\(\int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx\) [1410]

Optimal result

Integrand size = 35, antiderivative size = 579 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {\left (a^2 b^4 (145 A-192 C)-3 a^4 b^2 (25 A-187 C)-315 a^6 C-8 b^6 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{20 b^5 \left (a^2-b^2\right )^2 d}+\frac {a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 b^6 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (35 A b^6-a^2 b^4 (38 A-99 C)+15 a^4 b^2 (A-10 C)+63 a^6 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^6 (a+b)^3 d}-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (7 A b^4-a^2 b^2 (A-15 C)-9 a^4 C\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right ) \sin (c+d x)}{20 b^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \]

[Out]

-1/2*(A*b^2+C*a^2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2/sec(d*x+c)^(7/2)+1/4*(7*A*b^4-a^2*b^2*(A-15*C)-
9*a^4*C)*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))/sec(d*x+c)^(5/2)+1/20*(a^2*b^2*(15*A-101*C)-b^4*(45*A-8
*C)+63*a^4*C)*sin(d*x+c)/b^3/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)+1/4*a*(b^4*(11*A-8*C)-5*a^2*b^2*(A-7*C)-21*a^4*C)*
sin(d*x+c)/b^4/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)-1/20*(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))*(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))*cos(d*x+c)
^(1/2)*sec(d*x+c)^(1/2)/b^5/(a^2-b^2)^2/d+1/4*a*(a^2*b^4*(33*A-64*C)-3*a^4*b^2*(5*A-43*C)-63*a^6*C-8*b^6*(3*A+
C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec
(d*x+c)^(1/2)/b^6/(a^2-b^2)^2/d+1/4*a^2*(35*A*b^6-a^2*b^4*(38*A-99*C)+15*a^4*b^2*(A-10*C)+63*a^6*C)*(cos(1/2*d
*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*
x+c)^(1/2)/(a-b)^2/b^6/(a+b)^3/d

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4306, 3127, 3126, 3128, 3138, 2719, 3081, 2720, 2884} \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{2 b d \left (a^2-b^2\right ) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac {a \left (-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right ) \sin (c+d x)}{4 b^4 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)}}+\frac {\left (-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right ) \sin (c+d x)}{4 b^2 d \left (a^2-b^2\right )^2 \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac {\left (63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right ) \sin (c+d x)}{20 b^3 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x)}+\frac {a \left (-63 a^6 C-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 b^6 d \left (a^2-b^2\right )^2}+\frac {a^2 \left (63 a^6 C+15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+35 A b^6\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 b^6 d (a-b)^2 (a+b)^3}-\frac {\left (-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{20 b^5 d \left (a^2-b^2\right )^2} \]

[In]

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

[Out]

-1/20*((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]])/(b^5*(a^2 - b^2)^2*d) + (a*(a^2*b^4*(33*A - 64*C) - 3*a^4*b^2*(
5*A - 43*C) - 63*a^6*C - 8*b^6*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*
b^6*(a^2 - b^2)^2*d) + (a^2*(35*A*b^6 - 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^2*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 - 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)) + ((a^2*b^2*(1
5*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)) + (a*(b^
4*(11*A - 8*C) - 5*a^2*b^2*(A - 7*C) - 21*a^4*C)*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])

Rule 2719

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

Rule 2720

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

Rule 2884

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

Rule 3081

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3126

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) +
(c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*
c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n +
1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3127

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Si
n[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n
+ 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2
*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d
^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3138

