3.1115 \(\int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=609 \[ \frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{12 a^3 d \left (a^2-b^2\right )^2}-\frac {\sin (c+d x) \left (-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right )}{4 a^2 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\sin (c+d x) \left (a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right )}{12 a^3 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x)}+\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{4 a^4 d \left (a^2-b^2\right )^2}-\frac {\sin (c+d x) \left (-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right )}{4 a^4 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}+\frac {\left (15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3} \]

[Out]

1/4*(35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B+3*a^4*b*(8*A-3*C)-a^2*b^3*(65*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))/a^4/(a^2-b^2)^2/d+1/12*(35*A*b^4+33*a^3*b*B-15*a*b
^3*B+a^4*(8*A-21*C)-a^2*b^2*(61*A-3*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))/a^3/(a^2-b^2)^2/d+1/4*(35*A*b^6-35*a^5*b*B+38*a^3*b^3*B-15*a*b^5*B-a^2*b^4*(86*A-3*C)+3*a^4*b^
2*(21*A-2*C)+15*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^4/(a-b)^2/(a+b)^3/d+1/12*(35*A*b^4+33*a^3*b*B-15*a*b^3*B+a^4*(8*A-21*C)-a^2*b^2*(61*A-3*C))*sin(d*
x+c)/a^3/(a^2-b^2)^2/d/cos(d*x+c)^(3/2)+1/2*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/(a^2-b^2)/d/cos(d*x+c)^(3/2)/(a+b
*cos(d*x+c))^2-1/4*(7*A*b^4+9*a^3*b*B-3*a*b^3*B-5*a^4*C-a^2*b^2*(13*A+C))*sin(d*x+c)/a^2/(a^2-b^2)^2/d/cos(d*x
+c)^(3/2)/(a+b*cos(d*x+c))-1/4*(35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B+3*a^4*b*(8*A-3*C)-a^2*b^3*(65*A-3*C))
*sin(d*x+c)/a^4/(a^2-b^2)^2/d/cos(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.49, antiderivative size = 609, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3055, 3059, 2639, 3002, 2641, 2805} \[ \frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right )}{12 a^3 d \left (a^2-b^2\right )^2}+\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac {\left (3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 C-15 a b^5 B+35 A b^6\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3}-\frac {\sin (c+d x) \left (-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 C-3 a b^3 B+7 A b^4\right )}{4 a^2 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\sin (c+d x) \left (-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right )}{12 a^3 d \left (a^2-b^2\right )^2 \cos ^{\frac {3}{2}}(c+d x)}+\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\sin (c+d x) \left (-a^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right )}{4 a^4 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]

[Out]

((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*EllipticE[(c +
d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A
- 3*C))*EllipticF[(c + d*x)/2, 2])/(12*a^3*(a^2 - b^2)^2*d) + ((35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^
5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*
a^4*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*S
in[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B +
3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + ((A*b^2
- a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) - ((7*A*b^4 + 9*
a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*(13*A + C))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*(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 3055

