3.1501 \(\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=714 \[ -\frac{\sin (c+d x) \left (-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right )}{20 b^3 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left (-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{12 b^4 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left (-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right )}{12 b^6 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac{a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3} \]

[Out]

-((175*a^5*b*B - 325*a^3*b^3*B + 120*a*b^5*B + a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C
- 8*b^6*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(20*b^5*(a^2 - b^2)^2*d)
+ ((105*a^6*b*B - 223*a^4*b^3*B + 128*a^2*b^5*B + 8*b^7*B + 3*a^3*b^4*(33*A - 64*C) - 9*a^5*b^2*(5*A - 43*C)
- 189*a^7*C - 24*a*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^6*(a^
2 - b^2)^2*d) + (a^2*(35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(
A - 10*C) + 63*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a -
b)^2*b^6*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[
c + d*x]^(7/2)) + ((7*A*b^4 + 5*a^3*b*B - 11*a*b^3*B - a^2*b^2*(A - 15*C) - 9*a^4*C)*Sin[c + d*x])/(4*b^2*(a^2
- b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) - ((35*a^3*b*B - 65*a*b^3*B - a^2*b^2*(15*A - 101*C) + b^
4*(45*A - 8*C) - 63*a^4*C)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) + ((35*a^4*b*B - 61*a^2*b
^3*B + 8*b^5*B + 3*a*b^4*(11*A - 8*C) - 15*a^3*b^2*(A - 7*C) - 63*a^5*C)*Sin[c + d*x])/(12*b^4*(a^2 - b^2)^2*d
*Sqrt[Sec[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 3.04849, antiderivative size = 714, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4221, 3047, 3049, 3059, 2639, 3002, 2641, 2805} \[ -\frac{\sin (c+d x) \left (-a^2 b^2 (15 A-101 C)+35 a^3 b B-63 a^4 C-65 a b^3 B+b^4 (45 A-8 C)\right )}{20 b^3 d \left (a^2-b^2\right )^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left (-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+35 a^4 b B-63 a^5 C+3 a b^4 (11 A-8 C)+8 b^5 B\right )}{12 b^4 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left (-a^2 b^2 (A-15 C)+5 a^3 b B-9 a^4 C-11 a b^3 B+7 A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-9 a^5 b^2 (5 A-43 C)+3 a^3 b^4 (33 A-64 C)-223 a^4 b^3 B+128 a^2 b^5 B+105 a^6 b B-189 a^7 C-24 a b^6 (3 A+C)+8 b^7 B\right )}{12 b^6 d \left (a^2-b^2\right )^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-325 a^3 b^3 B+175 a^5 b B-315 a^6 C+120 a b^5 B-8 b^6 (5 A+3 C)\right )}{20 b^5 d \left (a^2-b^2\right )^2}+\frac{a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+86 a^3 b^3 B-35 a^5 b B+63 a^6 C-63 a b^5 B+35 A b^6\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^6 d (a-b)^2 (a+b)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]

[Out]

-((175*a^5*b*B - 325*a^3*b^3*B + 120*a*b^5*B + a^2*b^4*(145*A - 192*C) - 3*a^4*b^2*(25*A - 187*C) - 315*a^6*C
- 8*b^6*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(20*b^5*(a^2 - b^2)^2*d)
+ ((105*a^6*b*B - 223*a^4*b^3*B + 128*a^2*b^5*B + 8*b^7*B + 3*a^3*b^4*(33*A - 64*C) - 9*a^5*b^2*(5*A - 43*C)
- 189*a^7*C - 24*a*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^6*(a^
2 - b^2)^2*d) + (a^2*(35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B - a^2*b^4*(38*A - 99*C) + 15*a^4*b^2*(
A - 10*C) + 63*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a -
b)^2*b^6*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2*Sec[
c + d*x]^(7/2)) + ((7*A*b^4 + 5*a^3*b*B - 11*a*b^3*B - a^2*b^2*(A - 15*C) - 9*a^4*C)*Sin[c + d*x])/(4*b^2*(a^2
- b^2)^2*d*(a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)) - ((35*a^3*b*B - 65*a*b^3*B - a^2*b^2*(15*A - 101*C) + b^
4*(45*A - 8*C) - 63*a^4*C)*Sin[c + d*x])/(20*b^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) + ((35*a^4*b*B - 61*a^2*b
^3*B + 8*b^5*B + 3*a*b^4*(11*A - 8*C) - 15*a^3*b^2*(A - 7*C) - 63*a^5*C)*Sin[c + d*x])/(12*b^4*(a^2 - b^2)^2*d
*Sqrt[Sec[c + d*x]])

