3.428 \(\int (a+a \sin (e+f x)) (c+d \sin (e+f x)) \, dx\)

Optimal. Leaf size=48 \[ -\frac{a (c+d) \cos (e+f x)}{f}+\frac{1}{2} a x (2 c+d)-\frac{a d \sin (e+f x) \cos (e+f x)}{2 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.023605, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2734} \[ -\frac{a (c+d) \cos (e+f x)}{f}+\frac{1}{2} a x (2 c+d)-\frac{a d \sin (e+f x) \cos (e+f x)}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(2*c + d)*x)/2 - (a*(c + d)*Cos[e + f*x])/f - (a*d*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+d \sin (e+f x)) \, dx &=\frac{1}{2} a (2 c+d) x-\frac{a (c+d) \cos (e+f x)}{f}-\frac{a d \cos (e+f x) \sin (e+f x)}{2 f}\\ \end{align*}

Mathematica [A]  time = 0.112512, size = 45, normalized size = 0.94 \[ \frac{a (-4 (c+d) \cos (e+f x)+4 c f x-d \sin (2 (e+f x))+2 d e+2 d f x)}{4 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.029, size = 59, normalized size = 1.2 \begin{align*}{\frac{1}{f} \left ( da \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -ca\cos \left ( fx+e \right ) -da\cos \left ( fx+e \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+d*sin(f*x+e)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.10094, size = 77, normalized size = 1.6 \begin{align*} \frac{4 \,{\left (f x + e\right )} a c +{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a d - 4 \, a c \cos \left (f x + e\right ) - 4 \, a d \cos \left (f x + 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+d*sin(f*x+e)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.06934, size = 120, normalized size = 2.5 \begin{align*} -\frac{a d \cos \left (f x + e\right ) \sin \left (f x + e\right ) -{\left (2 \, a c + a d\right )} f x + 2 \,{\left (a c + a d\right )} \cos \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+d*sin(f*x+e)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.331355, size = 94, normalized size = 1.96 \begin{align*} \begin{cases} a c x - \frac{a c \cos{\left (e + f x \right )}}{f} + \frac{a d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{a d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{a d \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{a d \cos{\left (e + f x \right )}}{f} & \text{for}\: f \neq 0 \\x \left (c + d \sin{\left (e \right )}\right ) \left (a \sin{\left (e \right )} + a\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+d*sin(f*x+e)),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.25065, size = 74, normalized size = 1.54 \begin{align*} a c x + \frac{1}{2} \, a d x - \frac{a c \cos \left (f x + e\right )}{f} - \frac{a d \cos \left (f x + e\right )}{f} - \frac{a d \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+d*sin(f*x+e)),x, algorithm="giac")

[Out]

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