3.69 \(\int \sin (x) \sin (m x) \, dx\)

Optimal. Leaf size=35 \[ \frac{\sin ((1-m) x)}{2 (1-m)}-\frac{\sin ((m+1) x)}{2 (m+1)} \]

[Out]

Sin[(1 - m)*x]/(2*(1 - m)) - Sin[(1 + m)*x]/(2*(1 + m))

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Rubi [A]  time = 0.0308315, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4569, 2637} \[ \frac{\sin ((1-m) x)}{2 (1-m)}-\frac{\sin ((m+1) x)}{2 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]*Sin[m*x],x]

[Out]

Sin[(1 - m)*x]/(2*(1 - m)) - Sin[(1 + m)*x]/(2*(1 + m))

Rule 4569

Int[Sin[v_]^(p_.)*Sin[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Sin[w]^q, x], x] /; ((PolynomialQ[
v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])) && IGtQ[p, 0] && IGtQ[q
, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sin (x) \sin (m x) \, dx &=\int \left (\frac{1}{2} \cos ((1-m) x)-\frac{1}{2} \cos ((1+m) x)\right ) \, dx\\ &=\frac{1}{2} \int \cos ((1-m) x) \, dx-\frac{1}{2} \int \cos ((1+m) x) \, dx\\ &=\frac{\sin ((1-m) x)}{2 (1-m)}-\frac{\sin ((1+m) x)}{2 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0433652, size = 25, normalized size = 0.71 \[ \frac{\cos (x) \sin (m x)-m \sin (x) \cos (m x)}{m^2-1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]*Sin[m*x],x]

[Out]

(-(m*Cos[m*x]*Sin[x]) + Cos[x]*Sin[m*x])/(-1 + m^2)

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Maple [A]  time = 0.013, size = 28, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( \left ( m-1 \right ) x \right ) }{2\,m-2}}-{\frac{\sin \left ( \left ( 1+m \right ) x \right ) }{2+2\,m}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)*sin(m*x),x)

[Out]

1/2/(m-1)*sin((m-1)*x)-1/2*sin((1+m)*x)/(1+m)

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Maxima [A]  time = 0.968175, size = 38, normalized size = 1.09 \begin{align*} -\frac{\sin \left ({\left (m + 1\right )} x\right )}{2 \,{\left (m + 1\right )}} - \frac{\sin \left (-{\left (m - 1\right )} x\right )}{2 \,{\left (m - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*sin(m*x),x, algorithm="maxima")

[Out]

-1/2*sin((m + 1)*x)/(m + 1) - 1/2*sin(-(m - 1)*x)/(m - 1)

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Fricas [A]  time = 2.27251, size = 68, normalized size = 1.94 \begin{align*} -\frac{m \cos \left (m x\right ) \sin \left (x\right ) - \cos \left (x\right ) \sin \left (m x\right )}{m^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*sin(m*x),x, algorithm="fricas")

[Out]

-(m*cos(m*x)*sin(x) - cos(x)*sin(m*x))/(m^2 - 1)

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Sympy [A]  time = 5.02167, size = 78, normalized size = 2.23 \begin{align*} \begin{cases} - \frac{x \sin ^{2}{\left (x \right )}}{2} - \frac{x \cos ^{2}{\left (x \right )}}{2} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} & \text{for}\: m = -1 \\\frac{x \sin ^{2}{\left (x \right )}}{2} + \frac{x \cos ^{2}{\left (x \right )}}{2} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} & \text{for}\: m = 1 \\- \frac{m \sin{\left (x \right )} \cos{\left (m x \right )}}{m^{2} - 1} + \frac{\sin{\left (m x \right )} \cos{\left (x \right )}}{m^{2} - 1} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*sin(m*x),x)

[Out]

Piecewise((-x*sin(x)**2/2 - x*cos(x)**2/2 + sin(x)*cos(x)/2, Eq(m, -1)), (x*sin(x)**2/2 + x*cos(x)**2/2 - sin(
x)*cos(x)/2, Eq(m, 1)), (-m*sin(x)*cos(m*x)/(m**2 - 1) + sin(m*x)*cos(x)/(m**2 - 1), True))

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Giac [A]  time = 1.11365, size = 39, normalized size = 1.11 \begin{align*} -\frac{\sin \left (m x + x\right )}{2 \,{\left (m + 1\right )}} + \frac{\sin \left (m x - x\right )}{2 \,{\left (m - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)*sin(m*x),x, algorithm="giac")

[Out]

-1/2*sin(m*x + x)/(m + 1) + 1/2*sin(m*x - x)/(m - 1)