3.6 \(\int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=43 \[ -\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2} \]

[Out]

(a*x)/2 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)

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Rubi [A]  time = 0.0330301, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2669, 2635, 8} \[ -\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

(a*x)/2 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

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 8

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

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx &=-\frac{a \cos ^3(c+d x)}{3 d}+a \int \cos ^2(c+d x) \, dx\\ &=-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a \int 1 \, dx\\ &=\frac{a x}{2}-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.0484589, size = 46, normalized size = 1.07 \[ \frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{a \cos ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

(a*(c + d*x))/(2*d) - (a*Cos[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d)

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Maple [A]  time = 0.025, size = 41, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3}}+a \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+a*sin(d*x+c)),x)

[Out]

1/d*(-1/3*a*cos(d*x+c)^3+a*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [A]  time = 0.959197, size = 50, normalized size = 1.16 \begin{align*} -\frac{4 \, a \cos \left (d x + c\right )^{3} - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/12*(4*a*cos(d*x + c)^3 - 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*a)/d

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Fricas [A]  time = 1.65802, size = 96, normalized size = 2.23 \begin{align*} -\frac{2 \, a \cos \left (d x + c\right )^{3} - 3 \, a d x - 3 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/6*(2*a*cos(d*x + c)^3 - 3*a*d*x - 3*a*cos(d*x + c)*sin(d*x + c))/d

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Sympy [A]  time = 0.685955, size = 71, normalized size = 1.65 \begin{align*} \begin{cases} \frac{a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} - \frac{a \cos ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \cos ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+a*sin(d*x+c)),x)

[Out]

Piecewise((a*x*sin(c + d*x)**2/2 + a*x*cos(c + d*x)**2/2 + a*sin(c + d*x)*cos(c + d*x)/(2*d) - a*cos(c + d*x)*
*3/(3*d), Ne(d, 0)), (x*(a*sin(c) + a)*cos(c)**2, True))

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Giac [A]  time = 1.1155, size = 63, normalized size = 1.47 \begin{align*} \frac{1}{2} \, a x - \frac{a \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac{a \cos \left (d x + c\right )}{4 \, d} + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/2*a*x - 1/12*a*cos(3*d*x + 3*c)/d - 1/4*a*cos(d*x + c)/d + 1/4*a*sin(2*d*x + 2*c)/d