [A]  time = 0.027, size = 14, normalized size = 0.7 \begin{align*}{\frac{2}{3}\sin \left ({\frac{3\,x}{4}} \right ) }-{\frac{2}{5}\sin \left ({\frac{5\,x}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(1/4*x)*sin(x),x)

[Out]

2/3*sin(3/4*x)-2/5*sin(5/4*x)

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Maxima [A]  time = 0.924895, size = 18, normalized size = 0.86 \begin{align*} -\frac{2}{5} \, \sin \left (\frac{5}{4} \, x\right ) + \frac{2}{3} \, \sin \left (\frac{3}{4} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/4*x)*sin(x),x, algorithm="maxima")

[Out]

-2/5*sin(5/4*x) + 2/3*sin(3/4*x)

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Fricas [A]  time = 2.41178, size = 77, normalized size = 3.67 \begin{align*} -\frac{16}{15} \,{\left (6 \, \cos \left (\frac{1}{4} \, x\right )^{4} - 7 \, \cos \left (\frac{1}{4} \, x\right )^{2} + 1\right )} \sin \left (\frac{1}{4} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/4*x)*sin(x),x, algorithm="fricas")

[Out]

-16/15*(6*cos(1/4*x)^4 - 7*cos(1/4*x)^2 + 1)*sin(1/4*x)

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Sympy [A]  time = 0.539427, size = 22, normalized size = 1.05 \begin{align*} - \frac{16 \sin{\left (\frac{x}{4} \right )} \cos{\left (x \right )}}{15} + \frac{4 \sin{\left (x \right )} \cos{\left (\frac{x}{4} \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/4*x)*sin(x),x)

[Out]

-16*sin(x/4)*cos(x)/15 + 4*sin(x)*cos(x/4)/15

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Giac [A]  time = 1.04563, size = 18, normalized size = 0.86 \begin{align*} -\frac{2}{5} \, \sin \left (\frac{5}{4} \, x\right ) + \frac{2}{3} \, \sin \left (\frac{3}{4} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(1/4*x)*sin(x),x, algorithm="giac")

[Out]

-2/5*sin(5/4*x) + 2/3*sin(3/4*x)