3.2.6 \(\int \sin (2 x) \sin (5 x) \, dx\) [106]

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)

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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 = {4367} \begin {gather*} \frac {1}{6} \sin (3 x)-\frac {1}{14} \sin (7 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[3*x]/6 - Sin[7*x]/14

Rule 4367

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*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{6} \sin (3 x)-\frac {1}{14} \sin (7 x) \end {gather*}

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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Mathics [A]
time = 1.93, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\text {Sin}\left [7 x\right ]}{14}+\frac {\text {Sin}\left [3 x\right ]}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sin[5*x]*Sin[2*x],x]')

[Out]

-Sin[7 x] / 14 + Sin[3 x] / 6

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Maple [A]
time = 0.07, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {\sin \left (3 x \right )}{6}-\frac {\sin \left (7 x \right )}{14}\)	\(14\)
risch	\(\frac {\sin \left (3 x \right )}{6}-\frac {\sin \left (7 x \right )}{14}\)	\(14\)
norman	\(\frac {\frac {10 \tan \left (x \right ) \left (\tan ^{2}\left (\frac {5 x}{2}\right )\right )}{21}-\frac {4 \left (\tan ^{2}\left (x \right )\right ) \tan \left (\frac {5 x}{2}\right )}{21}-\frac {10 \tan \left (x \right )}{21}+\frac {4 \tan \left (\frac {5 x}{2}\right )}{21}}{\left (1+\tan ^{2}\left (x \right )\right ) \left (1+\tan ^{2}\left (\frac {5 x}{2}\right )\right )}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)*sin(5*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{14} \, \sin \left (7 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \end {gather*}

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 = 0.36, size = 24, normalized size = 1.41 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\)
time = 0.14, size = 26, normalized size = 1.53 \begin {gather*} - \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 {gather*}

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 = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {\sin \left (3 x\right )}{6}-\frac {\sin \left (7 x\right )}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*sin(7*x) + 1/6*sin(3*x)

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

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

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