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]

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

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Rubi [A]  time = 0.0313192, antiderivative size = 26, normalized size of antiderivative = 1., 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 = 0.184874, 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)

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Maple [B]  time = 0.048, size = 150, normalized size = 5.8 \begin{align*}{\frac{1}{5\,d} \left ({a}^{4}B \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \sin \left ( dx+c \right ) +16\,{a}^{4}B \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{14\,{a}^{4}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}-4\,{a}^{4}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) -11\,{a}^{4}B\sin \left ( dx+c \right ) -4\,{a}^{4}B \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [B]  time = 1.0555, size = 194, normalized size = 7.46 \begin{align*} \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} \end{align*}

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

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Fricas [B]  time = 1.48511, size = 167, normalized size = 6.42 \begin{align*} \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} \end{align*}

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

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Sympy [A]  time = 3.54753, size = 333, normalized size = 12.81 \begin{align*} \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{\left (c \right )} - \frac{4 B}{5}\right ) \left (a \cos{\left (c \right )} + a\right )^{4} & \text{otherwise} \end{cases} \end{align*}

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))

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Giac [B]  time = 1.82523, size = 119, normalized size = 4.58 \begin{align*} \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} \end{align*}

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