3.1059 \(\int \frac{(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=572 \[ -\frac{b \sin (c+d x) \left (-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}+\frac{b \sin (c+d x) \left (-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right )}{12 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{\left (-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{12 a^3 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{\left (-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{4 a^4 d \sqrt{a+b \cos (c+d x)}}-\frac{(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \]

[Out]

((105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) - 2*a^2*b^3*(85*A - 12*C))*Sqrt[a +
b*Cos[c + d*x]]*EllipticE[(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)]) + ((35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*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]]) + ((35*A*b^2 - 20*a*b*B + 4*a^2*
(A + 2*C))*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*C
os[c + d*x]]) + (b*(35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)
*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) -
2*a^2*b^3*(85*A - 12*C))*Sin[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((7*A*b - 4*a*B)*T
an[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))

________________________________________________________________________________________

Rubi [A]  time = 2.31224, antiderivative size = 572, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{b \sin (c+d x) \left (-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt{a+b \cos (c+d x)}}+\frac{b \sin (c+d x) \left (-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right )}{12 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{\left (-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{12 a^3 d \left (a^2-b^2\right ) \sqrt{a+b \cos (c+d x)}}+\frac{\left (-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right ) \sqrt{a+b \cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{4 a^4 d \sqrt{a+b \cos (c+d x)}}-\frac{(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) - 2*a^2*b^3*(85*A - 12*C))*Sqrt[a +
b*Cos[c + d*x]]*EllipticE[(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)]) + ((35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*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]]) + ((35*A*b^2 - 20*a*b*B + 4*a^2*
(A + 2*C))*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*C
os[c + d*x]]) + (b*(35*A*b^3 + 12*a^3*B - 20*a*b^2*B - a^2*(27*A*b - 8*b*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)
*d*(a + b*Cos[c + d*x])^(3/2)) - (b*(105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B + a^4*b*(33*A - 56*C) -
2*a^2*b^3*(85*A - 12*C))*Sin[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((7*A*b - 4*a*B)*T
an[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 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 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 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 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 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 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 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 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{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \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+2 C) \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 (35 A b^2-20 a b B+4 a^2 (A+2 C)\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 (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (35 A b^2-20 a b B+4 a^2 (A+2 C)\right )-\frac{3}{4} a b \left (7 A b^2-4 a b B-a^2 (3 A-4 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\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 (35 A b^2-20 a b B+4 a^2 (A+2 C)\right )+\frac{1}{8} a b \left (35 A b^4+36 a^3 b B-20 a b^3 B+3 a^4 (A-8 C)-2 a^2 b^2 (27 A-4 C)\right ) \cos (c+d x)+\frac{1}{16} b \left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B-2 a^2 b^3 (85 A-12 C)+a^4 (33 A b-56 b C)\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 (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\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 (35 A b^2-20 a b B+4 a^2 (A+2 C)\right )-\frac{1}{16} a b \left (a^2-b^2\right ) \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \int \sqrt{a+b \cos (c+d x)} \, dx}{24 a^4 \left (a^2-b^2\right )^2}\\ &=\frac{b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\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 (35 A b^2-20 a b B+4 a^2 (A+2 C)\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx}{8 a^4}+\frac{\left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )}+\frac{\left (\left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\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 (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\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 (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\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 (35 A b^2-20 a b B+4 a^2 (A+2 C)\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 (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\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 (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (35 A b^2-20 a b B+4 a^2 (A+2 C)\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 (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\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 (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\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*}

Mathematica [C]  time = 7.76397, size = 922, normalized size = 1.61 \[ \frac{\frac{2 \left (12 A b a^5-96 b C a^5+144 b^2 B a^4-216 A b^3 a^3+32 b^3 C a^3-80 b^4 B a^2+140 A b^5 a\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left (24 A a^6+48 C a^6-132 b B a^5+195 A b^2 a^4-152 b^2 C a^4+344 b^3 B a^3-566 A b^4 a^2+72 b^4 C a^2-180 b^5 B a+315 A b^6\right ) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right )}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left (105 A b^6-60 a B b^5-170 a^2 A b^4+24 a^2 C b^4+104 a^3 B b^3+33 a^4 A b^2-56 a^4 C b^2-12 a^5 B b\right ) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{\cos (c+d x) b+b}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 (a+b \cos (c+d x)) a-b^2+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-4 (a+b \cos (c+d x)) a-b^2+2 (a+b \cos (c+d x))^2\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{a+b \cos (c+d x)} \left (\frac{\sec (c+d x) (4 a B \sin (c+d x)-11 A b \sin (c+d x))}{4 a^4}+\frac{2 \left (A \sin (c+d x) b^4-a B \sin (c+d x) b^3+a^2 C \sin (c+d x) b^2\right )}{3 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{2 \left (-9 A \sin (c+d x) b^6+6 a B \sin (c+d x) b^5+13 a^2 A \sin (c+d x) b^4-3 a^2 C \sin (c+d x) b^4-10 a^3 B \sin (c+d x) b^3+7 a^4 C \sin (c+d x) b^2\right )}{3 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{A \sec (c+d x) \tan (c+d x)}{2 a^3}\right )}{d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*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 - 96*a^5*b*C + 32*a^3*b^3*C)*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 + 48*a^6*C - 152*a^4*b
^2*C + 72*a^2*b^4*C)*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 - 5
6*a^4*b^2*C + 24*a^2*b^4*C)*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*EllipticPi[(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*Sqrt[-(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] + a^2*b^2*C*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] + 7*a^4*b^2*C*Sin[
c + d*x] - 3*a^2*b^4*C*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

________________________________________________________________________________________

Maple [B]  time = 7.628, size = 2019, normalized size = 3.5 \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)+C*cos(d*x+c)^2)*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*(3*A*b^2-2*B*a*b+C*a^2)/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*b*sin(1/2*d*x+1/2*c)^4+(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)*Ellip
ticE(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*(A*b^2-B*a*b+C*a^
2)*b/a^3*(1/6/b/(a-b)/(a+b)*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(c
os(1/2*d*x+1/2*c)^2+1/2*(a-b)/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*b*sin(1/2*d*x+1/2*c)^4+(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)^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*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(c
os(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*(-2*A*b+B*a)/a^3*(-
1/a*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(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*b*sin(1/2*d*x+1/2*c)^4+(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*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*El
lipticE(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*b*sin(1/2*d*x+1/2*c)^4+(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*b*sin
(1/2*d*x+1/2*c)^4+(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/a
^2*(-1/2/a*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(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*b*sin(1/2*d*x+1/2*c)^4+(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*b*sin(1
/2*d*x+1/2*c)^4+(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*b*sin(1/2*d*x+1/2*c)^4+(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*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*El
lipticE(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*b*sin(1/2*d*x+1/2*c)^4+(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*b*sin(
1/2*d*x+1/2*c)^4+(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*
(3*A*b^2-2*B*a*b+C*a^2)/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*b*si
n(1/2*d*x+1/2*c)^4+(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., 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)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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