3.781 \(\int (a+a \cos (c+d x))^4 (-\frac {4 B}{5}+B \cos (c+d x)) \, dx\)

Optimal. Leaf size=26 \[ \frac {B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]

[Out]

1/5*B*(a+a*cos(d*x+c))^4*sin(d*x+c)/d

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2749} \[ \frac {B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^4*((-4*B)/5 + B*Cos[c + d*x]),x]

[Out]

(B*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)

Rule 2749

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

Rubi steps

\begin {align*} \int (a+a \cos (c+d x))^4 \left (-\frac {4 B}{5}+B \cos (c+d x)\right ) \, dx &=\frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 31, normalized size = 1.19 \[ \frac {a^4 B \sin ^9(c+d x) \csc ^8\left (\frac {1}{2} (c+d x)\right )}{80 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[c + d*x])^4*((-4*B)/5 + B*Cos[c + d*x]),x]

[Out]

(a^4*B*Csc[(c + d*x)/2]^8*Sin[c + d*x]^9)/(80*d)

________________________________________________________________________________________

fricas [B]  time = 0.82, size = 70, normalized size = 2.69 \[ \frac {{\left (B a^{4} \cos \left (d x + c\right )^{4} + 4 \, B a^{4} \cos \left (d x + c\right )^{3} + 6 \, B a^{4} \cos \left (d x + c\right )^{2} + 4 \, B a^{4} \cos \left (d x + c\right ) + B a^{4}\right )} \sin \left (d x + c\right )}{5 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4*(-4/5*B+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/5*(B*a^4*cos(d*x + c)^4 + 4*B*a^4*cos(d*x + c)^3 + 6*B*a^4*cos(d*x + c)^2 + 4*B*a^4*cos(d*x + c) + B*a^4)*si
n(d*x + c)/d

________________________________________________________________________________________

giac [B]  time = 0.40, size = 88, normalized size = 3.38 \[ \frac {B a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {B a^{4} \sin \left (4 \, d x + 4 \, c\right )}{10 \, d} + \frac {27 \, B a^{4} \sin \left (3 \, d x + 3 \, c\right )}{80 \, d} + \frac {3 \, B a^{4} \sin \left (2 \, d x + 2 \, c\right )}{5 \, d} + \frac {21 \, B a^{4} \sin \left (d x + c\right )}{40 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4*(-4/5*B+B*cos(d*x+c)),x, algorithm="giac")

[Out]

1/80*B*a^4*sin(5*d*x + 5*c)/d + 1/10*B*a^4*sin(4*d*x + 4*c)/d + 27/80*B*a^4*sin(3*d*x + 3*c)/d + 3/5*B*a^4*sin
(2*d*x + 2*c)/d + 21/40*B*a^4*sin(d*x + c)/d

________________________________________________________________________________________

maple [B]  time = 0.06, size = 150, normalized size = 5.77 \[ \frac {a^{4} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )+16 a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {14 a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}-4 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-11 a^{4} B \sin \left (d x +c \right )-4 a^{4} B \left (d x +c \right )}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cos(d*x+c))^4*(-4/5*B+B*cos(d*x+c)),x)

[Out]

1/5/d*(a^4*B*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+16*a^4*B*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*
x+c)+3/8*d*x+3/8*c)+14/3*a^4*B*(2+cos(d*x+c)^2)*sin(d*x+c)-4*a^4*B*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)-1
1*a^4*B*sin(d*x+c)-4*a^4*B*(d*x+c))

________________________________________________________________________________________

maxima [B]  time = 0.34, size = 144, normalized size = 5.54 \[ \frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 28 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 6 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 24 \, {\left (d x + c\right )} B a^{4} - 66 \, B a^{4} \sin \left (d x + c\right )}{30 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4*(-4/5*B+B*cos(d*x+c)),x, algorithm="maxima")

[Out]

1/30*(2*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 - 28*(sin(d*x + c)^3 - 3*sin(d*x + c))*
B*a^4 + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 6*(2*d*x + 2*c + sin(2*d*x + 2*c))*B
*a^4 - 24*(d*x + c)*B*a^4 - 66*B*a^4*sin(d*x + c))/d

________________________________________________________________________________________

mupad [B]  time = 0.68, size = 29, normalized size = 1.12 \[ \frac {32\,B\,a^4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((4*B)/5 - B*cos(c + d*x))*(a + a*cos(c + d*x))^4,x)

[Out]

(32*B*a^4*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2))/(5*d)

________________________________________________________________________________________

sympy [A]  time = 2.46, size = 333, normalized size = 12.81 \[ \begin {cases} \frac {6 B a^{4} x \sin ^{4}{\left (c + d x \right )}}{5} + \frac {12 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5} - \frac {2 B a^{4} x \sin ^{2}{\left (c + d x \right )}}{5} + \frac {6 B a^{4} x \cos ^{4}{\left (c + d x \right )}}{5} - \frac {2 B a^{4} x \cos ^{2}{\left (c + d x \right )}}{5} - \frac {4 B a^{4} x}{5} + \frac {8 B a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {6 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{5 d} + \frac {28 B a^{4} \sin ^{3}{\left (c + d x \right )}}{15 d} + \frac {B a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {2 B a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} + \frac {14 B a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} - \frac {2 B a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{5 d} - \frac {11 B a^{4} \sin {\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (B \cos {\relax (c )} - \frac {4 B}{5}\right ) \left (a \cos {\relax (c )} + a\right )^{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))**4*(-4/5*B+B*cos(d*x+c)),x)

[Out]

Piecewise((6*B*a**4*x*sin(c + d*x)**4/5 + 12*B*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/5 - 2*B*a**4*x*sin(c + d
*x)**2/5 + 6*B*a**4*x*cos(c + d*x)**4/5 - 2*B*a**4*x*cos(c + d*x)**2/5 - 4*B*a**4*x/5 + 8*B*a**4*sin(c + d*x)*
*5/(15*d) + 4*B*a**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 6*B*a**4*sin(c + d*x)**3*cos(c + d*x)/(5*d) + 28*
B*a**4*sin(c + d*x)**3/(15*d) + B*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 2*B*a**4*sin(c + d*x)*cos(c + d*x)**3/
d + 14*B*a**4*sin(c + d*x)*cos(c + d*x)**2/(5*d) - 2*B*a**4*sin(c + d*x)*cos(c + d*x)/(5*d) - 11*B*a**4*sin(c
+ d*x)/(5*d), Ne(d, 0)), (x*(B*cos(c) - 4*B/5)*(a*cos(c) + a)**4, True))

________________________________________________________________________________________