\(\int \frac {1}{(a+a \cos (c+d x))^5} \, dx\) [89]

Optimal result

Integrand size = 12, antiderivative size = 143 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {\sin (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {4 \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {4 \sin (c+d x)}{105 a^2 d (a+a \cos (c+d x))^3}+\frac {8 \sin (c+d x)}{315 a d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {8 \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]

[Out]

1/9*sin(d*x+c)/d/(a+a*cos(d*x+c))^5+4/63*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^4+4/105*sin(d*x+c)/a^2/d/(a+a*cos(d*x
+c))^3+8/315*sin(d*x+c)/a/d/(a^2+a^2*cos(d*x+c))^2+8/315*sin(d*x+c)/d/(a^5+a^5*cos(d*x+c))

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2729, 2727} \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {8 \sin (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac {8 \sin (c+d x)}{315 a d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {4 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}+\frac {4 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}+\frac {\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5} \]

[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 2727

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]

Rule 2729

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (126 \sin \left (\frac {1}{2} (c+d x)\right )+84 \sin \left (\frac {3}{2} (c+d x)\right )+36 \sin \left (\frac {5}{2} (c+d x)\right )+9 \sin \left (\frac {7}{2} (c+d x)\right )+\sin \left (\frac {9}{2} (c+d x)\right )\right )}{315 a^5 d (1+\cos (c+d x))^5} \]

[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)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.48

method	result	size
risch	\(\frac {16 i \left (126 \,{\mathrm e}^{4 i \left (d x +c \right )}+84 \,{\mathrm e}^{3 i \left (d x +c \right )}+36 \,{\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\)	\(69\)
derivativedivides	\(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\)	\(71\)
default	\(\frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}}\)	\(71\)
parallelrisch	\(\frac {35 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+180 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+378 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+420 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5040 a^{5} d}\)	\(73\)
norman	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d a}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d a}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{28 d a}+\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 d a}}{a^{4}}\)	\(99\)

[In]

int(1/(a+cos(d*x+c)*a)^5,x,method=_RETURNVERBOSE)

[Out]

16/315*I*(126*exp(4*I*(d*x+c))+84*exp(3*I*(d*x+c))+36*exp(2*I*(d*x+c))+9*exp(I*(d*x+c))+1)/d/a^5/(exp(I*(d*x+c
))+1)^9

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {{\left (8 \, \cos \left (d x + c\right )^{4} + 40 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} + 100 \, \cos \left (d x + c\right ) + 83\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 2.76 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {\tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{144 a^{5} d} + \frac {\tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{28 a^{5} d} + \frac {3 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{5} d} + \frac {\tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{12 a^{5} d} + \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {420 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {180 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{5040 \, a^{5} d} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.50 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 180 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{5040 \, a^{5} d} \]

[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)

Mupad [B] (verification not implemented)

Time = 14.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(a+a \cos (c+d x))^5} \, dx=\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+378\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+180\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

[In]

int(1/(a + a*cos(c + d*x))^5,x)

[Out]

(sin(c/2 + (d*x)/2)*(315*cos(c/2 + (d*x)/2)^8 + 35*sin(c/2 + (d*x)/2)^8 + 180*cos(c/2 + (d*x)/2)^2*sin(c/2 + (
d*x)/2)^6 + 378*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^4 + 420*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^2))/(5
040*a^5*d*cos(c/2 + (d*x)/2)^9)