3.1109 \(\int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=654 \[ -\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right )}{20 b^3 d \left (a^2-b^2\right )^2}+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{12 b^4 d \left (a^2-b^2\right )^2}-\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac {a^2 \left (63 a^6 C-35 a^5 b B+15 a^4 b^2 (A-10 C)+86 a^3 b^3 B-a^2 b^4 (38 A-99 C)-63 a b^5 B+35 A b^6\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3}+\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-189 a^7 C+105 a^6 b B-9 a^5 b^2 (5 A-43 C)-223 a^4 b^3 B+3 a^3 b^4 (33 A-64 C)+128 a^2 b^5 B-24 a b^6 (3 A+C)+8 b^7 B\right )}{12 b^6 d \left (a^2-b^2\right )^2} \]

[Out]

-1/20*(175*a^5*b*B-325*a^3*b^3*B+120*a*b^5*B+a^2*b^4*(145*A-192*C)-3*a^4*b^2*(25*A-187*C)-315*a^6*C-8*b^6*(5*A
+3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/b^5/(a^2-b^2)^2/d
+1/12*(105*a^6*b*B-223*a^4*b^3*B+128*a^2*b^5*B+8*b^7*B+3*a^3*b^4*(33*A-64*C)-9*a^5*b^2*(5*A-43*C)-189*a^7*C-24
*a*b^6*(3*A+C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/b^6/(a^2
-b^2)^2/d+1/4*a^2*(35*A*b^6-35*a^5*b*B+86*a^3*b^3*B-63*a*b^5*B-a^2*b^4*(38*A-99*C)+15*a^4*b^2*(A-10*C)+63*a^6*
C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/(a-b)^2/b^
6/(a+b)^3/d-1/20*(35*a^3*b*B-65*a*b^3*B-a^2*b^2*(15*A-101*C)+b^4*(45*A-8*C)-63*a^4*C)*cos(d*x+c)^(3/2)*sin(d*x
+c)/b^3/(a^2-b^2)^2/d-1/2*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^(7/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/4
*(7*A*b^4+5*a^3*b*B-11*a*b^3*B-a^2*b^2*(A-15*C)-9*a^4*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*co
s(d*x+c))+1/12*(35*a^4*b*B-61*a^2*b^3*B+8*b^5*B+3*a*b^4*(11*A-8*C)-15*a^3*b^2*(A-7*C)-63*a^5*C)*sin(d*x+c)*cos
(d*x+c)^(1/2)/b^4/(a^2-b^2)^2/d

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Rubi [A]  time = 2.64, antiderivative size = 654, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3047, 3049, 3059, 2639, 3002, 2641, 2805} \[ \frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right )}{12 b^6 d \left (a^2-b^2\right )^2}-\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac {a^2 \left (15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right )}{20 b^3 d \left (a^2-b^2\right )^2}+\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{12 b^4 d \left (a^2-b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]

[Out]

-((175*a^5*b*B - 325*a^3*b^3*B + 120*a*b^5*B + a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C
- 8*b^6*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(20*b^5*(a^2 - b^2)^2*d) + ((105*a^6*b*B - 223*a^4*b^3*B + 128
*a^2*b^5*B + 8*b^7*B + 3*a^3*b^4*(33*A - 64*C) - 9*a^5*b^2*(5*A - 43*C) - 189*a^7*C - 24*a*b^6*(3*A + C))*Elli
pticF[(c + d*x)/2, 2])/(12*b^6*(a^2 - b^2)^2*d) + (a^2*(35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B - a^
2*b^4*(38*A - 99*C) + 15*a^4*b^2*(A - 10*C) + 63*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^
2*b^6*(a + b)^3*d) + ((35*a^4*b*B - 61*a^2*b^3*B + 8*b^5*B + 3*a*b^4*(11*A - 8*C) - 15*a^3*b^2*(A - 7*C) - 63*
a^5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^4*(a^2 - b^2)^2*d) - ((35*a^3*b*B - 65*a*b^3*B - a^2*b^2*(15*A -
101*C) + b^4*(45*A - 8*C) - 63*a^4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a
*(b*B - a*C))*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((7*A*b^4 + 5*a^3*
b*B - 11*a*b^3*B - a^2*b^2*(A - 15*C) - 9*a^4*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a +
b*Cos[c + d*x]))