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[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 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 {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx &=\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {\int \frac {\frac {1}{2} \left (-7 A b^2+3 a b B+a^2 (4 A-3 C)\right )-2 a (A b-a B+b C) \cos (c+d x)+\frac {5}{2} \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right )+a \left (A b^3+2 a^3 B+a b^2 B-a^2 b (4 A+3 C)\right ) \cos (c+d x)-\frac {3}{4} \left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\int \frac {-\frac {3}{8} \left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B-a^2 b^3 (65 A-3 C)+a^4 (24 A b-9 b C)\right )-\frac {1}{2} a \left (7 A b^4+12 a^3 b B-3 a b^3 B-2 a^4 (A+3 C)-a^2 b^2 (14 A+3 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{3 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {2 \int \frac {\frac {1}{16} \left (105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right )+\frac {1}{4} a \left (35 A b^5-6 a^5 B+30 a^3 b^2 B-15 a b^4 B+4 a^4 b (5 A-3 C)-a^2 b^3 (64 A-3 C)\right ) \cos (c+d x)+\frac {3}{16} b \left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac {2 \int \frac {-\frac {1}{16} b \left (105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right )-\frac {1}{16} a b^2 \left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^4 b \left (a^2-b^2\right )^2}+\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )^2}+\frac {\left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (35 A b^6-35 a^5 b B+38 a^3 b^3 B-15 a b^5 B-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 C\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 (a-b)^2 (a+b)^3 d}+\frac {\left (35 A b^4+33 a^3 b B-15 a b^3 B+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B+3 a^4 b (8 A-3 C)-a^2 b^3 (65 A-3 C)\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac {\left (7 A b^4+9 a^3 b B-3 a b^3 B-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 7.76, size = 672, normalized size = 1.10 \[ \frac {\sqrt {\cos (c+d x)} \left (\frac {2 \sec (c+d x) (a B \sin (c+d x)-3 A b \sin (c+d x))}{a^4}+\frac {2 A \tan (c+d x) \sec (c+d x)}{3 a^3}+\frac {a^2 b^2 C \sin (c+d x)-a b^3 B \sin (c+d x)+A b^4 \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {9 a^4 b^2 C \sin (c+d x)-13 a^3 b^3 B \sin (c+d x)+17 a^2 A b^4 \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)+7 a b^5 B \sin (c+d x)-11 A b^6 \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}\right )}{d}+\frac {\frac {2 \sin (c+d x) \cos (2 (c+d x)) \left (-24 a^5 b B+72 a^4 A b^2-27 a^4 b^2 C+87 a^3 b^3 B-195 a^2 A b^4+9 a^2 b^4 C-45 a b^5 B+105 A b^6\right ) \left (\left (b^2-2 a^2\right ) \Pi \left (-\frac {b}{a};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a b^2 \sqrt {1-\cos ^2(c+d x)} \left (2 \cos ^2(c+d x)-1\right )}+\frac {\left (-48 a^6 B+160 a^5 A b-96 a^5 b C+240 a^4 b^2 B-512 a^3 A b^3+24 a^3 b^3 C-120 a^2 b^4 B+280 a A b^5\right ) \left (2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-\frac {2 a \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}\right )}{b}+\frac {2 \left (16 a^6 A+48 a^6 C-168 a^5 b B+328 a^4 A b^2-57 a^4 b^2 C+285 a^3 b^3 B-641 a^2 A b^4+27 a^2 b^4 C-135 a b^5 B+315 A b^6\right ) \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}}{48 a^4 d (a-b)^2 (a+b)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]

[Out]

((2*(16*a^6*A + 328*a^4*A*b^2 - 641*a^2*A*b^4 + 315*A*b^6 - 168*a^5*b*B + 285*a^3*b^3*B - 135*a*b^5*B + 48*a^6
*C - 57*a^4*b^2*C + 27*a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((160*a^5*A*b - 512*a^3
*A*b^3 + 280*a*A*b^5 - 48*a^6*B + 240*a^4*b^2*B - 120*a^2*b^4*B - 96*a^5*b*C + 24*a^3*b^3*C)*(2*EllipticF[(c +
d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(72*a^4*A*b^2 - 195*a^2*A*b^4 +
105*A*b^6 - 24*a^5*b*B + 87*a^3*b^3*B - 45*a*b^5*B - 27*a^4*b^2*C + 9*a^2*b^4*C)*Cos[2*(c + d*x)]*(-2*a*b*Ell
ipticE[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), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Co
s[c + d*x]^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]*(-3*A*b*Sin[c + d*x] + a
*B*Sin[c + d*x]))/a^4 + (A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x])/(2*a^3*(a^2 - b^2
)*(a + b*Cos[c + d*x])^2) + (17*a^2*A*b^4*Sin[c + d*x] - 11*A*b^6*Sin[c + d*x] - 13*a^3*b^3*B*Sin[c + d*x] + 7
*a*b^5*B*Sin[c + d*x] + 9*a^4*b^2*C*Sin[c + d*x] - 3*a^2*b^4*C*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*(a + b*Cos[c
+ d*x])) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^3)))/d

________________________________________________________________________________________

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((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)^3*cos(d*x + c)^(5/2)), x)

________________________________________________________________________________________

maple [B]  time = 22.87, size = 2165, normalized size = 3.56 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(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*b^2*(3*A*b-B*a)/a^4/(-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)*Ellipt
icPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))+2*A/a^3*(-1/6*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)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+1/3*(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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(-3*A*b+B
*a)/a^4*(-(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2
*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+2*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*co
s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)+2*(A*b^2-B*a*b+C*a^2)/a
^2*(-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*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+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(c
os(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*b*(2*A*b-B*a)/a^3*(-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)*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)*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

________________________________________________________________________________________

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((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+b*cos(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^3),x)

[Out]

int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^3), x)

________________________________________________________________________________________

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((A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)**(5/2)/(a+b*cos(d*x+c))**3,x)

[Out]

Timed out

________________________________________________________________________________________