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 4306

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]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx \\ & = -\frac {\left (A b^2+a^2 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^2 C\right )-2 a b (A+C) \cos (c+d x)-\frac {1}{2} \left (5 A b^2+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^2 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-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-a^2 b^2 (A-15 C)-9 a^4 C\right )-a b \left (3 A b^2-\left (a^2-4 b^2\right ) C\right ) \cos (c+d x)+\frac {1}{4} \left (a^2 b^2 (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^2 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-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 (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 (a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)+63 a^4 C\right )-\frac {1}{2} b \left (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 {15}{8} a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 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^2 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-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 (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 {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 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 {15}{16} a^2 \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right )+\frac {3}{4} a b \left (a^2 b^2 (5 A-32 C)+21 a^4 C-4 b^4 (5 A+C)\right ) \cos (c+d x)-\frac {3}{16} \left (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^2 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-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 (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 {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 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 {15}{16} a^2 b \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right )-\frac {15}{16} a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^5 \left (a^2-b^2\right )^2}-\frac {\left (\left (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 (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^2 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-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 (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 {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (a^2 \left (35 A b^6-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 (a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 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}{8 b^6 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (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 {a \left (a^2 b^4 (33 A-64 C)-3 a^4 b^2 (5 A-43 C)-63 a^6 C-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 b^6 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (35 A b^6-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)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^6 (a+b)^3 d}-\frac {\left (A b^2+a^2 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-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 (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 {a \left (b^4 (11 A-8 C)-5 a^2 b^2 (A-7 C)-21 a^4 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 7.88 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.60 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {\frac {2 \left (25 a^4 A b^2-35 a^2 A b^4+40 A b^6+105 a^6 C-211 a^4 b^2 C+112 a^2 b^4 C+24 b^6 C\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (40 a^3 A b^3-160 a A b^5+168 a^5 b C-256 a^3 b^3 C-32 a b^5 C\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (75 a^4 A b^2-145 a^2 A b^4+40 A b^6+315 a^6 C-561 a^4 b^2 C+192 a^2 b^4 C+24 b^6 C\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 (2 a-b) b \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 a^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\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 )}}{80 (a-b)^2 b^4 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (-\frac {\left (35 a^4 A b^2-65 a^2 A b^4+75 a^6 C-107 a^4 b^2 C+4 a^2 b^4 C-2 b^6 C\right ) \sin (c+d x)}{20 b^5 \left (a^2-b^2\right )^2}-\frac {-a^4 A b^2 \sin (c+d x)-a^6 C \sin (c+d x)}{2 b^5 \left (-a^2+b^2\right ) (a+b \cos (c+d x))^2}+\frac {9 a^5 A b^2 \sin (c+d x)-15 a^3 A b^4 \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 \left (-a^2+b^2\right )^2 (a+b \cos (c+d x))}-\frac {a C \sin (2 (c+d x))}{b^4}+\frac {C \sin (3 (c+d x))}{10 b^3}\right )}{d} \]

[In]

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

[Out]

((2*(25*a^4*A*b^2 - 35*a^2*A*b^4 + 40*A*b^6 + 105*a^6*C - 211*a^4*b^2*C + 112*a^2*b^4*C + 24*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*S
ec[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)) + (2*(40*a^3
*A*b^3 - 160*a*A*b^5 + 168*a^5*b*C - 256*a^3*b^3*C - 32*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 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 -
Cos[c + d*x]^2)) + ((75*a^4*A*b^2 - 145*a^2*A*b^4 + 40*A*b^6 + 315*a^6*C - 561*a^4*b^2*C + 192*a^2*b^4*C + 24*
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*EllipticF[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[S
ec[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)))/(80*(a - b)^2*b^4*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/20*((35*a^4*A*b^2 -
65*a^2*A*b^4 + 75*a^6*C - 107*a^4*b^2*C + 4*a^2*b^4*C - 2*b^6*C)*Sin[c + d*x])/(b^5*(a^2 - b^2)^2) - (-(a^4*A
*b^2*Sin[c + d*x]) - a^6*C*Sin[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] + 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])) - (a*C*Sin[2*(c + d*x)])/b^4 + (C*Sin[3*(c + d*x)])/(10*b^3)))/d

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2465\) vs. \(2(623)=1246\).

Time = 9.45 (sec) , antiderivative size = 2466, normalized size of antiderivative = 4.26

method	result	size
default	\(\text {Expression too large to display}\)	\(2466\)

[In]

int((A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-4/5*C/b^3/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/
2*c)^2)^(1/2)*(4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-14*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+6*sin(1/2*
d*x+1/2*c)^2*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*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+
1/2*c),2^(1/2)))-4*C/b^4*(a+b)*(2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2
*c)+2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-2*sin(1/2
*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a^4*(A*b^2+C*a^2)/b^6*(-1/2*b^2/a/(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*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/
2*d*x+1/2*c),2^(1/2))*b+3/8/(a+b)/(a^2-b^2)/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)
/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b^2-9/8*b/(a^2-b^2
)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)
^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+3/8*b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1
/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(
1/2))+9/8*b/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^
4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(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(co
s(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)))-12/b^5*a^2*(2*A*b^2+5*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+s
in(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))-4*a^3/b^6*(2*A*b^2+3*C*a^2)*(-b^2
/a/(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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2*b/a/(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*b/a/(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*(si
n(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/b^5*(A*b^2+6*C*a^2+6*C*a*b+3*C*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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-2*(3*A*a*b^2+A*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)))/sin(1/2*d*x+1/2*c)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((A+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)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]

[In]

int((A + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(7/2)*(a + b*cos(c + d*x))^3),x)

[Out]

int((A + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(7/2)*(a + b*cos(c + d*x))^3), x)