3.230 \(\int (a+a \sin (e+f x)) (c-c \sin (e+f x)) \, dx\)

Optimal. Leaf size=29 \[ \frac{a c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a c x}{2} \]

[Out]

(a*c*x)/2 + (a*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)

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Rubi [A]  time = 0.0183067, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {2734} \[ \frac{a c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a c x}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x]),x]

[Out]

(a*c*x)/2 + (a*c*Cos[e + f*x]*Sin[e + f*x])/(2*f)

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x)) (c-c \sin (e+f x)) \, dx &=\frac{a c x}{2}+\frac{a c \cos (e+f x) \sin (e+f x)}{2 f}\\ \end{align*}

Mathematica [A]  time = 0.0241253, size = 25, normalized size = 0.86 \[ \frac{a c (2 (e+f x)+\sin (2 (e+f x)))}{4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])*(c - c*Sin[e + f*x]),x]

[Out]

(a*c*(2*(e + f*x) + Sin[2*(e + f*x)]))/(4*f)

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Maple [A]  time = 0.013, size = 40, normalized size = 1.4 \begin{align*}{\frac{1}{f} \left ( -ca \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) +ca \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))*(c-c*sin(f*x+e)),x)

[Out]

1/f*(-c*a*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+c*a*(f*x+e))

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Maxima [A]  time = 1.18982, size = 50, normalized size = 1.72 \begin{align*} -\frac{{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a c - 4 \,{\left (f x + e\right )} a c}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e)),x, algorithm="maxima")

[Out]

-1/4*((2*f*x + 2*e - sin(2*f*x + 2*e))*a*c - 4*(f*x + e)*a*c)/f

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Fricas [A]  time = 1.50889, size = 66, normalized size = 2.28 \begin{align*} \frac{a c f x + a c \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e)),x, algorithm="fricas")

[Out]

1/2*(a*c*f*x + a*c*cos(f*x + e)*sin(f*x + e))/f

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Sympy [A]  time = 0.406081, size = 70, normalized size = 2.41 \begin{align*} \begin{cases} - \frac{a c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac{a c x \cos ^{2}{\left (e + f x \right )}}{2} + a c x + \frac{a c \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} & \text{for}\: f \neq 0 \\x \left (a \sin{\left (e \right )} + a\right ) \left (- c \sin{\left (e \right )} + c\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e)),x)

[Out]

Piecewise((-a*c*x*sin(e + f*x)**2/2 - a*c*x*cos(e + f*x)**2/2 + a*c*x + a*c*sin(e + f*x)*cos(e + f*x)/(2*f), N
e(f, 0)), (x*(a*sin(e) + a)*(-c*sin(e) + c), True))

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Giac [A]  time = 1.9982, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{2} \, a c x + \frac{a c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e)),x, algorithm="giac")

[Out]

1/2*a*c*x + 1/4*a*c*sin(2*f*x + 2*e)/f