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

Optimal. Leaf size=53 \[ -\frac{(a d+b c) \cos (e+f x)}{f}+\frac{1}{2} x (2 a c+b d)-\frac{b d \sin (e+f x) \cos (e+f x)}{2 f} \]

[Out]

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

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Rubi [A]  time = 0.022245, antiderivative size = 53, 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} \[ -\frac{(a d+b c) \cos (e+f x)}{f}+\frac{1}{2} x (2 a c+b d)-\frac{b d \sin (e+f x) \cos (e+f x)}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0907934, size = 52, normalized size = 0.98 \[ \frac{-4 (a d+b c) \cos (e+f x)+4 a c f x-b d \sin (2 (e+f x))+2 b d e+2 b d f x}{4 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.022, size = 59, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ( bd \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -da\cos \left ( fx+e \right ) -cb\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+b*sin(f*x+e))*(c+d*sin(f*x+e)),x)

[Out]

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

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Maxima [A]  time = 1.03329, size = 77, normalized size = 1.45 \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 )} b d - 4 \, b 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+b*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))*b*d - 4*b*c*cos(f*x + e) - 4*a*d*cos(f*x + e))/f

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Fricas [A]  time = 1.59674, size = 120, normalized size = 2.26 \begin{align*} -\frac{b d \cos \left (f x + e\right ) \sin \left (f x + e\right ) -{\left (2 \, a c + b d\right )} f x + 2 \,{\left (b 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+b*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.337631, size = 94, normalized size = 1.77 \begin{align*} \begin{cases} a c x - \frac{a d \cos{\left (e + f x \right )}}{f} - \frac{b c \cos{\left (e + f x \right )}}{f} + \frac{b d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{b d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{b d \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} & \text{for}\: f \neq 0 \\x \left (a + b \sin{\left (e \right )}\right ) \left (c + d \sin{\left (e \right )}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15234, size = 65, normalized size = 1.23 \begin{align*} \frac{1}{2} \,{\left (2 \, a c + b d\right )} x - \frac{b d \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} - \frac{{\left (b c + a d\right )} \cos \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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