\(\int \frac {(a+a \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x))}{\cos ^{\frac {15}{2}}(c+d x)} \, dx\) [197]

Optimal result

Integrand size = 37, antiderivative size = 313 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)} \]

[Out]

10/143*a*A*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(11/2)+2/13*A*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d/co
s(d*x+c)^(13/2)+2/9009*a^3*(2224*A+2717*C)*sin(d*x+c)/d/cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^(1/2)+2/15015*a^3*(8
368*A+10439*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(1/2)+8/45045*a^3*(8368*A+10439*C)*sin(d*x+c)/d/
cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(1/2)+16/45045*a^3*(8368*A+10439*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)/(a+a*cos(d
*x+c))^(1/2)+2/1287*a^2*(136*A+143*C)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(9/2)

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {3123, 3054, 3059, 2851, 2850} \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (136 A+143 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac {13}{2}}(c+d x)} \]

[In]

Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]

[Out]

(2*a^3*(2224*A + 2717*C)*Sin[c + d*x])/(9009*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(8368*A +
10439*C)*Sin[c + d*x])/(15015*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sin[
c + d*x])/(45045*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45
045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d
*x])/(1287*d*Cos[c + d*x]^(9/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)
) + (2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2))

Rule 2850

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*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]

Rule 2851

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x]
+ Dist[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), 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[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rule 3054

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d
*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x
])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n
+ 1) - B*(b*c*m - a*d*(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] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*
n] || EqQ[c, 0])

Rule 3059

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n +
1)*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), 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[n, -1]

Rule 3123

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^{5/2} \left (\frac {5 a A}{2}+\frac {1}{2} a (6 A+13 C) \cos (c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx}{13 a} \\ & = \frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {1}{4} a^2 (136 A+143 C)+\frac {1}{4} a^2 (96 A+143 C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{143 a} \\ & = \frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)}+\frac {8 \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {1}{8} a^3 (2224 A+2717 C)+\frac {15}{8} a^3 (112 A+143 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{1287 a} \\ & = \frac {2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)}+\frac {\left (a^2 (8368 A+10439 C)\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{3003} \\ & = \frac {2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)}+\frac {\left (4 a^2 (8368 A+10439 C)\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{15015} \\ & = \frac {2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)}+\frac {\left (8 a^2 (8368 A+10439 C)\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{45045} \\ & = \frac {2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (136 A+143 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d \cos ^{\frac {13}{2}}(c+d x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.55 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (343612 A+322751 C+1120 (347 A+286 C) \cos (c+d x)+14 (30334 A+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+141570 C \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+156585 C \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+20878 C \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+20878 C \cos (6 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{180180 d \cos ^{\frac {13}{2}}(c+d x)} \]

[In]

Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(343612*A + 322751*C + 1120*(347*A + 286*C)*Cos[c + d*x] + 14*(30334*A + 32747
*C)*Cos[2*(c + d*x)] + 125520*A*Cos[3*(c + d*x)] + 141570*C*Cos[3*(c + d*x)] + 125520*A*Cos[4*(c + d*x)] + 156
585*C*Cos[4*(c + d*x)] + 16736*A*Cos[5*(c + d*x)] + 20878*C*Cos[5*(c + d*x)] + 16736*A*Cos[6*(c + d*x)] + 2087
8*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/(180180*d*Cos[c + d*x]^(13/2))

Maple [A] (verified)

Time = 11.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.54

method	result	size
default	\(\frac {2 a^{2} \sin \left (d x +c \right ) \left (66944 A \left (\cos ^{6}\left (d x +c \right )\right )+83512 C \left (\cos ^{6}\left (d x +c \right )\right )+33472 A \left (\cos ^{5}\left (d x +c \right )\right )+41756 C \left (\cos ^{5}\left (d x +c \right )\right )+25104 A \left (\cos ^{4}\left (d x +c \right )\right )+31317 C \left (\cos ^{4}\left (d x +c \right )\right )+20920 A \left (\cos ^{3}\left (d x +c \right )\right )+18590 C \left (\cos ^{3}\left (d x +c \right )\right )+18305 A \left (\cos ^{2}\left (d x +c \right )\right )+5005 C \left (\cos ^{2}\left (d x +c \right )\right )+11970 A \cos \left (d x +c \right )+3465 A \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{45045 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {13}{2}}}\)	\(168\)
parts	\(\frac {2 A \sin \left (d x +c \right ) \left (66944 \left (\cos ^{6}\left (d x +c \right )\right )+33472 \left (\cos ^{5}\left (d x +c \right )\right )+25104 \left (\cos ^{4}\left (d x +c \right )\right )+20920 \left (\cos ^{3}\left (d x +c \right )\right )+18305 \left (\cos ^{2}\left (d x +c \right )\right )+11970 \cos \left (d x +c \right )+3465\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{45045 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {13}{2}}}+\frac {2 C \sin \left (d x +c \right ) \left (584 \left (\cos ^{4}\left (d x +c \right )\right )+292 \left (\cos ^{3}\left (d x +c \right )\right )+219 \left (\cos ^{2}\left (d x +c \right )\right )+130 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{315 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {9}{2}}}\)	\(192\)

[In]

int((a+cos(d*x+c)*a)^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x,method=_RETURNVERBOSE)

[Out]

