3.561 \(\int (a+b \cos (c+d x)) (a^2-b^2 \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=92 \[ \frac{2 b \left (2 a^2-b^2\right ) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left (2 a^2-b^2\right )+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]

[Out]

(a*(2*a^2 - b^2)*x)/2 + (2*b*(2*a^2 - b^2)*Sin[c + d*x])/(3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*
(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.112424, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3016, 2753, 2734} \[ \frac{2 b \left (2 a^2-b^2\right ) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left (2 a^2-b^2\right )+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(2*a^2 - b^2)*x)/2 + (2*b*(2*a^2 - b^2)*Sin[c + d*x])/(3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(6*d) - (b*
(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)

Rule 3016

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[
C/b^2, Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[-a + b*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x
] && EqQ[A*b^2 + a^2*C, 0]

Rule 2753

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[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

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 (a+b \cos (c+d x)) \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx &=-\int (-a+b \cos (c+d x)) (a+b \cos (c+d x))^2 \, dx\\ &=-\frac{b (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}-\frac{1}{3} \int (a+b \cos (c+d x)) \left (-3 a^2+2 b^2-a b \cos (c+d x)\right ) \, dx\\ &=\frac{1}{2} a \left (2 a^2-b^2\right ) x+\frac{2 b \left (2 a^2-b^2\right ) \sin (c+d x)}{3 d}+\frac{a b^2 \cos (c+d x) \sin (c+d x)}{6 d}-\frac{b (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.168031, size = 75, normalized size = 0.82 \[ -\frac{\left (9 b^3-12 a^2 b\right ) \sin (c+d x)-12 a^3 d x+3 a b^2 \sin (2 (c+d x))+6 a b^2 c+6 a b^2 d x+b^3 \sin (3 (c+d x))}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*a*b^2*c - 12*a^3*d*x + 6*a*b^2*d*x + (-12*a^2*b + 9*b^3)*Sin[c + d*x] + 3*a*b^2*Sin[2*(c + d*x)] + b^3*Sin
[3*(c + d*x)])/(12*d)

________________________________________________________________________________________

Maple [A]  time = 0.018, size = 75, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{{b}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}-a{b}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{2}b\sin \left ( dx+c \right ) +{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.994795, size = 99, normalized size = 1.08 \begin{align*} \frac{12 \,{\left (d x + c\right )} a^{3} - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{2} + 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b^{3} + 12 \, a^{2} b \sin \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.38727, size = 149, normalized size = 1.62 \begin{align*} \frac{3 \,{\left (2 \, a^{3} - a b^{2}\right )} d x -{\left (2 \, b^{3} \cos \left (d x + c\right )^{2} + 3 \, a b^{2} \cos \left (d x + c\right ) - 6 \, a^{2} b + 4 \, b^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*(2*a^3 - a*b^2)*d*x - (2*b^3*cos(d*x + c)^2 + 3*a*b^2*cos(d*x + c) - 6*a^2*b + 4*b^3)*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 0.962799, size = 131, normalized size = 1.42 \begin{align*} \begin{cases} a^{3} x + \frac{a^{2} b \sin{\left (c + d x \right )}}{d} - \frac{a b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} - \frac{a b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} - \frac{a b^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} - \frac{2 b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right ) \left (a^{2} - b^{2} \cos ^{2}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.38413, size = 100, normalized size = 1.09 \begin{align*} -\frac{b^{3} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac{a b^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{1}{2} \,{\left (2 \, a^{3} - a b^{2}\right )} x + \frac{{\left (4 \, a^{2} b - 3 \, b^{3}\right )} \sin \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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