3.89 \(\int \frac{1}{(a+a \cos (c+d x))^5} \, dx\)

Optimal. Leaf size=143 \[ \frac{8 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac{8 \sin (c+d x)}{315 a d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{4 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}+\frac{4 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}+\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]

[Out]

Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) + (4*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + (4*Sin[c + d*x]
)/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (8*Sin[c + d*x])/(315*a*d*(a^2 + a^2*Cos[c + d*x])^2) + (8*Sin[c + d*x]
)/(315*d*(a^5 + a^5*Cos[c + d*x]))

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Rubi [A]  time = 0.0913571, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2650, 2648} \[ \frac{8 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac{8 \sin (c+d x)}{315 a d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac{4 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}+\frac{4 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}+\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^(-5),x]

[Out]

Sin[c + d*x]/(9*d*(a + a*Cos[c + d*x])^5) + (4*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4) + (4*Sin[c + d*x]
)/(105*a^2*d*(a + a*Cos[c + d*x])^3) + (8*Sin[c + d*x])/(315*a*d*(a^2 + a^2*Cos[c + d*x])^2) + (8*Sin[c + d*x]
)/(315*d*(a^5 + a^5*Cos[c + d*x]))

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{1}{(a+a \cos (c+d x))^5} \, dx &=\frac{\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac{4 \int \frac{1}{(a+a \cos (c+d x))^4} \, dx}{9 a}\\ &=\frac{\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac{4 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac{4 \int \frac{1}{(a+a \cos (c+d x))^3} \, dx}{21 a^2}\\ &=\frac{\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac{4 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac{4 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac{8 \int \frac{1}{(a+a \cos (c+d x))^2} \, dx}{105 a^3}\\ &=\frac{\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac{4 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac{4 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac{8 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac{8 \int \frac{1}{a+a \cos (c+d x)} \, dx}{315 a^4}\\ &=\frac{\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac{4 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac{4 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac{8 \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac{8 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.15382, size = 89, normalized size = 0.62 \[ \frac{\left (126 \sin \left (\frac{1}{2} (c+d x)\right )+84 \sin \left (\frac{3}{2} (c+d x)\right )+36 \sin \left (\frac{5}{2} (c+d x)\right )+9 \sin \left (\frac{7}{2} (c+d x)\right )+\sin \left (\frac{9}{2} (c+d x)\right )\right ) \cos \left (\frac{1}{2} (c+d x)\right )}{315 a^5 d (\cos (c+d x)+1)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(c + d*x)/2]*(126*Sin[(c + d*x)/2] + 84*Sin[(3*(c + d*x))/2] + 36*Sin[(5*(c + d*x))/2] + 9*Sin[(7*(c + d*
x))/2] + Sin[(9*(c + d*x))/2]))/(315*a^5*d*(1 + Cos[c + d*x])^5)

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Maple [A]  time = 0.034, size = 71, normalized size = 0.5 \begin{align*}{\frac{1}{16\,d{a}^{5}} \left ({\frac{1}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}+{\frac{4}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{6}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{4}{3} \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(1/(a+cos(d*x+c)*a)^5,x)

[Out]

1/16/d/a^5*(1/9*tan(1/2*d*x+1/2*c)^9+4/7*tan(1/2*d*x+1/2*c)^7+6/5*tan(1/2*d*x+1/2*c)^5+4/3*tan(1/2*d*x+1/2*c)^
3+tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.13397, size = 144, normalized size = 1.01 \begin{align*} \frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{420 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{180 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{5040 \, a^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c))^5,x, algorithm="maxima")

[Out]

1/5040*(315*sin(d*x + c)/(cos(d*x + c) + 1) + 420*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 378*sin(d*x + c)^5/(co
s(d*x + c) + 1)^5 + 180*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/(a^5*d)

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Fricas [A]  time = 1.54747, size = 316, normalized size = 2.21 \begin{align*} \frac{{\left (8 \, \cos \left (d x + c\right )^{4} + 40 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} + 100 \, \cos \left (d x + c\right ) + 83\right )} \sin \left (d x + c\right )}{315 \,{\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

1/315*(8*cos(d*x + c)^4 + 40*cos(d*x + c)^3 + 84*cos(d*x + c)^2 + 100*cos(d*x + c) + 83)*sin(d*x + c)/(a^5*d*c
os(d*x + c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x + c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x +
c) + a^5*d)

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Sympy [A]  time = 15.2683, size = 102, normalized size = 0.71 \begin{align*} \begin{cases} \frac{\tan ^{9}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{144 a^{5} d} + \frac{\tan ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{28 a^{5} d} + \frac{3 \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{40 a^{5} d} + \frac{\tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{12 a^{5} d} + \frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{16 a^{5} d} & \text{for}\: d \neq 0 \\\frac{x}{\left (a \cos{\left (c \right )} + a\right )^{5}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c))**5,x)

[Out]

Piecewise((tan(c/2 + d*x/2)**9/(144*a**5*d) + tan(c/2 + d*x/2)**7/(28*a**5*d) + 3*tan(c/2 + d*x/2)**5/(40*a**5
*d) + tan(c/2 + d*x/2)**3/(12*a**5*d) + tan(c/2 + d*x/2)/(16*a**5*d), Ne(d, 0)), (x/(a*cos(c) + a)**5, True))

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Giac [A]  time = 1.43179, size = 97, normalized size = 0.68 \begin{align*} \frac{35 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 180 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 378 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 420 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{5040 \, a^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c))^5,x, algorithm="giac")

[Out]

1/5040*(35*tan(1/2*d*x + 1/2*c)^9 + 180*tan(1/2*d*x + 1/2*c)^7 + 378*tan(1/2*d*x + 1/2*c)^5 + 420*tan(1/2*d*x
+ 1/2*c)^3 + 315*tan(1/2*d*x + 1/2*c))/(a^5*d)