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

Optimal. Leaf size=57 \[ -\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+a^2 x \]

[Out]

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

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Rubi [A]  time = 0.0414491, antiderivative size = 69, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2751, 2644} \[ \frac{4 a^2 \sin (c+d x)}{3 d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{3 d}+a^2 x+\frac{\sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2644

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[((2*a^2 + b^2)*x)/2, x] + (-Simp[(2*a*b*Cos[c
+ d*x])/d, x] - Simp[(b^2*Cos[c + d*x]*Sin[c + d*x])/(2*d), x]) /; FreeQ[{a, b, c, d}, x]

Rubi steps

\begin{align*} \int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx &=\frac{(a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{2}{3} \int (a+a \cos (c+d x))^2 \, dx\\ &=a^2 x+\frac{4 a^2 \sin (c+d x)}{3 d}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{3 d}+\frac{(a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.079044, size = 41, normalized size = 0.72 \[ \frac{a^2 (21 \sin (c+d x)+6 \sin (2 (c+d x))+\sin (3 (c+d x))+12 d x)}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(12*d*x + 21*Sin[c + d*x] + 6*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(12*d)

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Maple [A]  time = 0.039, size = 64, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,{a}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{2}\sin \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12368, size = 82, normalized size = 1.44 \begin{align*} -\frac{2 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 6 \, a^{2} \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)*(a+a*cos(d*x+c))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.70289, size = 113, normalized size = 1.98 \begin{align*} \frac{3 \, a^{2} d x +{\left (a^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} \cos \left (d x + c\right ) + 5 \, a^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*a^2*d*x + (a^2*cos(d*x + c)^2 + 3*a^2*cos(d*x + c) + 5*a^2)*sin(d*x + c))/d

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Sympy [A]  time = 0.609382, size = 107, normalized size = 1.88 \begin{align*} \begin{cases} a^{2} x \sin ^{2}{\left (c + d x \right )} + a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac{2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{a^{2} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\right )^{2} \cos{\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)*(a+a*cos(d*x+c))**2,x)

[Out]

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

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Giac [A]  time = 1.34546, size = 73, normalized size = 1.28 \begin{align*} a^{2} x + \frac{a^{2} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{a^{2} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac{7 \, a^{2} \sin \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x + 1/12*a^2*sin(3*d*x + 3*c)/d + 1/2*a^2*sin(2*d*x + 2*c)/d + 7/4*a^2*sin(d*x + c)/d