\(\int \frac {\cos ^{\frac {5}{2}}(c+d x) (A+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^3} \, dx\) [721]

Optimal result

Integrand size = 35, antiderivative size = 433 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=-\frac {a \left (a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)+35 a^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 b^5 \left (a^2-b^2\right )^2 d}-\frac {a \left (15 A b^6+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^5 (a+b)^3 d}+\frac {\left (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (5 A b^4-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]

[Out]

-1/4*a*(a^2*b^2*(3*A-65*C)-3*b^4*(3*A-8*C)+35*a^4*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^4/(a^2-b^2)^2/d+1/12*(a^4*b^2*(9*A-223*C)-a^2*b^4*(15*A-128*C)+105*a^6*C+8*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^5/(a^2-b^2)^2
/d-1/4*a*(15*A*b^6+a^4*b^2*(3*A-86*C)-3*a^2*b^4*(2*A-21*C)+35*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^5/(a+b)^3/d-1/2*(A*b^2+C*a^2)*cos(d*x+c)^(5/
2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/4*(5*A*b^4-7*a^4*C+a^2*b^2*(A+13*C))*cos(d*x+c)^(3/2)*sin(d*x
+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))+1/12*(a^2*b^2*(3*A-61*C)-b^4*(21*A-8*C)+35*a^4*C)*sin(d*x+c)*cos(d*x+c)
^(1/2)/b^3/(a^2-b^2)^2/d

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3127, 3126, 3128, 3138, 2719, 3081, 2720, 2884} \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {a \left (35 a^4 C+a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 d \left (a^2-b^2\right )^2}+\frac {\left (-7 a^4 C+a^2 b^2 (A+13 C)+5 A b^4\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {\left (35 a^4 C+a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{12 b^3 d \left (a^2-b^2\right )^2}+\frac {\left (105 a^6 C+a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+8 b^6 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 b^5 d \left (a^2-b^2\right )^2}-\frac {a \left (35 a^6 C+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+15 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 b^5 d (a-b)^2 (a+b)^3} \]

[In]

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

[Out]

-1/4*(a*(a^2*b^2*(3*A - 65*C) - 3*b^4*(3*A - 8*C) + 35*a^4*C)*EllipticE[(c + d*x)/2, 2])/(b^4*(a^2 - b^2)^2*d)
+ ((a^4*b^2*(9*A - 223*C) - a^2*b^4*(15*A - 128*C) + 105*a^6*C + 8*b^6*(3*A + C))*EllipticF[(c + d*x)/2, 2])/
(12*b^5*(a^2 - b^2)^2*d) - (a*(15*A*b^6 + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*EllipticPi
[(2*b)/(a + b), (c + d*x)/2, 2])/(4*(a - b)^2*b^5*(a + b)^3*d) + ((a^2*b^2*(3*A - 61*C) - b^4*(21*A - 8*C) + 3
5*a^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^(5/2)*Sin[c
+ d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) + ((5*A*b^4 - 7*a^4*C + a^2*b^2*(A + 13*C))*Cos[c + d*x]^(
3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))

Rule 2719

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

Rule 2720

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

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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 3081

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 3126

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 3127

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/(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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n
+ 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(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, 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 3128

