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

Optimal. Leaf size=532 \[ -\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {\left (4 a^2 A-20 a b B+35 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (-12 a^3 B+27 a^2 A b+20 a b^2 B-35 A b^3\right ) \sin (c+d x)}{12 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\left (-12 a^3 B+27 a^2 A b+20 a b^2 B-35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^3 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {b \left (-12 a^5 B+33 a^4 A b+104 a^3 b^2 B-170 a^2 A b^3-60 a b^4 B+105 A b^5\right ) \sin (c+d x)}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {\left (-12 a^5 B+33 a^4 A b+104 a^3 b^2 B-170 a^2 A b^3-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \]

[Out]

-1/12*b*(27*A*a^2*b-35*A*b^3-12*B*a^3+20*B*a*b^2)*sin(d*x+c)/a^3/(a^2-b^2)/d/(a+b*cos(d*x+c))^(3/2)-1/12*b*(33
*A*a^4*b-170*A*a^2*b^3+105*A*b^5-12*B*a^5+104*B*a^3*b^2-60*B*a*b^4)*sin(d*x+c)/a^4/(a^2-b^2)^2/d/(a+b*cos(d*x+
c))^(1/2)+1/12*(33*A*a^4*b-170*A*a^2*b^3+105*A*b^5-12*B*a^5+104*B*a^3*b^2-60*B*a*b^4)*(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/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/a^4/(a^2-
b^2)^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-1/12*(27*A*a^2*b-35*A*b^3-12*B*a^3+20*B*a*b^2)*(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/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/
a^3/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)+1/4*(4*A*a^2+35*A*b^2-20*B*a*b)*(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),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/a^4/d/(a+b*co
s(d*x+c))^(1/2)-1/4*(7*A*b-4*B*a)*tan(d*x+c)/a^2/d/(a+b*cos(d*x+c))^(3/2)+1/2*A*sec(d*x+c)*tan(d*x+c)/a/d/(a+b
*cos(d*x+c))^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 1.93, antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3000, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {b \left (-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right ) \sin (c+d x)}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {b \left (27 a^2 A b-12 a^3 B+20 a b^2 B-35 A b^3\right ) \sin (c+d x)}{12 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\left (27 a^2 A b-12 a^3 B+20 a b^2 B-35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^3 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\left (-170 a^2 A b^3+33 a^4 A b+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (4 a^2 A-20 a b B+35 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Ell
ipticE[(c + d*x)/2, (2*b)/(a + b)])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((27*a^2*A*b
- 35*A*b^3 - 12*a^3*B + 20*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])
/(12*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*A + 35*A*b^2 - 20*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/
(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(4*a^4*d*Sqrt[a + b*Cos[c + d*x]]) - (b*(27*a^2*A*b - 35*A
*b^3 - 12*a^3*B + 20*a*b^2*B)*Sin[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(33*a^4*A*b
- 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*Sin[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sq
rt[a + b*Cos[c + d*x]]) - ((7*A*b - 4*a*B)*Tan[c + d*x])/(4*a^2*d*(a + b*Cos[c + d*x])^(3/2)) + (A*Sec[c + d*x
]*Tan[c + d*x])/(2*a*d*(a + b*Cos[c + d*x])^(3/2))

