3.44 \(\int \cos (a+b x) \sin (a+b x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\sin ^2(a+b x)}{2 b} \]

[Out]

Sin[a + b*x]^2/(2*b)

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Rubi [A]  time = 0.0108554, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2564, 30} \[ \frac{\sin ^2(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]*Sin[a + b*x],x]

[Out]

Sin[a + b*x]^2/(2*b)

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \cos (a+b x) \sin (a+b x) \, dx &=\frac{\operatorname{Subst}(\int x \, dx,x,\sin (a+b x))}{b}\\ &=\frac{\sin ^2(a+b x)}{2 b}\\ \end{align*}

Mathematica [B]  time = 0.0126988, size = 37, normalized size = 2.47 \[ \frac{1}{2} \left (\frac{\sin (2 a) \sin (2 b x)}{2 b}-\frac{\cos (2 a) \cos (2 b x)}{2 b}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]*Sin[a + b*x],x]

[Out]

(-(Cos[2*a]*Cos[2*b*x])/(2*b) + (Sin[2*a]*Sin[2*b*x])/(2*b))/2

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)*sin(b*x+a),x)

[Out]

1/2*sin(b*x+a)^2/b

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Maxima [A]  time = 1.01269, size = 18, normalized size = 1.2 \begin{align*} -\frac{\cos \left (b x + a\right )^{2}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/2*cos(b*x + a)^2/b

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Fricas [A]  time = 1.67621, size = 31, normalized size = 2.07 \begin{align*} -\frac{\cos \left (b x + a\right )^{2}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*sin(b*x+a),x, algorithm="fricas")

[Out]

-1/2*cos(b*x + a)^2/b

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Sympy [A]  time = 0.23076, size = 19, normalized size = 1.27 \begin{align*} \begin{cases} \frac{\sin ^{2}{\left (a + b x \right )}}{2 b} & \text{for}\: b \neq 0 \\x \sin{\left (a \right )} \cos{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*sin(b*x+a),x)

[Out]

Piecewise((sin(a + b*x)**2/(2*b), Ne(b, 0)), (x*sin(a)*cos(a), True))

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Giac [A]  time = 1.14891, size = 18, normalized size = 1.2 \begin{align*} \frac{\sin \left (b x + a\right )^{2}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*sin(b*x+a),x, algorithm="giac")

[Out]

1/2*sin(b*x + a)^2/b