3.409 \(\int \cos (c+d x) (a+b \cos (c+d x)) \, dx\)

Optimal. Leaf size=38 \[ \frac {a \sin (c+d x)}{d}+\frac {b \sin (c+d x) \cos (c+d x)}{2 d}+\frac {b x}{2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2734} \[ \frac {a \sin (c+d x)}{d}+\frac {b \sin (c+d x) \cos (c+d x)}{2 d}+\frac {b x}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.06, size = 35, normalized size = 0.92 \[ \frac {4 a \sin (c+d x)+b (2 (c+d x)+\sin (2 (c+d x)))}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.90, size = 29, normalized size = 0.76 \[ \frac {b d x + {\left (b \cos \left (d x + c\right ) + 2 \, a\right )} \sin \left (d x + c\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.43, size = 31, normalized size = 0.82 \[ \frac {1}{2} \, b x + \frac {b \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {a \sin \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 38, normalized size = 1.00 \[ \frac {b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\sin \left (d x +c \right ) a}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(a+b*cos(d*x+c)),x)

[Out]

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

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maxima [A]  time = 0.65, size = 34, normalized size = 0.89 \[ \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} b + 4 \, a \sin \left (d x + c\right )}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.52, size = 31, normalized size = 0.82 \[ \frac {b\,x}{2}+\frac {b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {a\,\sin \left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)*(a + b*cos(c + d*x)),x)

[Out]

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

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sympy [A]  time = 0.20, size = 66, normalized size = 1.74 \[ \begin {cases} \frac {a \sin {\left (c + d x \right )}}{d} + \frac {b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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