### 3.106 $$\int \sin (2 x) \sin (5 x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{6} \sin (3 x)-\frac{1}{14} \sin (7 x)$

[Out]

Sin[3*x]/6 - Sin[7*x]/14

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Rubi [A]  time = 0.0084189, antiderivative size = 17, 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} $\frac{1}{6} \sin (3 x)-\frac{1}{14} \sin (7 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sin[2*x]*Sin[5*x],x]

[Out]

Sin[3*x]/6 - Sin[7*x]/14

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 (2 x) \sin (5 x) \, dx &=\frac{1}{6} \sin (3 x)-\frac{1}{14} \sin (7 x)\\ \end{align*}

Mathematica [A]  time = 0.0065277, size = 17, normalized size = 1. $\frac{1}{6} \sin (3 x)-\frac{1}{14} \sin (7 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sin[2*x]*Sin[5*x],x]

[Out]

Sin[3*x]/6 - Sin[7*x]/14

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Maple [A]  time = 0.046, size = 14, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( 3\,x \right ) }{6}}-{\frac{\sin \left ( 7\,x \right ) }{14}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)*sin(5*x),x)

[Out]

1/6*sin(3*x)-1/14*sin(7*x)

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Maxima [A]  time = 0.933001, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{14} \, \sin \left (7 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)*sin(5*x),x, algorithm="maxima")

[Out]

-1/14*sin(7*x) + 1/6*sin(3*x)

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Fricas [A]  time = 2.08164, size = 81, normalized size = 4.76 \begin{align*} -\frac{2}{21} \,{\left (48 \, \cos \left (x\right )^{6} - 60 \, \cos \left (x\right )^{4} + 11 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)*sin(5*x),x, algorithm="fricas")

[Out]

-2/21*(48*cos(x)^6 - 60*cos(x)^4 + 11*cos(x)^2 + 1)*sin(x)

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Sympy [B]  time = 0.503333, size = 26, normalized size = 1.53 \begin{align*} - \frac{5 \sin{\left (2 x \right )} \cos{\left (5 x \right )}}{21} + \frac{2 \sin{\left (5 x \right )} \cos{\left (2 x \right )}}{21} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)*sin(5*x),x)

[Out]

-5*sin(2*x)*cos(5*x)/21 + 2*sin(5*x)*cos(2*x)/21

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Giac [A]  time = 1.04559, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{14} \, \sin \left (7 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)*sin(5*x),x, algorithm="giac")

[Out]

-1/14*sin(7*x) + 1/6*sin(3*x)