\(\int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx\) [63]

Optimal result

Integrand size = 21, antiderivative size = 153 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {13 x}{2 a^3}-\frac {152 \sin (c+d x)}{15 a^3 d}+\frac {13 \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {11 \cos ^3(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {76 \cos ^2(c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )} \]

[Out]

13/2*x/a^3-152/15*sin(d*x+c)/a^3/d+13/2*cos(d*x+c)*sin(d*x+c)/a^3/d-1/5*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x
+c))^3-11/15*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^2-76/15*cos(d*x+c)^2*sin(d*x+c)/d/(a^3+a^3*cos(d*x+c
))

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2844, 3056, 2813} \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {152 \sin (c+d x)}{15 a^3 d}-\frac {76 \sin (c+d x) \cos ^2(c+d x)}{15 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {13 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac {13 x}{2 a^3}-\frac {\sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac {11 \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2} \]

[In]

Int[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^3,x]

[Out]

(13*x)/(2*a^3) - (152*Sin[c + d*x])/(15*a^3*d) + (13*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (Cos[c + d*x]^4*Si
n[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (11*Cos[c + d*x]^3*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) -
(76*Cos[c + d*x]^2*Sin[c + d*x])/(15*d*(a^3 + a^3*Cos[c + d*x]))

Rule 2813

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

Rule 2844

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 3056

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) (4 a-7 a \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {11 \cos ^3(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (33 a^2-43 a^2 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4} \\ & = -\frac {\cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {11 \cos ^3(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {76 \cos ^2(c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {\int \cos (c+d x) \left (152 a^3-195 a^3 \cos (c+d x)\right ) \, dx}{15 a^6} \\ & = \frac {13 x}{2 a^3}-\frac {152 \sin (c+d x)}{15 a^3 d}+\frac {13 \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {11 \cos ^3(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {76 \cos ^2(c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \cos (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \csc ^6(c+d x) \sin ^7\left (\frac {1}{2} (c+d x)\right ) \left (12480 \arcsin (\cos (c+d x)) \cos ^6\left (\frac {1}{2} (c+d x)\right )+(4303+6006 \cos (c+d x)+1856 \cos (2 (c+d x))+90 \cos (3 (c+d x))-15 \cos (4 (c+d x))) \sqrt {\sin ^2(c+d x)}\right )}{15 a^3 d \sqrt {\sin ^2(c+d x)}} \]

[In]

Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^3,x]

[Out]

-1/15*(Cos[(c + d*x)/2]*Csc[c + d*x]^6*Sin[(c + d*x)/2]^7*(12480*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^6 + (43
03 + 6006*Cos[c + d*x] + 1856*Cos[2*(c + d*x)] + 90*Cos[3*(c + d*x)] - 15*Cos[4*(c + d*x)])*Sqrt[Sin[c + d*x]^
2]))/(a^3*d*Sqrt[Sin[c + d*x]^2])

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.50

method	result	size
parallelrisch	\(\frac {-\frac {1001 \left (\cos \left (d x +c \right )+\frac {928 \cos \left (2 d x +2 c \right )}{3003}+\frac {15 \cos \left (3 d x +3 c \right )}{1001}-\frac {5 \cos \left (4 d x +4 c \right )}{2002}+\frac {331}{462}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160}+\frac {13 d x}{2}}{a^{3} d}\)	\(77\)
derivativedivides	\(\frac {-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-28 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+52 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\)	\(101\)
default	\(\frac {-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-28 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+52 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\)	\(101\)
risch	\(\frac {13 x}{2 a^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {2 i \left (150 \,{\mathrm e}^{4 i \left (d x +c \right )}+525 \,{\mathrm e}^{3 i \left (d x +c \right )}+745 \,{\mathrm e}^{2 i \left (d x +c \right )}+485 \,{\mathrm e}^{i \left (d x +c \right )}+127\right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\)	\(148\)
norman	\(\frac {\frac {13 x}{2 a}-\frac {51 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a}-\frac {721 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {6613 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d a}-\frac {1165 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {475 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {59 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}+\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d a}-\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{20 d a}+\frac {65 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {65 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {65 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {65 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {13 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{2}}\)	\(262\)

