3.4 \(\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx\)

Optimal. Leaf size=47 \[ \frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (2 A+B)+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d} \]

[Out]

(a*(2*A + B)*x)/2 + (a*(A + B)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)

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Rubi [A]  time = 0.020611, antiderivative size = 47, 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} \[ \frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (2 A+B)+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]

[Out]

(a*(2*A + B)*x)/2 + (a*(A + B)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)

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 \cos (c+d x)) (A+B \cos (c+d x)) \, dx &=\frac{1}{2} a (2 A+B) x+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a B \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0979611, size = 44, normalized size = 0.94 \[ \frac{a (4 (A+B) \sin (c+d x)+4 A d x+B \sin (2 (c+d x))+2 B c+2 B d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]

[Out]

(a*(2*B*c + 4*A*d*x + 2*B*d*x + 4*(A + B)*Sin[c + d*x] + B*Sin[2*(c + d*x)]))/(4*d)

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Maple [A]  time = 0.046, size = 57, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( aB \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aA\sin \left ( dx+c \right ) +aB\sin \left ( dx+c \right ) +aA \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+cos(d*x+c)*a)*(A+B*cos(d*x+c)),x)

[Out]

1/d*(a*B*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+a*A*sin(d*x+c)+a*B*sin(d*x+c)+a*A*(d*x+c))

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Maxima [A]  time = 0.986873, size = 74, normalized size = 1.57 \begin{align*} \frac{4 \,{\left (d x + c\right )} A a +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 4 \, A a \sin \left (d x + c\right ) + 4 \, B a \sin \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))*(A+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

1/4*(4*(d*x + c)*A*a + (2*d*x + 2*c + sin(2*d*x + 2*c))*B*a + 4*A*a*sin(d*x + c) + 4*B*a*sin(d*x + c))/d

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Fricas [A]  time = 1.33308, size = 99, normalized size = 2.11 \begin{align*} \frac{{\left (2 \, A + B\right )} a d x +{\left (B a \cos \left (d x + c\right ) + 2 \,{\left (A + B\right )} a\right )} \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))*(A+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/2*((2*A + B)*a*d*x + (B*a*cos(d*x + c) + 2*(A + B)*a)*sin(d*x + c))/d

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Sympy [A]  time = 0.411819, size = 94, normalized size = 2. \begin{align*} \begin{cases} A a x + \frac{A a \sin{\left (c + d x \right )}}{d} + \frac{B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{B a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{B a \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + a\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))*(A+B*cos(d*x+c)),x)

[Out]

Piecewise((A*a*x + A*a*sin(c + d*x)/d + B*a*x*sin(c + d*x)**2/2 + B*a*x*cos(c + d*x)**2/2 + B*a*sin(c + d*x)*c
os(c + d*x)/(2*d) + B*a*sin(c + d*x)/d, Ne(d, 0)), (x*(A + B*cos(c))*(a*cos(c) + a), True))

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Giac [A]  time = 1.16775, size = 61, normalized size = 1.3 \begin{align*} \frac{1}{2} \,{\left (2 \, A a + B a\right )} x + \frac{B a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (A a + B a\right )} \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

1/2*(2*A*a + B*a)*x + 1/4*B*a*sin(2*d*x + 2*c)/d + (A*a + B*a)*sin(d*x + c)/d