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

Optimal. Leaf size=52 \[ \frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (2 a A+b B)+\frac{b B \sin (c+d x) \cos (c+d x)}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0228202, antiderivative size = 52, 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} \[ \frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (2 a A+b B)+\frac{b B \sin (c+d x) \cos (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0839046, size = 51, normalized size = 0.98 \[ \frac{4 (a B+A b) \sin (c+d x)+4 a A d x+b B \sin (2 (c+d x))+2 b B c+2 b B d x}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.038, size = 57, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( Bb \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Ab\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+b*cos(d*x+c))*(A+B*cos(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.01024, size = 74, normalized size = 1.42 \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 b + 4 \, B a \sin \left (d x + c\right ) + 4 \, A b \sin \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*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*b + 4*B*a*sin(d*x + c) + 4*A*b*sin(d*x + c))/d

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Fricas [A]  time = 1.35626, size = 104, normalized size = 2. \begin{align*} \frac{{\left (2 \, A a + B b\right )} d x +{\left (B b \cos \left (d x + c\right ) + 2 \, B a + 2 \, A b\right )} \sin \left (d x + c\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.316311, size = 94, normalized size = 1.81 \begin{align*} \begin{cases} A a x + \frac{A b \sin{\left (c + d x \right )}}{d} + \frac{B a \sin{\left (c + d x \right )}}{d} + \frac{B b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{B b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{B b \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a + b \cos{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.36641, size = 61, normalized size = 1.17 \begin{align*} \frac{1}{2} \,{\left (2 \, A a + B b\right )} x + \frac{B b \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (B a + A b\right )} \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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