[In]

int(cos(d*x+c)^5/(a+cos(d*x+c)*a)^3,x,method=_RETURNVERBOSE)

[Out]

13/160*(-77*(cos(d*x+c)+928/3003*cos(2*d*x+2*c)+15/1001*cos(3*d*x+3*c)-5/2002*cos(4*d*x+4*c)+331/462)*tan(1/2*
d*x+1/2*c)*sec(1/2*d*x+1/2*c)^4+80*d*x)/a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {195 \, d x \cos \left (d x + c\right )^{3} + 585 \, d x \cos \left (d x + c\right )^{2} + 585 \, d x \cos \left (d x + c\right ) + 195 \, d x + {\left (15 \, \cos \left (d x + c\right )^{4} - 45 \, \cos \left (d x + c\right )^{3} - 479 \, \cos \left (d x + c\right )^{2} - 717 \, \cos \left (d x + c\right ) - 304\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

1/30*(195*d*x*cos(d*x + c)^3 + 585*d*x*cos(d*x + c)^2 + 585*d*x*cos(d*x + c) + 195*d*x + (15*cos(d*x + c)^4 -
45*cos(d*x + c)^3 - 479*cos(d*x + c)^2 - 717*cos(d*x + c) - 304)*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d
*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (143) = 286\).

Time = 3.89 (sec) , antiderivative size = 473, normalized size of antiderivative = 3.09 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\begin {cases} \frac {390 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 120 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} + \frac {780 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 120 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} + \frac {390 d x}{60 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 120 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {3 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 120 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} + \frac {34 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 120 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {388 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 120 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {1310 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 120 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} - \frac {765 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{60 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 120 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 60 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5/(a+a*cos(d*x+c))**3,x)

[Out]

Piecewise((390*d*x*tan(c/2 + d*x/2)**4/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a*
*3*d) + 780*d*x*tan(c/2 + d*x/2)**2/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*
d) + 390*d*x/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 3*tan(c/2 + d*x/2)
**9/(60*a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 34*tan(c/2 + d*x/2)**7/(60*
a**3*d*tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 388*tan(c/2 + d*x/2)**5/(60*a**3*d*
tan(c/2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 1310*tan(c/2 + d*x/2)**3/(60*a**3*d*tan(c/
2 + d*x/2)**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 765*tan(c/2 + d*x/2)/(60*a**3*d*tan(c/2 + d*x/2)
**4 + 120*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d), Ne(d, 0)), (x*cos(c)**5/(a*cos(c) + a)**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {780 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{60 \, d} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/60*(60*(5*sin(d*x + c)/(cos(d*x + c) + 1) + 7*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^3 + 2*a^3*sin(d*x + c
)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (465*sin(d*x + c)/(cos(d*x + c) + 1) - 4
0*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 780*arctan(sin(d*x + c)/(
cos(d*x + c) + 1))/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.74 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {\frac {390 \, {\left (d x + c\right )}}{a^{3}} - \frac {60 \, {\left (7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} - \frac {3 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 465 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]

[In]

integrate(cos(d*x+c)^5/(a+a*cos(d*x+c))^3,x, algorithm="giac")

[Out]

1/60*(390*(d*x + c)/a^3 - 60*(7*tan(1/2*d*x + 1/2*c)^3 + 5*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)
^2*a^3) - (3*a^12*tan(1/2*d*x + 1/2*c)^5 - 40*a^12*tan(1/2*d*x + 1/2*c)^3 + 465*a^12*tan(1/2*d*x + 1/2*c))/a^1
5)/d

Mupad [B] (verification not implemented)

Time = 14.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \frac {\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-46\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+508\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-390\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{60\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]

[In]

int(cos(c + d*x)^5/(a + a*cos(c + d*x))^3,x)

[Out]

-(3*sin(c/2 + (d*x)/2) - 46*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 508*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)
/2) + 420*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) - 120*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) - 390*cos(c/2
+ (d*x)/2)^5*(c + d*x))/(60*a^3*d*cos(c/2 + (d*x)/2)^5)