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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = 0.01, size = 17, normalized size = 1.00 \[ \frac {1}{6} \sin (3 x)-\frac {1}{14} \sin (7 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[3*x]/6 - Sin[7*x]/14

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 24, normalized size = 1.41 \[ -\frac {2}{21} \, {\left (48 \, \cos \relax (x)^{6} - 60 \, \cos \relax (x)^{4} + 11 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.88, size = 13, normalized size = 0.76 \[ -\frac {1}{14} \, \sin \left (7 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \]

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)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 14, normalized size = 0.82 \[ \frac {\sin \left (3 x \right )}{6}-\frac {\sin \left (7 x \right )}{14} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 13, normalized size = 0.76 \[ -\frac {1}{14} \, \sin \left (7 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 13, normalized size = 0.76 \[ \frac {\sin \left (3\,x\right )}{6}-\frac {\sin \left (7\,x\right )}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(3*x)/6 - sin(7*x)/14

________________________________________________________________________________________

sympy [B]  time = 0.57, size = 26, normalized size = 1.53 \[ - \frac {5 \sin {\left (2 x \right )} \cos {\left (5 x \right )}}{21} + \frac {2 \sin {\left (5 x \right )} \cos {\left (2 x \right )}}{21} \]

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

________________________________________________________________________________________