3.74 \(\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=127 \[ -\frac{43 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}+\frac{11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}+\frac{x}{a^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^2(c+d x)}{7 a d (a \cos (c+d x)+a)^3} \]

[Out]

x/a^4 + (11*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])^2) - (43*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])) -
(Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^2*Sin[c + d*x])/(7*a*d*(a + a*Cos
[c + d*x])^3)

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Rubi [A]  time = 0.282547, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2765, 2977, 2968, 3019, 2735, 2648} \[ -\frac{43 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}+\frac{11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}+\frac{x}{a^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^2(c+d x)}{7 a d (a \cos (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a^4 + (11*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])^2) - (43*Sin[c + d*x])/(21*a^4*d*(1 + Cos[c + d*x])) -
(Cos[c + d*x]^3*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - (2*Cos[c + d*x]^2*Sin[c + d*x])/(7*a*d*(a + a*Cos
[c + d*x])^3)

Rule 2765

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

Rule 2977

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

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3019

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

Rule 2735

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

Rule 2648

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

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{\int \frac{\cos ^2(c+d x) (3 a-7 a \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac{\int \frac{\cos (c+d x) \left (20 a^2-35 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac{\int \frac{20 a^2 \cos (c+d x)-35 a^2 \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac{11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac{\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{-110 a^3+105 a^3 \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac{x}{a^4}+\frac{11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac{\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac{43 \int \frac{1}{a+a \cos (c+d x)} \, dx}{21 a^3}\\ &=\frac{x}{a^4}+\frac{11 \sin (c+d x)}{21 a^4 d (1+\cos (c+d x))^2}-\frac{\cos ^3(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 \cos ^2(c+d x) \sin (c+d x)}{7 a d (a+a \cos (c+d x))^3}-\frac{43 \sin (c+d x)}{21 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.333164, size = 224, normalized size = 1.76 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right ) \left (1652 \sin \left (c+\frac{d x}{2}\right )-1428 \sin \left (c+\frac{3 d x}{2}\right )+756 \sin \left (2 c+\frac{3 d x}{2}\right )-560 \sin \left (2 c+\frac{5 d x}{2}\right )+168 \sin \left (3 c+\frac{5 d x}{2}\right )-104 \sin \left (3 c+\frac{7 d x}{2}\right )+735 d x \cos \left (c+\frac{d x}{2}\right )+441 d x \cos \left (c+\frac{3 d x}{2}\right )+441 d x \cos \left (2 c+\frac{3 d x}{2}\right )+147 d x \cos \left (2 c+\frac{5 d x}{2}\right )+147 d x \cos \left (3 c+\frac{5 d x}{2}\right )+21 d x \cos \left (3 c+\frac{7 d x}{2}\right )+21 d x \cos \left (4 c+\frac{7 d x}{2}\right )-1988 \sin \left (\frac{d x}{2}\right )+735 d x \cos \left (\frac{d x}{2}\right )\right )}{2688 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^7*(735*d*x*Cos[(d*x)/2] + 735*d*x*Cos[c + (d*x)/2] + 441*d*x*Cos[c + (3*d*x)/2] + 4
41*d*x*Cos[2*c + (3*d*x)/2] + 147*d*x*Cos[2*c + (5*d*x)/2] + 147*d*x*Cos[3*c + (5*d*x)/2] + 21*d*x*Cos[3*c + (
7*d*x)/2] + 21*d*x*Cos[4*c + (7*d*x)/2] - 1988*Sin[(d*x)/2] + 1652*Sin[c + (d*x)/2] - 1428*Sin[c + (3*d*x)/2]
+ 756*Sin[2*c + (3*d*x)/2] - 560*Sin[2*c + (5*d*x)/2] + 168*Sin[3*c + (5*d*x)/2] - 104*Sin[3*c + (7*d*x)/2]))/
(2688*a^4*d)

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Maple [A]  time = 0.046, size = 94, normalized size = 0.7 \begin{align*}{\frac{1}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{11}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{15}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/56/d/a^4*tan(1/2*d*x+1/2*c)^7-1/8/d/a^4*tan(1/2*d*x+1/2*c)^5+11/24/d/a^4*tan(1/2*d*x+1/2*c)^3-15/8/d/a^4*tan
(1/2*d*x+1/2*c)+2/d/a^4*arctan(tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.71711, size = 151, normalized size = 1.19 \begin{align*} -\frac{\frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{336 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}}{168 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/168*((315*sin(d*x + c)/(cos(d*x + c) + 1) - 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos
(d*x + c) + 1)^5 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 336*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^
4)/d

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Fricas [A]  time = 1.58009, size = 397, normalized size = 3.13 \begin{align*} \frac{21 \, d x \cos \left (d x + c\right )^{4} + 84 \, d x \cos \left (d x + c\right )^{3} + 126 \, d x \cos \left (d x + c\right )^{2} + 84 \, d x \cos \left (d x + c\right ) + 21 \, d x -{\left (52 \, \cos \left (d x + c\right )^{3} + 124 \, \cos \left (d x + c\right )^{2} + 107 \, \cos \left (d x + c\right ) + 32\right )} \sin \left (d x + c\right )}{21 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(21*d*x*cos(d*x + c)^4 + 84*d*x*cos(d*x + c)^3 + 126*d*x*cos(d*x + c)^2 + 84*d*x*cos(d*x + c) + 21*d*x -
(52*cos(d*x + c)^3 + 124*cos(d*x + c)^2 + 107*cos(d*x + c) + 32)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d
*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.46203, size = 112, normalized size = 0.88 \begin{align*} \frac{\frac{168 \,{\left (d x + c\right )}}{a^{4}} + \frac{3 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 21 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 77 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 315 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{168 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168*(168*(d*x + c)/a^4 + (3*a^24*tan(1/2*d*x + 1/2*c)^7 - 21*a^24*tan(1/2*d*x + 1/2*c)^5 + 77*a^24*tan(1/2*d
*x + 1/2*c)^3 - 315*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d