3.441 \(\int (a+b \cos (c+d x))^4 \, dx\)

Optimal. Leaf size=137 \[ \frac{a b \left (19 a^2+16 b^2\right ) \sin (c+d x)}{6 d}+\frac{b^2 \left (26 a^2+9 b^2\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (24 a^2 b^2+8 a^4+3 b^4\right )+\frac{b \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{7 a b \sin (c+d x) (a+b \cos (c+d x))^2}{12 d} \]

[Out]

((8*a^4 + 24*a^2*b^2 + 3*b^4)*x)/8 + (a*b*(19*a^2 + 16*b^2)*Sin[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Cos[c
+ d*x]*Sin[c + d*x])/(24*d) + (7*a*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*(a + b*Cos[c + d*x])^3*S
in[c + d*x])/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.146885, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2656, 2753, 2734} \[ \frac{a b \left (19 a^2+16 b^2\right ) \sin (c+d x)}{6 d}+\frac{b^2 \left (26 a^2+9 b^2\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (24 a^2 b^2+8 a^4+3 b^4\right )+\frac{b \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{7 a b \sin (c+d x) (a+b \cos (c+d x))^2}{12 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^4,x]

[Out]

((8*a^4 + 24*a^2*b^2 + 3*b^4)*x)/8 + (a*b*(19*a^2 + 16*b^2)*Sin[c + d*x])/(6*d) + (b^2*(26*a^2 + 9*b^2)*Cos[c
+ d*x]*Sin[c + d*x])/(24*d) + (7*a*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (b*(a + b*Cos[c + d*x])^3*S
in[c + d*x])/(4*d)

Rule 2656

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[1/n, Int[(a + b*Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*
x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]

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))^4 \, dx &=\frac{b (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{4} \int (a+b \cos (c+d x))^2 \left (4 a^2+3 b^2+7 a b \cos (c+d x)\right ) \, dx\\ &=\frac{7 a b (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{b (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{12} \int (a+b \cos (c+d x)) \left (a \left (12 a^2+23 b^2\right )+b \left (26 a^2+9 b^2\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{1}{8} \left (8 a^4+24 a^2 b^2+3 b^4\right ) x+\frac{a b \left (19 a^2+16 b^2\right ) \sin (c+d x)}{6 d}+\frac{b^2 \left (26 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{7 a b (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{b (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.209031, size = 104, normalized size = 0.76 \[ \frac{12 \left (24 a^2 b^2+8 a^4+3 b^4\right ) (c+d x)+24 b^2 \left (6 a^2+b^2\right ) \sin (2 (c+d x))+96 a b \left (4 a^2+3 b^2\right ) \sin (c+d x)+32 a b^3 \sin (3 (c+d x))+3 b^4 \sin (4 (c+d x))}{96 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[c + d*x])^4,x]

[Out]

(12*(8*a^4 + 24*a^2*b^2 + 3*b^4)*(c + d*x) + 96*a*b*(4*a^2 + 3*b^2)*Sin[c + d*x] + 24*b^2*(6*a^2 + b^2)*Sin[2*
(c + d*x)] + 32*a*b^3*Sin[3*(c + d*x)] + 3*b^4*Sin[4*(c + d*x)])/(96*d)

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 116, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{4} \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{4\,a{b}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+6\,{a}^{2}{b}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,{a}^{3}b\sin \left ( dx+c \right ) +{a}^{4} \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))^4,x)

[Out]

1/d*(b^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4/3*a*b^3*(2+cos(d*x+c)^2)*sin(d*x+c)+6*
a^2*b^2*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+4*a^3*b*sin(d*x+c)+a^4*(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 0.993018, size = 150, normalized size = 1.09 \begin{align*} a^{4} x + \frac{3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b^{2}}{2 \, d} - \frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a b^{3}}{3 \, d} + \frac{{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{4}}{32 \, d} + \frac{4 \, a^{3} b \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

a^4*x + 3/2*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^2*b^2/d - 4/3*(sin(d*x + c)^3 - 3*sin(d*x + c))*a*b^3/d + 1/32*
(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*b^4/d + 4*a^3*b*sin(d*x + c)/d

________________________________________________________________________________________

Fricas [A]  time = 1.96055, size = 224, normalized size = 1.64 \begin{align*} \frac{3 \,{\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x +{\left (6 \, b^{4} \cos \left (d x + c\right )^{3} + 32 \, a b^{3} \cos \left (d x + c\right )^{2} + 96 \, a^{3} b + 64 \, a b^{3} + 9 \,{\left (8 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

1/24*(3*(8*a^4 + 24*a^2*b^2 + 3*b^4)*d*x + (6*b^4*cos(d*x + c)^3 + 32*a*b^3*cos(d*x + c)^2 + 96*a^3*b + 64*a*b
^3 + 9*(8*a^2*b^2 + b^4)*cos(d*x + c))*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 1.28512, size = 240, normalized size = 1.75 \begin{align*} \begin{cases} a^{4} x + \frac{4 a^{3} b \sin{\left (c + d x \right )}}{d} + 3 a^{2} b^{2} x \sin ^{2}{\left (c + d x \right )} + 3 a^{2} b^{2} x \cos ^{2}{\left (c + d x \right )} + \frac{3 a^{2} b^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{8 a b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{4 a b^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 b^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 b^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 b^{4} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{5 b^{4} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right )^{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**4,x)

[Out]

Piecewise((a**4*x + 4*a**3*b*sin(c + d*x)/d + 3*a**2*b**2*x*sin(c + d*x)**2 + 3*a**2*b**2*x*cos(c + d*x)**2 +
3*a**2*b**2*sin(c + d*x)*cos(c + d*x)/d + 8*a*b**3*sin(c + d*x)**3/(3*d) + 4*a*b**3*sin(c + d*x)*cos(c + d*x)*
*2/d + 3*b**4*x*sin(c + d*x)**4/8 + 3*b**4*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*b**4*x*cos(c + d*x)**4/8 +
3*b**4*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 5*b**4*sin(c + d*x)*cos(c + d*x)**3/(8*d), Ne(d, 0)), (x*(a + b*co
s(c))**4, True))

________________________________________________________________________________________

Giac [A]  time = 1.32937, size = 144, normalized size = 1.05 \begin{align*} \frac{b^{4} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{a b^{3} \sin \left (3 \, d x + 3 \, c\right )}{3 \, d} + \frac{1}{8} \,{\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4}\right )} x + \frac{{\left (6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/32*b^4*sin(4*d*x + 4*c)/d + 1/3*a*b^3*sin(3*d*x + 3*c)/d + 1/8*(8*a^4 + 24*a^2*b^2 + 3*b^4)*x + 1/4*(6*a^2*b
^2 + b^4)*sin(2*d*x + 2*c)/d + (4*a^3*b + 3*a*b^3)*sin(d*x + c)/d