3.115 \(\int \cos ^2(a+b x) \, dx\)

Optimal. Leaf size=25 \[ \frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2635, 8} \[ \frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^2,x]

[Out]

x/2 + (Cos[a + b*x]*Sin[a + b*x])/(2*b)

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]

Rubi steps

\begin {align*} \int \cos ^2(a+b x) \, dx &=\frac {\cos (a+b x) \sin (a+b x)}{2 b}+\frac {\int 1 \, dx}{2}\\ &=\frac {x}{2}+\frac {\cos (a+b x) \sin (a+b x)}{2 b}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.92 \[ \frac {2 (a+b x)+\sin (2 (a+b x))}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^2,x]

[Out]

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

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fricas [A]  time = 0.44, size = 22, normalized size = 0.88 \[ \frac {b x + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.06, size = 18, normalized size = 0.72 \[ \frac {1}{2} \, x + \frac {\sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2,x, algorithm="giac")

[Out]

1/2*x + 1/4*sin(2*b*x + 2*a)/b

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maple [A]  time = 0.03, size = 27, normalized size = 1.08 \[ \frac {\frac {b x}{2}+\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {a}{2}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^2,x)

[Out]

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

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maxima [A]  time = 0.43, size = 22, normalized size = 0.88 \[ \frac {2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.18, size = 18, normalized size = 0.72 \[ \frac {x}{2}+\frac {\sin \left (2\,a+2\,b\,x\right )}{4\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^2,x)

[Out]

x/2 + sin(2*a + 2*b*x)/(4*b)

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sympy [A]  time = 0.23, size = 46, normalized size = 1.84 \[ \begin {cases} \frac {x \sin ^{2}{\left (a + b x \right )}}{2} + \frac {x \cos ^{2}{\left (a + b x \right )}}{2} + \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \cos ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**2,x)

[Out]

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

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