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 3047

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 3049

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 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)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (\frac{7}{2} \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \cos (c+d x)-\frac{1}{2} \left (5 A b^2-5 a b B+9 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 (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{5}{4} \left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \cos (c+d x)-\frac{1}{4} \left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 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 b^2-a (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)} \left (-\frac{3}{8} a \left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right )+\frac{1}{2} b \left (5 a^3 b B-20 a b^3 B-9 a^4 C+2 b^4 (5 A+3 C)+a^2 b^2 (5 A+18 C)\right ) \cos (c+d x)+\frac{5}{8} \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{5 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{16} a \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right )-\frac{1}{4} b \left (35 a^4 b B-70 a^2 b^3 B-10 b^5 B-3 a^3 b^2 (5 A-32 C)-63 a^5 C+12 a b^4 (5 A+C)\right ) \cos (c+d x)-\frac{3}{16} \left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}-\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{5}{16} a b \left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right )-\frac{5}{16} \left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^5 \left (a^2-b^2\right )^2}-\frac{\left (\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{40 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{20 b^5 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}+\frac{\left (a^2 \left (35 A b^6-35 a^5 b B+86 a^3 b^3 B-63 a b^5 B-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 b^6 \left (a^2-b^2\right )^2}+\frac{\left (\left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{24 b^6 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (175 a^5 b B-325 a^3 b^3 B+120 a b^5 B+a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac{\left (105 a^6 b B-223 a^4 b^3 B+128 a^2 b^5 B+8 b^7 B+3 a^3 b^4 (33 A-64 C)-9 a^5 b^2 (5 A-43 C)-189 a^7 C-24 a b^6 (3 A+C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{12 b^6 \left (a^2-b^2\right )^2 d}+\frac{a^2 \left (35 A b^6-35 a^5 b B+86 a^3 b^3 B-63 a b^5 B-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{4 (a-b)^2 b^6 (a+b)^3 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (7 A b^4+5 a^3 b B-11 a b^3 B-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (35 a^3 b B-65 a b^3 B-a^2 b^2 (15 A-101 C)+b^4 (45 A-8 C)-63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (35 a^4 b B-61 a^2 b^3 B+8 b^5 B+3 a b^4 (11 A-8 C)-15 a^3 b^2 (A-7 C)-63 a^5 C\right ) \sin (c+d x)}{12 b^4 \left (a^2-b^2\right )^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 7.9263, size = 1065, normalized size = 1.49 \[ \frac{-\frac{2 \left (80 B b^6-480 a A b^5-96 a C b^5+560 a^2 B b^4+120 a^3 A b^3-768 a^3 C b^3-280 a^4 B b^2+504 a^5 C b\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{2 \left (315 C a^6-175 b B a^5+75 A b^2 a^4-633 b^2 C a^4+365 b^3 B a^3-105 A b^4 a^2+336 b^4 C a^2-280 b^5 B a+120 A b^6+72 b^6 C\right ) \left (F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right ) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{\left (945 C a^6-525 b B a^5+225 A b^2 a^4-1683 b^2 C a^4+975 b^3 B a^3-435 A b^4 a^2+576 b^4 C a^2-360 b^5 B a+120 A b^6+72 b^6 C\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (4 \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}-2 b^2 \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{240 (a-b)^2 b^4 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left (-\frac{\left (75 C a^6-55 b B a^5+35 A b^2 a^4-107 b^2 C a^4+85 b^3 B a^3-65 A b^4 a^2+4 b^4 C a^2-2 b^6 C\right ) \sin (c+d x)}{20 b^5 \left (a^2-b^2\right )^2}-\frac{-C \sin (c+d x) a^6+b B \sin (c+d x) a^5-A b^2 \sin (c+d x) a^4}{2 b^5 \left (b^2-a^2\right ) (a+b \cos (c+d x))^2}+\frac{17 C \sin (c+d x) a^7-13 b B \sin (c+d x) a^6+9 A b^2 \sin (c+d x) a^5-23 b^2 C \sin (c+d x) a^5+19 b^3 B \sin (c+d x) a^4-15 A b^4 \sin (c+d x) a^3}{4 b^5 \left (b^2-a^2\right )^2 (a+b \cos (c+d x))}+\frac{(b B-3 a C) \sin (2 (c+d x))}{3 b^4}+\frac{C \sin (3 (c+d x))}{10 b^3}\right )}{d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]

[Out]

