\(\int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx\) [82]

Optimal result

Integrand size = 21, antiderivative size = 225 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {31 x}{2 a^5}-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \cos ^2(c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]

[Out]

31/2*x/a^5-7664/315*sin(d*x+c)/a^5/d+31/2*cos(d*x+c)*sin(d*x+c)/a^5/d-1/9*cos(d*x+c)^6*sin(d*x+c)/d/(a+a*cos(d
*x+c))^5-17/63*cos(d*x+c)^5*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^4-28/45*cos(d*x+c)^4*sin(d*x+c)/a^2/d/(a+a*cos(d*x
+c))^3-577/315*cos(d*x+c)^3*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^2-3832/315*cos(d*x+c)^2*sin(d*x+c)/d/(a^5+a^5*co
s(d*x+c))

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2844, 3056, 2813} \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {7664 \sin (c+d x)}{315 a^5 d}-\frac {3832 \sin (c+d x) \cos ^2(c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}+\frac {31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}+\frac {31 x}{2 a^5}-\frac {577 \sin (c+d x) \cos ^3(c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac {28 \sin (c+d x) \cos ^4(c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac {\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac {17 \sin (c+d x) \cos ^5(c+d x)}{63 a d (a \cos (c+d x)+a)^4} \]

[In]

Int[Cos[c + d*x]^7/(a + a*Cos[c + d*x])^5,x]

[Out]

(31*x)/(2*a^5) - (7664*Sin[c + d*x])/(315*a^5*d) + (31*Cos[c + d*x]*Sin[c + d*x])/(2*a^5*d) - (Cos[c + d*x]^6*
Sin[c + d*x])/(9*d*(a + a*Cos[c + d*x])^5) - (17*Cos[c + d*x]^5*Sin[c + d*x])/(63*a*d*(a + a*Cos[c + d*x])^4)
- (28*Cos[c + d*x]^4*Sin[c + d*x])/(45*a^2*d*(a + a*Cos[c + d*x])^3) - (577*Cos[c + d*x]^3*Sin[c + d*x])/(315*
a^3*d*(a + a*Cos[c + d*x])^2) - (3832*Cos[c + d*x]^2*Sin[c + d*x])/(315*d*(a^5 + a^5*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 ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {\int \frac {\cos ^5(c+d x) (6 a-11 a \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx}{9 a^2} \\ & = -\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {\int \frac {\cos ^4(c+d x) \left (85 a^2-111 a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{63 a^4} \\ & = -\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {\int \frac {\cos ^3(c+d x) \left (784 a^3-947 a^3 \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{315 a^6} \\ & = -\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (5193 a^4-6303 a^4 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{945 a^8} \\ & = -\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \cos ^2(c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}-\frac {\int \cos (c+d x) \left (22992 a^5-29295 a^5 \cos (c+d x)\right ) \, dx}{945 a^{10}} \\ & = \frac {31 x}{2 a^5}-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos ^6(c+d x) \sin (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \cos ^5(c+d x) \sin (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \cos ^4(c+d x) \sin (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \cos ^3(c+d x) \sin (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \cos ^2(c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.64 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \csc ^{10}(c+d x) \sin ^9\left (\frac {1}{2} (c+d x)\right ) \left (984312+1035321 \cos (c+d x)-484476 \cos (2 (c+d x))-933309 \cos (3 (c+d x))-491576 \cos (4 (c+d x))-106807 \cos (5 (c+d x))-3780 \cos (6 (c+d x))+315 \cos (7 (c+d x))+9999360 \arcsin (\cos (c+d x)) \cos ^8\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{1260 a^5 d} \]

[In]

Integrate[Cos[c + d*x]^7/(a + a*Cos[c + d*x])^5,x]

[Out]

-1/1260*(Cos[(c + d*x)/2]*Csc[c + d*x]^10*Sin[(c + d*x)/2]^9*(984312 + 1035321*Cos[c + d*x] - 484476*Cos[2*(c
+ d*x)] - 933309*Cos[3*(c + d*x)] - 491576*Cos[4*(c + d*x)] - 106807*Cos[5*(c + d*x)] - 3780*Cos[6*(c + d*x)]
+ 315*Cos[7*(c + d*x)] + 9999360*ArcSin[Cos[c + d*x]]*Cos[(c + d*x)/2]^8*Sqrt[Sin[c + d*x]^2]))/(a^5*d)

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.44

method	result	size
parallelrisch	\(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (6 d x +6 c \right )-\frac {854012 \cos \left (d x +c \right )}{63}-\frac {2250427 \cos \left (2 d x +2 c \right )}{315}-\frac {143054 \cos \left (3 d x +3 c \right )}{63}-\frac {113422 \cos \left (4 d x +4 c \right )}{315}-10 \cos \left (5 d x +5 c \right )-\frac {2627186}{315}\right ) \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15872 d x}{1024 a^{5} d}\)	\(98\)
derivativedivides	\(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-176 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+496 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\)	\(127\)
default	\(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-176 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+496 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\)	\(127\)
risch	\(\frac {31 x}{2 a^{5}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{5} d}+\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{5} d}-\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{5} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{5} d}-\frac {2 i \left (11025 \,{\mathrm e}^{8 i \left (d x +c \right )}+77175 \,{\mathrm e}^{7 i \left (d x +c \right )}+247695 \,{\mathrm e}^{6 i \left (d x +c \right )}+465255 \,{\mathrm e}^{5 i \left (d x +c \right )}+557109 \,{\mathrm e}^{4 i \left (d x +c \right )}+433881 \,{\mathrm e}^{3 i \left (d x +c \right )}+214929 \,{\mathrm e}^{2 i \left (d x +c \right )}+62001 \,{\mathrm e}^{i \left (d x +c \right )}+8114\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\)	\(192\)

