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

Optimal. Leaf size=523 \[ \frac{\left (-61 a^2 A b^2+8 a^4 A+33 a^3 b B-15 a b^3 B+35 A b^4\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 a^3 d \left (a^2-b^2\right )^2}+\frac{\left (-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac{b \left (-86 a^2 A b^3+63 a^4 A b+38 a^3 b^2 B-35 a^5 B-15 a b^4 B+35 A b^5\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{b \left (13 a^2 A b-9 a^3 B+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left (-61 a^2 A b^2+8 a^4 A+33 a^3 b B-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{12 a^3 d \left (a^2-b^2\right )^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{\left (-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{4 a^4 d \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)}} \]

[Out]

((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*EllipticE[(c + d*x)/2, 2])/(4*a^
4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*EllipticF[(c + d*x)/2, 2])
/(12*a^3*(a^2 - b^2)^2*d) + (b*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*E
llipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4
+ 33*a^3*b*B - 15*a*b^3*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((24*a^4*A*b - 65*a^2*A
*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]
) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) + (b*(13*a^2*A*
b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*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]
))

________________________________________________________________________________________

Rubi [A]  time = 1.95256, antiderivative size = 523, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3000, 3055, 3059, 2639, 3002, 2641, 2805} \[ \frac{\left (-61 a^2 A b^2+8 a^4 A+33 a^3 b B-15 a b^3 B+35 A b^4\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 a^3 d \left (a^2-b^2\right )^2}+\frac{\left (-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac{b \left (-86 a^2 A b^3+63 a^4 A b+38 a^3 b^2 B-35 a^5 B-15 a b^4 B+35 A b^5\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{b \left (13 a^2 A b-9 a^3 B+3 a b^2 B-7 A b^3\right ) \sin (c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left (-61 a^2 A b^2+8 a^4 A+33 a^3 b B-15 a b^3 B+35 A b^4\right ) \sin (c+d x)}{12 a^3 d \left (a^2-b^2\right )^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{b (A b-a B) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{\left (-65 a^2 A b^3+24 a^4 A b+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right ) \sin (c+d x)}{4 a^4 d \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*EllipticE[(c + d*x)/2, 2])/(4*a^
4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*EllipticF[(c + d*x)/2, 2])
/(12*a^3*(a^2 - b^2)^2*d) + (b*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*E
llipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4
+ 33*a^3*b*B - 15*a*b^3*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)) - ((24*a^4*A*b - 65*a^2*A
*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]
) + (b*(A*b - a*B)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2) + (b*(13*a^2*A*
b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*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 3000

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^2 - a*b*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*
Sin[e + f*x])^(1 + n))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[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[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m
+ n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^
2, 0] && RationalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n,
-1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 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]

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 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 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]

Rubi steps

\begin{align*} \int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx &=\frac{b (A b-a B) \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 (4 a^2 A-7 A b^2+3 a b B\right )-2 a (A b-a B) \cos (c+d x)+\frac{5}{2} b (A b-a B) \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{b (A b-a B) \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{b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\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 (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right )-a \left (4 a^2 A b-A b^3-2 a^3 B-a b^2 B\right ) \cos (c+d x)+\frac{3}{4} b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\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 (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b (A b-a B) \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{b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\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 (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right )+\frac{1}{2} a \left (2 a^4 A+14 a^2 A b^2-7 A b^4-12 a^3 b B+3 a b^3 B\right ) \cos (c+d x)+\frac{1}{8} b \left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\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 (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\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 (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)}}+\frac{b (A b-a B) \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{b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\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 (8 a^6 A+128 a^4 A b^2-223 a^2 A b^4+105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B\right )+\frac{1}{4} a \left (20 a^4 A b-64 a^2 A b^3+35 A b^5-6 a^5 B+30 a^3 b^2 B-15 a b^4 B\right ) \cos (c+d x)+\frac{3}{16} b \left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\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 (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\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 (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)}}+\frac{b (A b-a B) \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{b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\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 (8 a^6 A+128 a^4 A b^2-223 a^2 A b^4+105 A b^6-72 a^5 b B+99 a^3 b^3 B-45 a b^5 B\right )-\frac{1}{16} a b^2 \left (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\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 (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\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 (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\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 (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)}}+\frac{b (A b-a B) \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{b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\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 (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )^2}+\frac{\left (b \left (63 a^4 A b-86 a^2 A b^3+35 A b^5-35 a^5 B+38 a^3 b^2 B-15 a b^4 B\right )\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 (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\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 (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\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{b \left (63 a^4 A b-86 a^2 A b^3+35 A b^5-35 a^5 B+38 a^3 b^2 B-15 a b^4 B\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 (8 a^4 A-61 a^2 A b^2+35 A b^4+33 a^3 b B-15 a b^3 B\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 (24 a^4 A b-65 a^2 A b^3+35 A b^5-8 a^5 B+29 a^3 b^2 B-15 a b^4 B\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)}}+\frac{b (A b-a B) \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{b \left (13 a^2 A b-7 A b^3-9 a^3 B+3 a b^2 B\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 = 7.16701, size = 574, normalized size = 1.1 \[ \frac{\sqrt{\cos (c+d x)} \left (\frac{A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{17 a^2 A b^4 \sin (c+d x)-13 a^3 b^3 B \sin (c+d x)+7 a b^5 B \sin (c+d x)-11 A b^6 \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{2 \sec (c+d x) (a B \sin (c+d x)-3 A b \sin (c+d x))}{a^4}+\frac{2 A \tan (c+d x) \sec (c+d x)}{3 a^3}\right )}{d}+\frac{\frac{2 \left (328 a^4 A b^2-641 a^2 A b^4+16 a^6 A+285 a^3 b^3 B-168 a^5 b B-135 a b^5 B+315 A b^6\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+\frac{\left (-512 a^3 A b^3+160 a^5 A b+240 a^4 b^2 B-120 a^2 b^4 B-48 a^6 B+280 a A b^5\right ) \left (2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-\frac{2 a \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}\right )}{b}+\frac{2 \left (-195 a^2 A b^4+72 a^4 A b^2+87 a^3 b^3 B-24 a^5 b B-45 a b^5 B+105 A b^6\right ) \sin (c+d x) \cos (2 (c+d x)) \left (\left (2 a^2-b^2\right ) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) F\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a b^2 \sqrt{1-\cos ^2(c+d x)} \left (2 \cos ^2(c+d x)-1\right )}}{48 a^4 d (a-b)^2 (a+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cos[c + d*x])/(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)*Ellipti
cPi[(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)*(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)*Cos[2*(c + d*x)]
*(-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])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2
]*(-1 + 2*Cos[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])/(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])/(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

________________________________________________________________________________________

Maple [B]  time = 27.283, size = 2158, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c))/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*(-3*A*b+B*a)/a^4*(-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(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))+2*
(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(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*b*(2*A*b-B*a)/a^3*(-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)*EllipticF(cos(1/2*d*x+1/2*c
),2^(1/2))-1/2*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))+1/2*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)*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)))+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)/(cos(1/2*d*x+1/2*c)^2-1/2)^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)*Ell
ipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(A*b-B*a)*b/a^2*(-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*c
os(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)*El
lipticF(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))))/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., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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