3.94 \(\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx\)

Optimal. Leaf size=176 \[ \frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)}+\frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^2}-\frac{241 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^3}+\frac{9 \sin (c+d x)}{77 a^2 d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{4 \sin (c+d x) \cos ^2(c+d x)}{33 a d (a \cos (c+d x)+a)^5} \]

[Out]

(-241*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])^3) + (61*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])^2) +
(61*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^3*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6
) - (4*Cos[c + d*x]^2*Sin[c + d*x])/(33*a*d*(a + a*Cos[c + d*x])^5) + (9*Sin[c + d*x])/(77*a^2*d*(a + a*Cos[c
+ d*x])^4)

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Rubi [A]  time = 0.318357, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2765, 2977, 2968, 3019, 2750, 2650, 2648} \[ \frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)}+\frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^2}-\frac{241 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^3}+\frac{9 \sin (c+d x)}{77 a^2 d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{4 \sin (c+d x) \cos ^2(c+d x)}{33 a d (a \cos (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^6,x]

[Out]

(-241*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])^3) + (61*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])^2) +
(61*Sin[c + d*x])/(1155*a^6*d*(1 + Cos[c + d*x])) - (Cos[c + d*x]^3*Sin[c + d*x])/(11*d*(a + a*Cos[c + d*x])^6
) - (4*Cos[c + d*x]^2*Sin[c + d*x])/(33*a*d*(a + a*Cos[c + d*x])^5) + (9*Sin[c + d*x])/(77*a^2*d*(a + a*Cos[c
+ d*x])^4)

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 2750

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

Rule 2650

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

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))^6} \, dx &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{\int \frac{\cos ^2(c+d x) (3 a-9 a \cos (c+d x))}{(a+a \cos (c+d x))^5} \, dx}{11 a^2}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}-\frac{\int \frac{\cos (c+d x) \left (24 a^2-57 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx}{99 a^4}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}-\frac{\int \frac{24 a^2 \cos (c+d x)-57 a^2 \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx}{99 a^4}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac{9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4}+\frac{\int \frac{-324 a^3+399 a^3 \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx}{693 a^6}\\ &=-\frac{241 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^3}-\frac{\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac{9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4}+\frac{61 \int \frac{1}{(a+a \cos (c+d x))^2} \, dx}{385 a^4}\\ &=-\frac{241 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^3}-\frac{\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac{9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4}+\frac{61 \sin (c+d x)}{1155 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac{61 \int \frac{1}{a+a \cos (c+d x)} \, dx}{1155 a^5}\\ &=-\frac{241 \sin (c+d x)}{1155 a^6 d (1+\cos (c+d x))^3}-\frac{\cos ^3(c+d x) \sin (c+d x)}{11 d (a+a \cos (c+d x))^6}-\frac{4 \cos ^2(c+d x) \sin (c+d x)}{33 a d (a+a \cos (c+d x))^5}+\frac{9 \sin (c+d x)}{77 a^2 d (a+a \cos (c+d x))^4}+\frac{61 \sin (c+d x)}{1155 d \left (a^3+a^3 \cos (c+d x)\right )^2}+\frac{61 \sin (c+d x)}{1155 d \left (a^6+a^6 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.297256, size = 151, normalized size = 0.86 \[ \frac{\sec \left (\frac{c}{2}\right ) \left (-12936 \sin \left (c+\frac{d x}{2}\right )+10890 \sin \left (c+\frac{3 d x}{2}\right )-9240 \sin \left (2 c+\frac{3 d x}{2}\right )+6600 \sin \left (2 c+\frac{5 d x}{2}\right )-3465 \sin \left (3 c+\frac{5 d x}{2}\right )+2200 \sin \left (3 c+\frac{7 d x}{2}\right )-1155 \sin \left (4 c+\frac{7 d x}{2}\right )+671 \sin \left (4 c+\frac{9 d x}{2}\right )+61 \sin \left (5 c+\frac{11 d x}{2}\right )+15246 \sin \left (\frac{d x}{2}\right )\right ) \sec ^{11}\left (\frac{1}{2} (c+d x)\right )}{1182720 a^6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^6,x]