[In]

int(cos(d*x+c)^7/(a+cos(d*x+c)*a)^5,x,method=_RETURNVERBOSE)

[Out]

1/1024*(tan(1/2*d*x+1/2*c)*(cos(6*d*x+6*c)-854012/63*cos(d*x+c)-2250427/315*cos(2*d*x+2*c)-143054/63*cos(3*d*x
+3*c)-113422/315*cos(4*d*x+4*c)-10*cos(5*d*x+5*c)-2627186/315)*sec(1/2*d*x+1/2*c)^8+15872*d*x)/a^5/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {9765 \, d x \cos \left (d x + c\right )^{5} + 48825 \, d x \cos \left (d x + c\right )^{4} + 97650 \, d x \cos \left (d x + c\right )^{3} + 97650 \, d x \cos \left (d x + c\right )^{2} + 48825 \, d x \cos \left (d x + c\right ) + 9765 \, d x + {\left (315 \, \cos \left (d x + c\right )^{6} - 1575 \, \cos \left (d x + c\right )^{5} - 28828 \, \cos \left (d x + c\right )^{4} - 87440 \, \cos \left (d x + c\right )^{3} - 112119 \, \cos \left (d x + c\right )^{2} - 66875 \, \cos \left (d x + c\right ) - 15328\right )} \sin \left (d x + c\right )}{630 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]

[In]

integrate(cos(d*x+c)^7/(a+a*cos(d*x+c))^5,x, algorithm="fricas")

[Out]

1/630*(9765*d*x*cos(d*x + c)^5 + 48825*d*x*cos(d*x + c)^4 + 97650*d*x*cos(d*x + c)^3 + 97650*d*x*cos(d*x + c)^
2 + 48825*d*x*cos(d*x + c) + 9765*d*x + (315*cos(d*x + c)^6 - 1575*cos(d*x + c)^5 - 28828*cos(d*x + c)^4 - 874
40*cos(d*x + c)^3 - 112119*cos(d*x + c)^2 - 66875*cos(d*x + c) - 15328)*sin(d*x + c))/(a^5*d*cos(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (214) = 428\).

Time = 19.70 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.61 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\begin {cases} \frac {78120 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {156240 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {78120 d x}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {35 \tan ^{13}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {380 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {2159 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} + \frac {10152 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {82089 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {260820 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} - \frac {155925 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{5040 a^{5} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 10080 a^{5} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 5040 a^{5} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{7}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{5}} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**7/(a+a*cos(d*x+c))**5,x)

[Out]

Piecewise((78120*d*x*tan(c/2 + d*x/2)**4/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 +
5040*a**5*d) + 156240*d*x*tan(c/2 + d*x/2)**2/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2
)**2 + 5040*a**5*d) + 78120*d*x/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**
5*d) - 35*tan(c/2 + d*x/2)**13/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**5
*d) + 380*tan(c/2 + d*x/2)**11/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**5
*d) - 2159*tan(c/2 + d*x/2)**9/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**5
*d) + 10152*tan(c/2 + d*x/2)**7/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a**
5*d) - 82089*tan(c/2 + d*x/2)**5/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a*
*5*d) - 260820*tan(c/2 + d*x/2)**3/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*
a**5*d) - 155925*tan(c/2 + d*x/2)/(5040*a**5*d*tan(c/2 + d*x/2)**4 + 10080*a**5*d*tan(c/2 + d*x/2)**2 + 5040*a
**5*d), Ne(d, 0)), (x*cos(c)**7/(a*cos(c) + a)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {\frac {5040 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} + \frac {2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{5}} - \frac {156240 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \]

[In]

integrate(cos(d*x+c)^7/(a+a*cos(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/5040*(5040*(9*sin(d*x + c)/(cos(d*x + c) + 1) + 11*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^5 + 2*a^5*sin(d*
x + c)^2/(cos(d*x + c) + 1)^2 + a^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (110565*sin(d*x + c)/(cos(d*x + c)
+ 1) - 15750*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3024*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 450*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 - 156240*arctan(sin(d*x + c)/(cos(d*x +
c) + 1))/a^5)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.64 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=\frac {\frac {78120 \, {\left (d x + c\right )}}{a^{5}} - \frac {5040 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 450 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15750 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \]

[In]

integrate(cos(d*x+c)^7/(a+a*cos(d*x+c))^5,x, algorithm="giac")

[Out]

1/5040*(78120*(d*x + c)/a^5 - 5040*(11*tan(1/2*d*x + 1/2*c)^3 + 9*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)
^2 + 1)^2*a^5) - (35*a^40*tan(1/2*d*x + 1/2*c)^9 - 450*a^40*tan(1/2*d*x + 1/2*c)^7 + 3024*a^40*tan(1/2*d*x + 1
/2*c)^5 - 15750*a^40*tan(1/2*d*x + 1/2*c)^3 + 110565*a^40*tan(1/2*d*x + 1/2*c))/a^45)/d

Mupad [B] (verification not implemented)

Time = 15.43 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx=-\frac {35\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-590\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+4584\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-23288\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+129824\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+55440\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-10080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]

[In]

int(cos(c + d*x)^7/(a + a*cos(c + d*x))^5,x)

[Out]

-(35*sin(c/2 + (d*x)/2) - 590*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) + 4584*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d
*x)/2) - 23288*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) + 129824*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2) + 5544
0*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) - 10080*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2) - 78120*cos(c/2 +
(d*x)/2)^9*(c + d*x))/(5040*a^5*d*cos(c/2 + (d*x)/2)^9)