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 3002

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3047

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

Rule 3049

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

Rule 3059

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps

\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7}{2} \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \cos (c+d x)-\frac {1}{2} \left (5 A b^2-5 a b B+9 a^2 C-4 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {5}{4} \left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \cos (c+d x)-\frac {1}{4} \left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-\frac {3}{8} a \left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right )+\frac {1}{2} b \left (5 a^3 b B-20 a b^3 B-9 a^4 C+2 b^4 (5 A+3 C)+a^2 b^2 (5 A+18 C)\right ) \cos (c+d x)+\frac {5}{8} \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{5 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {2 \int \frac {\frac {5}{16} a \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right )-\frac {1}{4} b \left (35 a^4 b B-70 a^2 b^3 B-10 b^5 B-3 a^3 b^2 (5 A-32 C)-63 a^5 C+12 a b^4 (5 A+C)\right ) \cos (c+d x)-\frac {3}{16} \left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {2 \int \frac {-\frac {5}{16} a b \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right )-\frac {5}{16} \left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^5 \left (a^2-b^2\right )^2}-\frac {\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \int \sqrt {\cos (c+d x)} \, dx}{40 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac {\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\left (a^2 \left (35 A b^6-35 a^5 b B+86 a^3 b^3 B-63 a b^5 B-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 b^6 \left (a^2-b^2\right )^2}+\frac {\left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 b^6 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac {\left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 b^6 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (35 A b^6-35 a^5 b B+86 a^3 b^3 B-63 a b^5 B-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 (a-b)^2 b^6 (a+b)^3 d}+\frac {\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 7.69, size = 551, normalized size = 0.84 \[ \frac {4 \sqrt {\cos (c+d x)} \left (\frac {30 a^3 \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{\left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {15 a^2 \sin (c+d x) \left (15 a^4 C-11 a^3 b B+7 a^2 b^2 (A-3 C)+17 a b^3 B-13 A b^4\right )}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+40 (b B-3 a C) \sin (c+d x)+12 b C \sin (2 (c+d x))\right )+\frac {\frac {16 \left (63 a^5 C-35 a^4 b B+3 a^3 b^2 (5 A-32 C)+70 a^2 b^3 B-12 a b^4 (5 A+C)+10 b^5 B\right ) \left ((a+b) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-a \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{a+b}+\frac {2 \left (315 a^6 C-175 a^5 b B+3 a^4 b^2 (25 A-211 C)+365 a^3 b^3 B-21 a^2 b^4 (5 A-16 C)-280 a b^5 B+24 b^6 (5 A+3 C)\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+\frac {6 \sin (c+d x) \left (315 a^6 C-175 a^5 b B+3 a^4 b^2 (25 A-187 C)+325 a^3 b^3 B+a^2 b^4 (192 C-145 A)-120 a b^5 B+8 b^6 (5 A+3 C)\right ) \left (\left (b^2-2 a^2\right ) \Pi \left (-\frac {b}{a};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a b^2 \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{240 b^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]

[Out]

(((2*(-175*a^5*b*B + 365*a^3*b^3*B - 280*a*b^5*B + 3*a^4*b^2*(25*A - 211*C) - 21*a^2*b^4*(5*A - 16*C) + 315*a^
6*C + 24*b^6*(5*A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(-35*a^4*b*B + 70*a^2*b^3*B
+ 10*b^5*B + 3*a^3*b^2*(5*A - 32*C) + 63*a^5*C - 12*a*b^4*(5*A + C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*E
llipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-175*a^5*b*B + 325*a^3*b^3*B - 120*a*b^5*B + 3*a^4*b^
2*(25*A - 187*C) + 315*a^6*C + 8*b^6*(5*A + 3*C) + a^2*b^4*(-145*A + 192*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos
[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), A
rcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2) + 4*Sqrt[Cos
[c + d*x]]*(40*(b*B - 3*a*C)*Sin[c + d*x] + (30*a^3*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a^2 - b^2)*(a +
b*Cos[c + d*x])^2) - (15*a^2*(-13*A*b^4 - 11*a^3*b*B + 17*a*b^3*B + 7*a^2*b^2*(A - 3*C) + 15*a^4*C)*Sin[c + d
*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])) + 12*b*C*Sin[2*(c + d*x)]))/(240*b^4*d)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(7/2)/(b*cos(d*x + c) + a)^3, x)

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maple [B]  time = 16.68, size = 2520, normalized size = 3.85 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x)

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4/5/b^3*C*(-4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^6+14*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)
*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-9*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipti
cE(cos(1/2*d*x+1/2*c),2^(1/2))-6*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x
+1/2*c)^2)^(1/2)+4/3/b^4*(B*b-3*C*a-3*C*b)*(2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+2*(sin(1/2*d*x+1/2*c)^2)
^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(
2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c))
/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2/b^5*(A*b^2-3*B*a*b-2*B*b^2+6*C*a^2+6*C*a*b+3*C*b^2)*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1
/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-2*(3*A*a*b^2+A*b^3-6*B*a^2*b
-3*B*a*b^2-B*b^3+10*C*a^3+6*C*a^2*b+3*C*a*b^2+C*b^3)/b^6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2
+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-4/b^5*a^2
*(6*A*b^2-10*B*a*b+15*C*a^2)/(-2*a*b+2*b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2
*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))+2*a^4*(A*b
^2-B*a*b+C*a^2)/b^6*(-1/2*b^2/a/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1
/2)/(2*cos(1/2*d*x+1/2*c)^2*b+a-b)^2-3/4*b^2*(3*a^2-b^2)/a^2/(a^2-b^2)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/
2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*b+a-b)-7/8/(a+b)/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(
1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*
d*x+1/2*c),2^(1/2))+1/4/(a+b)/(a^2-b^2)/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*s
in(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b+3/8/(a+b)/(a^2-b^2)/a^
2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2
)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b^2-9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d
*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)
)+3/8*b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)
^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/
2*d*x+1/2*c),2^(1/2))-3/8*b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(
-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-15/4*a^2/(a^2-b^2)^2
/(-2*a*b+2*b^2)*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(
1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))+3/2/(a^2-b^2)^2/(-2*a*b+2*b^2)*b^3*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(
1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))-3/4/a^2/(a^2-b^2)^2/(-2*a*b+2*b^2)*b^5*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi
(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2)))-2*a^3/b^6*(4*A*b^2-5*B*a*b+6*C*a^2)*(-b^2/a/(a^2-b^2)*cos(1/2*d*x+1/2
*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*b+a-b)-1/2/(a+b)/a*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Elli
pticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2*b/a/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(
1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/2*b/a/(a^2-b
^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)
^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-3*a/(a^2-b^2)/(-2*a*b+2*b^2)*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-
2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/
2*c),-2*b/(a-b),2^(1/2))+1/a/(a^2-b^2)/(-2*a*b+2*b^2)*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^
2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/
2))))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3,x)

[Out]

int((cos(c + d*x)^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**3,x)

[Out]

Timed out

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