((-2*(120*a^3*A*b^3 - 480*a*A*b^5 - 280*a^4*b^2*B + 560*a^2*b^4*B + 80*b^6*B + 504*a^5*b*C - 768*a^3*b^3*C - 9
6*a*b^5*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), -ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Se
c[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(75*a^4*A*b^2 - 105*a^2*A*b^4 +
120*A*b^6 - 175*a^5*b*B + 365*a^3*b^3*B - 280*a*b^5*B + 315*a^6*C - 633*a^4*b^2*C + 336*a^2*b^4*C + 72*b^6*C)
*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + EllipticPi[-(a/b), -ArcSin[Sqrt[Sec[c + d*x]]], -
1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2))
+ ((225*a^4*A*b^2 - 435*a^2*A*b^4 + 120*A*b^6 - 525*a^5*b*B + 975*a^3*b^3*B - 360*a*b^5*B + 945*a^6*C - 1683*a
^4*b^2*C + 576*a^2*b^4*C + 72*b^6*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*
a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*Elli
pticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 4*a^2*EllipticPi[-(a/b), -
ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*b^2*EllipticPi[-(a/b), -ArcSin
[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x
])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(240*(a - b)^2*b^4*(a + b)^2*d) + (Sqrt[Sec[
c + d*x]]*(-((35*a^4*A*b^2 - 65*a^2*A*b^4 - 55*a^5*b*B + 85*a^3*b^3*B + 75*a^6*C - 107*a^4*b^2*C + 4*a^2*b^4*C
- 2*b^6*C)*Sin[c + d*x])/(20*b^5*(a^2 - b^2)^2) - (-(a^4*A*b^2*Sin[c + d*x]) + a^5*b*B*Sin[c + d*x] - a^6*C*S
in[c + d*x])/(2*b^5*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (9*a^5*A*b^2*Sin[c + d*x] - 15*a^3*A*b^4*Sin[c + d*
x] - 13*a^6*b*B*Sin[c + d*x] + 19*a^4*b^3*B*Sin[c + d*x] + 17*a^7*C*Sin[c + d*x] - 23*a^5*b^2*C*Sin[c + d*x])/
(4*b^5*(-a^2 + b^2)^2*(a + b*Cos[c + d*x])) + ((b*B - 3*a*C)*Sin[2*(c + d*x)])/(3*b^4) + (C*Sin[3*(c + d*x)])/
(10*b^3)))/d

________________________________________________________________________________________

Maple [B]  time = 9.429, size = 2520, 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)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x)

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4/5*C/b^3*(-4*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2
*c)+14*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)
*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-9*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2
*sin(1/2*d*x+1/2*c)^2-1)^(1/2)-6*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)^4+sin(1/2*d*x
+1/2*c)^2)^(1/2)+4/3/b^4*(B*b-3*C*a-3*C*b)*(2*sin(1/2*d*x+1/2*c)^4*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3*EllipticE(cos(1/2*d*x+1/2*c),2
^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)-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)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2/b^5*(A*b^2-3*B*a*b-2*B*b^2+6*C*a^2+6*C*a*b+3*C*b^2)*(s
in(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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-2*(3*A*a*b^2+A*b^3-6*B*a^2*b
-3*B*a*b^2-B*b^3+10*C*a^3+6*C*a^2*b+3*C*a*b^2+C*b^3)/b^6*(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))-4/b^5*a^2
*(6*A*b^2-10*B*a*b+15*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))+2*a^4*(A*b
^2-B*a*b+C*a^2)/b^6*(-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*cos(1/2*d*x+1/2*c)^2*b+a-b)^2-3/4*b^2*(3*a^2-b^2)/a^2/(a^2-b^2)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/
2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*b+a-b)-7/8/(a+b)/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(
1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*
d*x+1/2*c),2^(1/2))+1/4/(a+b)/(a^2-b^2)/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*s
in(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b+3/8/(a+b)/(a^2-b^2)/a^
2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2
)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b^2-9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d
*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)
)+3/8*b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)
^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/
2*d*x+1/2*c),2^(1/2))-3/8*b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(
-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-15/4*a^2/(a^2-b^2)^2
/(-2*a*b+2*b^2)*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(
1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))+3/2/(a^2-b^2)^2/(-2*a*b+2*b^2)*b^3*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(
1/2)*EllipticPi(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)))-2/b^6*a^3*(4*A*b^2-5*B*a*b+6*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*cos(1/2*d*x+1/2*c)^2*b+a-b)-1/2/(a+b)/a*(sin(1/2
*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*El
lipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2/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))+1/2/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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-3*a/(a^2-b^2)/(-2*a*b+2*b^2)*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*
(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+
1/2*c),-2*b/(a-b),2^(1/2))+1/a/(a^2-b^2)/(-2*a*b+2*b^2)*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c
)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(
1/2))))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., 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)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),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)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),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)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**3/sec(d*x+c)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{7}{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)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)^3*sec(d*x + c)^(7/2)), x)