Rule 2653

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

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

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

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

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 2807

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

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 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)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx &=\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {1}{2} (-7 A b+4 a B)+a A \cos (c+d x)+\frac {5}{2} A b \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{2 a}\\ &=-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {1}{4} \left (4 a^2 A+35 A b^2-20 a b B\right )+\frac {5}{2} a A b \cos (c+d x)-\frac {3}{4} b (7 A b-4 a B) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{2 a^2}\\ &=-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {3}{8} \left (a^2-b^2\right ) \left (4 a^2 A+35 A b^2-20 a b B\right )+\frac {3}{4} a b \left (3 a^2 A-7 A b^2+4 a b B\right ) \cos (c+d x)-\frac {1}{8} b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 a^3 \left (a^2-b^2\right )}\\ &=-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {2 \int \frac {\left (\frac {3}{16} \left (a^2-b^2\right )^2 \left (4 a^2 A+35 A b^2-20 a b B\right )+\frac {1}{8} a b \left (3 a^4 A-54 a^2 A b^2+35 A b^4+36 a^3 b B-20 a b^3 B\right ) \cos (c+d x)+\frac {1}{16} b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {2 \int \frac {\left (-\frac {3}{16} b \left (a^2-b^2\right )^2 \left (4 a^2 A+35 A b^2-20 a b B\right )+\frac {1}{16} a b \left (a^2-b^2\right ) \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^4 b \left (a^2-b^2\right )^2}+\frac {\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{24 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\left (4 a^2 A+35 A b^2-20 a b B\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{8 a^4}-\frac {\left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )}+\frac {\left (\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{24 a^4 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\\ &=\frac {\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\left (\left (4 a^2 A+35 A b^2-20 a b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{8 a^4 \sqrt {a+b \cos (c+d x)}}-\frac {\left (\left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{24 a^3 \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\\ &=\frac {\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (4 a^2 A+35 A b^2-20 a b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 7.95, size = 820, normalized size = 1.54 \[ \frac {\frac {2 \left (12 A b a^5+144 b^2 B a^4-216 A b^3 a^3-80 b^4 B a^2+140 A b^5 a\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (24 A a^6-132 b B a^5+195 A b^2 a^4+344 b^3 B a^3-566 A b^4 a^2-180 b^5 B a+315 A b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (105 A b^6-60 a B b^5-170 a^2 A b^4+104 a^3 B b^3+33 a^4 A b^2-12 a^5 B b\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {\cos (c+d x) b+b}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-2 (a+b \cos (c+d x)) a-b^2+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-4 (a+b \cos (c+d x)) a-b^2+2 (a+b \cos (c+d x))^2\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {\sec (c+d x) (4 a B \sin (c+d x)-11 A b \sin (c+d x))}{4 a^4}-\frac {2 \left (a b^3 B \sin (c+d x)-A b^4 \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {2 \left (9 A \sin (c+d x) b^6-6 a B \sin (c+d x) b^5-13 a^2 A \sin (c+d x) b^4+10 a^3 B \sin (c+d x) b^3\right )}{3 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a^3}\right )}{d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((2*(12*a^5*A*b - 216*a^3*A*b^3 + 140*a*A*b^5 + 144*a^4*b^2*B - 80*a^2*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b
)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(24*a^6*A + 195*a^4*A*b^2 - 566*a^2*A*
b^4 + 315*A*b^6 - 132*a^5*b*B + 344*a^3*b^3*B - 180*a*b^5*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2,
(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(33*a^4*A*b^2 - 170*a^2*A*b^4 + 105*A*b^6 - 12*
a^5*b*B + 104*a^3*b^3*B - 60*a*b^5*B)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]
*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a -
b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*Elliptic
Pi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqr
t[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^
2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(48*a^4*(a - b)^2*(a + b)^2*d)
+ (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(-11*A*b*Sin[c + d*x] + 4*a*B*Sin[c + d*x]))/(4*a^4) - (2*(-(A*b^4*
Sin[c + d*x]) + a*b^3*B*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(-13*a^2*A*b^4*Sin[c +
d*x] + 9*A*b^6*Sin[c + d*x] + 10*a^3*b^3*B*Sin[c + d*x] - 6*a*b^5*B*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(a + b
*Cos[c + d*x])) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a^3)))/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((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 9.71, size = 2000, normalized size = 3.76 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b^2*(3*A*b-2*B*a)/a^4/sin(1/2*d*x+1/2*c)^2/
(-2*sin(1/2*d*x+1/2*c)^2*b+a+b)/(a^2-b^2)*(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((sin(1
/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/
(a-b))^(1/2))*a-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos
(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b+2*b*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2)-2*b*(3*A*b-2*B*a)/a^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)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*
d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-2*(A*b-B*a)*b^2/a^3*(1/6/b/(a-b)/(a+b)
*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2/b*(
a-b))^2+8/3*b*sin(1/2*d*x+1/2*c)^2/(a-b)^2/(a+b)^2*cos(1/2*d*x+1/2*c)*a/(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(
1/2*d*x+1/2*c)^2)^(1/2)+(3*a-b)/(3*a^3+3*a^2*b-3*a*b^2-3*b^3)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2
*c)^2*b+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1
/2*c),(-2*b/(a-b))^(1/2))-4/3*a/(a-b)/(a+b)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-
b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-
b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))))+2*A/a^2*(-1/2/a*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d
*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)^2+3/4*b/a^2*cos(1/2*d*x+1/2*c)*(-2*
sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)-1/8*b/a*(sin(1/2*d*x+1/2*c
)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)
^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+3/8/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2
*c)^2*b+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*EllipticE(cos(1/2*d*x
+1/2*c),(-2*b/(a-b))^(1/2))-3/8*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*b+a-b)/(a-b))^(1
/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/
2))-1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(
a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-3/8/a^2*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2
)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^2)+2*(-2*A*b+B*a)/a^3*(-1/a*cos(1/2*d*x+1/2*c)*(
-2*sin(1/2*d*x+1/2*c)^4*b+(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*(sin(1/2*d*x+1/2*c)
^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^
(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)
^2*b+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*
c),(-2*b/(a-b))^(1/2))+1/2/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*sin
(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/2/a
*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*
sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))))/sin(1/2*d*x+1/2*c)/(-2*sin(1
/2*d*x+1/2*c)^2*b+a+b)^(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((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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