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 3138

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*} \text {integral}& = -\frac {\left (A b^2+a^2 C\right ) \cos ^{\frac {5}{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 {3}{2}}(c+d x) \left (\frac {5}{2} \left (A b^2+a^2 C\right )-2 a b (A+C) \cos (c+d x)-\frac {1}{2} \left (3 A b^2+7 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^2 C\right ) \cos ^{\frac {5}{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 (5 A b^4-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{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}{4} \left (5 A b^4-7 a^4 C+a^2 b^2 (A+13 C)\right )-a b \left (3 A b^2-\left (a^2-4 b^2\right ) C\right ) \cos (c+d x)+\frac {1}{4} \left (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 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 (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \cos ^{\frac {5}{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 (5 A b^4-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{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 {\frac {1}{8} a \left (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 a^4 C\right )-\frac {1}{2} b \left (7 a^4 C-2 b^4 (3 A+C)-a^2 b^2 (3 A+14 C)\right ) \cos (c+d x)-\frac {3}{8} a \left (a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)+35 a^4 C\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 b^3 \left (a^2-b^2\right )^2} \\ & = \frac {\left (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \cos ^{\frac {5}{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 (5 A b^4-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{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 {-\frac {1}{8} a b \left (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 a^4 C\right )-\frac {1}{8} \left (a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 b^4 \left (a^2-b^2\right )^2}-\frac {\left (a \left (a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)+35 a^4 C\right )\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 b^4 \left (a^2-b^2\right )^2} \\ & = -\frac {a \left (a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)+35 a^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \cos ^{\frac {5}{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 (5 A b^4-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{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 \left (15 A b^6+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 b^5 \left (a^2-b^2\right )^2}+\frac {\left (a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 b^5 \left (a^2-b^2\right )^2} \\ & = -\frac {a \left (a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)+35 a^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 b^5 \left (a^2-b^2\right )^2 d}-\frac {a \left (15 A b^6+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^5 (a+b)^3 d}+\frac {\left (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 a^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \cos ^{\frac {5}{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 (5 A b^4-7 a^4 C+a^2 b^2 (A+13 C)\right ) \cos ^{\frac {3}{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*}

Mathematica [A] (verified)

Time = 5.42 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {4 \sqrt {\cos (c+d x)} \left (3 a^4 A b^2-21 a^2 A b^4+35 a^6 C-57 a^4 b^2 C+4 b^6 C+a b \left (a^2 b^2 (9 A-83 C)+49 a^4 C+b^4 (-27 A+16 C)\right ) \cos (c+d x)+4 \left (-a^2 b+b^3\right )^2 C \cos (2 (c+d x))\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac {\frac {2 \left (a^3 b^2 (3 A-73 C)+35 a^5 C+a b^4 (15 A+56 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}-\frac {16 \left (-7 a^4 C+2 b^4 (3 A+C)+a^2 b^2 (3 A+14 C)\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+\frac {6 \left (a^2 b^2 (3 A-65 C)+35 a^4 C+3 b^4 (-3 A+8 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{b^2 \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 b^3 d} \]

[In]

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

[Out]

((4*Sqrt[Cos[c + d*x]]*(3*a^4*A*b^2 - 21*a^2*A*b^4 + 35*a^6*C - 57*a^4*b^2*C + 4*b^6*C + a*b*(a^2*b^2*(9*A - 8
3*C) + 49*a^4*C + b^4*(-27*A + 16*C))*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/((
a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - ((2*(a^3*b^2*(3*A - 73*C) + 35*a^5*C + a*b^4*(15*A + 56*C))*EllipticPi[
(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (16*(-7*a^4*C + 2*b^4*(3*A + C) + a^2*b^2*(3*A + 14*C))*((a + b)*Ell
ipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(a^2*b^2*(3*A - 65*C) + 35
*a^4*C + 3*b^4*(-3*A + 8*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), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b
^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(48*b^3*d)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2239\) vs. \(2(493)=986\).

Time = 101.91 (sec) , antiderivative size = 2240, normalized size of antiderivative = 5.17

method	result	size
default	\(\text {Expression too large to display}\)	\(2240\)

[In]

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

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4/3/b^3*C*(2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*
c)-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)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ell
ipticF(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(c
os(1/2*d*x+1/2*c),2^(1/2)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)-2*C/b^4*(3*a+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)*(Elli
pticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(A*b^2+6*C*a^2+3*C*a*b+C*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+2*a^2/b^5*(3*A*b^2+5*C*a^2)*(-1/a*b^2/(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*b*cos(1/2*d*x+1/2*c)^2+a-b)-1/2/a/(a+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)*Elliptic
F(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/(a^2-b^2)*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/(a^2-b^2)*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)*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*co
s(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)))
-2*a^3*(A*b^2+C*a^2)/b^5*(-1/2/a*b^2/(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*b*cos(1/2*d*x+1/2*c)^2+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*b*cos(1/2*d*x+1/2*c)^2+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(c
os(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)*Ellip
ticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2)))+4*a/b^4*(3*A*b^2+10*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)))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]

[In]

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

[Out]

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