2/45045*a^2/d*sin(d*x+c)*(66944*A*cos(d*x+c)^6+83512*C*cos(d*x+c)^6+33472*A*cos(d*x+c)^5+41756*C*cos(d*x+c)^5+
25104*A*cos(d*x+c)^4+31317*C*cos(d*x+c)^4+20920*A*cos(d*x+c)^3+18590*C*cos(d*x+c)^3+18305*A*cos(d*x+c)^2+5005*
C*cos(d*x+c)^2+11970*A*cos(d*x+c)+3465*A)*(a*(1+cos(d*x+c)))^(1/2)/(1+cos(d*x+c))/cos(d*x+c)^(13/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.54 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\frac {2 \, {\left (8 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 4 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 3 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (2092 \, A + 1859 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (523 \, A + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 11970 \, A a^{2} \cos \left (d x + c\right ) + 3465 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{8} + d \cos \left (d x + c\right )^{7}\right )}} \]

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x, algorithm="fricas")

[Out]

2/45045*(8*(8368*A + 10439*C)*a^2*cos(d*x + c)^6 + 4*(8368*A + 10439*C)*a^2*cos(d*x + c)^5 + 3*(8368*A + 10439
*C)*a^2*cos(d*x + c)^4 + 10*(2092*A + 1859*C)*a^2*cos(d*x + c)^3 + 35*(523*A + 143*C)*a^2*cos(d*x + c)^2 + 119
70*A*a^2*cos(d*x + c) + 3465*A*a^2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*sin(d*x + c)/(d*cos(d*x + c)^8
+ d*cos(d*x + c)^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))**(5/2)*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(15/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (271) = 542\).

Time = 0.37 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.14 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x, algorithm="maxima")

[Out]

8/45045*(143*(315*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 945*sqrt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*
x + c) + 1)^3 + 1449*sqrt(2)*a^(5/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1287*sqrt(2)*a^(5/2)*sin(d*x + c)^7
/(cos(d*x + c) + 1)^7 + 572*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 104*sqrt(2)*a^(5/2)*sin(d*x
+ c)^11/(cos(d*x + c) + 1)^11)*C*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^3/((sin(d*x + c)/(cos(d*x + c) + 1)
+ 1)^(11/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x
+ c)^4/(cos(d*x + c) + 1)^4 + sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 1)) + (45045*sqrt(2)*a^(5/2)*sin(d*x + c)/
(cos(d*x + c) + 1) - 165165*sqrt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 414414*sqrt(2)*a^(5/2)*sin(d
*x + c)^5/(cos(d*x + c) + 1)^5 - 604890*sqrt(2)*a^(5/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 522665*sqrt(2)*a
^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 289185*sqrt(2)*a^(5/2)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 88
980*sqrt(2)*a^(5/2)*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - 11864*sqrt(2)*a^(5/2)*sin(d*x + c)^15/(cos(d*x + c
) + 1)^15)*A*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(15/2)*(-sin(d
*x + c)/(cos(d*x + c) + 1) + 1)^(15/2)*(5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*sin(d*x + c)^4/(cos(d*x + c
) + 1)^4 + 10*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + sin(d*x + c)^10/(c
os(d*x + c) + 1)^10 + 1)))/d

Giac [F(-1)]

Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(15/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 8.85 (sec) , antiderivative size = 911, normalized size of antiderivative = 2.91 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {15}{2}}(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2))/cos(c + d*x)^(15/2),x)

[Out]

((a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((a^2*(8368*A + 10439*C)*16i)/(45045*d) - (a^2*
exp(c*5i + d*x*5i)*(6*A + 23*C)*16i)/(15*d) + (a^2*exp(c*8i + d*x*8i)*(6*A + 23*C)*16i)/(15*d) + (a^2*exp(c*6i
+ d*x*6i)*(348*A + 379*C)*16i)/(105*d) - (a^2*exp(c*7i + d*x*7i)*(348*A + 379*C)*16i)/(105*d) + (a^2*exp(c*4i
+ d*x*4i)*(523*A + 554*C)*32i)/(315*d) - (a^2*exp(c*9i + d*x*9i)*(523*A + 554*C)*32i)/(315*d) + (a^2*exp(c*2i
+ d*x*2i)*(8368*A + 10439*C)*8i)/(3465*d) - (a^2*exp(c*11i + d*x*11i)*(8368*A + 10439*C)*8i)/(3465*d) - (a^2*
exp(c*13i + d*x*13i)*(8368*A + 10439*C)*16i)/(45045*d) - (C*a^2*exp(c*3i + d*x*3i)*8i)/(3*d) + (C*a^2*exp(c*10
i + d*x*10i)*8i)/(3*d)))/((exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + exp(c*1i + d*x*1i)*(exp(- c*
1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 6*exp(c*2i + d*x*2i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1
i)/2)^(1/2) + 6*exp(c*3i + d*x*3i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 15*exp(c*4i + d*x*4
i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 15*exp(c*5i + d*x*5i)*(exp(- c*1i - d*x*1i)/2 + exp
(c*1i + d*x*1i)/2)^(1/2) + 20*exp(c*6i + d*x*6i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 20*ex
p(c*7i + d*x*7i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 15*exp(c*8i + d*x*8i)*(exp(- c*1i - d
*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 15*exp(c*9i + d*x*9i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)
^(1/2) + 6*exp(c*10i + d*x*10i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 6*exp(c*11i + d*x*11i)
*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + exp(c*12i + d*x*12i)*(exp(- c*1i - d*x*1i)/2 + exp(c*
1i + d*x*1i)/2)^(1/2) + exp(c*13i + d*x*13i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2))