3.294 \(\int (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=41 \[ A x+\frac {B \sin (c+d x)}{d}+\frac {C \sin (c+d x) \cos (c+d x)}{2 d}+\frac {C x}{2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2637, 2635, 8} \[ A x+\frac {B \sin (c+d x)}{d}+\frac {C \sin (c+d x) \cos (c+d x)}{2 d}+\frac {C x}{2} \]

Antiderivative was successfully verified.

[In]

Int[A + B*Cos[c + d*x] + C*Cos[c + d*x]^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=A x+B \int \cos (c+d x) \, dx+C \int \cos ^2(c+d x) \, dx\\ &=A x+\frac {B \sin (c+d x)}{d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} C \int 1 \, dx\\ &=A x+\frac {C x}{2}+\frac {B \sin (c+d x)}{d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 55, normalized size = 1.34 \[ A x+\frac {B \sin (c) \cos (d x)}{d}+\frac {B \cos (c) \sin (d x)}{d}+\frac {C (c+d x)}{2 d}+\frac {C \sin (2 (c+d x))}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[A + B*Cos[c + d*x] + C*Cos[c + d*x]^2,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 33, normalized size = 0.80 \[ \frac {{\left (2 \, A + C\right )} d x + {\left (C \cos \left (d x + c\right ) + 2 \, B\right )} \sin \left (d x + c\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(A+B*cos(d*x+c)+C*cos(d*x+c)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.54, size = 35, normalized size = 0.85 \[ \frac {1}{4} \, C {\left (2 \, x + \frac {\sin \left (2 \, d x + 2 \, c\right )}{d}\right )} + A x + \frac {B \sin \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(A+B*cos(d*x+c)+C*cos(d*x+c)^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 43, normalized size = 1.05 \[ A x +\frac {B \sin \left (d x +c \right )}{d}+\frac {C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(A+B*cos(d*x+c)+C*cos(d*x+c)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 38, normalized size = 0.93 \[ A x + \frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C}{4 \, d} + \frac {B \sin \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(A+B*cos(d*x+c)+C*cos(d*x+c)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.06, size = 34, normalized size = 0.83 \[ A\,x+\frac {C\,x}{2}+\frac {C\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,\sin \left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(A + B*cos(c + d*x) + C*cos(c + d*x)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.23, size = 66, normalized size = 1.61 \[ A x + B \left (\begin {cases} \frac {\sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cos {\relax (c )} & \text {otherwise} \end {cases}\right ) + C \left (\begin {cases} \frac {x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {\sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(A+B*cos(d*x+c)+C*cos(d*x+c)**2,x)

[Out]

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

________________________________________________________________________________________