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^11*(15246*Sin[(d*x)/2] - 12936*Sin[c + (d*x)/2] + 10890*Sin[c + (3*d*x)/2] - 9240*S
in[2*c + (3*d*x)/2] + 6600*Sin[2*c + (5*d*x)/2] - 3465*Sin[3*c + (5*d*x)/2] + 2200*Sin[3*c + (7*d*x)/2] - 1155
*Sin[4*c + (7*d*x)/2] + 671*Sin[4*c + (9*d*x)/2] + 61*Sin[5*c + (11*d*x)/2]))/(1182720*a^6*d)

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Maple [A]  time = 0.04, size = 84, normalized size = 0.5 \begin{align*}{\frac{1}{32\,d{a}^{6}} \left ({\frac{1}{11} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{11}}-{\frac{1}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{2}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{2}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}- \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}+\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \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)^6,x)

[Out]

1/32/d/a^6*(1/11*tan(1/2*d*x+1/2*c)^11-1/3*tan(1/2*d*x+1/2*c)^9+2/7*tan(1/2*d*x+1/2*c)^7+2/5*tan(1/2*d*x+1/2*c
)^5-tan(1/2*d*x+1/2*c)^3+tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.14884, size = 171, normalized size = 0.97 \begin{align*} \frac{\frac{1155 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1155 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{462 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{330 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{385 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{105 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{36960 \, a^{6} 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))^6,x, algorithm="maxima")

[Out]

1/36960*(1155*sin(d*x + c)/(cos(d*x + c) + 1) - 1155*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 462*sin(d*x + c)^5/
(cos(d*x + c) + 1)^5 + 330*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 385*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 105
*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/(a^6*d)

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Fricas [A]  time = 1.59349, size = 383, normalized size = 2.18 \begin{align*} \frac{{\left (61 \, \cos \left (d x + c\right )^{5} + 366 \, \cos \left (d x + c\right )^{4} + 368 \, \cos \left (d x + c\right )^{3} + 248 \, \cos \left (d x + c\right )^{2} + 96 \, \cos \left (d x + c\right ) + 16\right )} \sin \left (d x + c\right )}{1155 \,{\left (a^{6} d \cos \left (d x + c\right )^{6} + 6 \, a^{6} d \cos \left (d x + c\right )^{5} + 15 \, a^{6} d \cos \left (d x + c\right )^{4} + 20 \, a^{6} d \cos \left (d x + c\right )^{3} + 15 \, a^{6} d \cos \left (d x + c\right )^{2} + 6 \, a^{6} d \cos \left (d x + c\right ) + a^{6} 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))^6,x, algorithm="fricas")

[Out]

1/1155*(61*cos(d*x + c)^5 + 366*cos(d*x + c)^4 + 368*cos(d*x + c)^3 + 248*cos(d*x + c)^2 + 96*cos(d*x + c) + 1
6)*sin(d*x + c)/(a^6*d*cos(d*x + c)^6 + 6*a^6*d*cos(d*x + c)^5 + 15*a^6*d*cos(d*x + c)^4 + 20*a^6*d*cos(d*x +
c)^3 + 15*a^6*d*cos(d*x + c)^2 + 6*a^6*d*cos(d*x + c) + a^6*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))**6,x)

[Out]

Timed out

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Giac [A]  time = 1.43857, size = 115, normalized size = 0.65 \begin{align*} \frac{105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 385 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 330 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 462 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1155 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1155 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{36960 \, a^{6} 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))^6,x, algorithm="giac")

[Out]

1/36960*(105*tan(1/2*d*x + 1/2*c)^11 - 385*tan(1/2*d*x + 1/2*c)^9 + 330*tan(1/2*d*x + 1/2*c)^7 + 462*tan(1/2*d
*x + 1/2*c)^5 - 1155*tan(1/2*d*x + 1/2*c)^3 + 1155*tan(1/2*d*x + 1/2*c))/(a^6*d)