3.13.24 \(\int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^2} \, dx\) [1224]

Optimal. Leaf size=258 \[ -\frac {7 (11 A-8 B+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}+\frac {5 (30 A-21 B+14 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^2 d}+\frac {5 (30 A-21 B+14 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 a^2 d}-\frac {7 (11 A-8 B+5 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}+\frac {(30 A-21 B+14 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 a^2 d}-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \]

[Out]

-7/5*(11*A-8*B+5*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))/a^2/
d+5/21*(30*A-21*B+14*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))/
a^2/d-7/15*(11*A-8*B+5*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/a^2/d+1/7*(30*A-21*B+14*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/a
^2/d-1/3*(11*A-8*B+5*C)*cos(d*x+c)^(7/2)*sin(d*x+c)/a^2/d/(1+cos(d*x+c))-1/3*(A-B+C)*cos(d*x+c)^(9/2)*sin(d*x+
c)/d/(a+a*cos(d*x+c))^2+5/21*(30*A-21*B+14*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/a^2/d

________________________________________________________________________________________

Rubi [A]
time = 0.33, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4197, 3120, 3056, 2827, 2715, 2719, 2720} \begin {gather*} \frac {5 (30 A-21 B+14 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^2 d}-\frac {7 (11 A-8 B+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac {(11 A-8 B+5 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac {(30 A-21 B+14 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 a^2 d}-\frac {7 (11 A-8 B+5 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{15 a^2 d}+\frac {5 (30 A-21 B+14 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 a^2 d}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(11*A - 8*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) + (5*(30*A - 21*B + 14*C)*EllipticF[(c + d*x)/2, 2
])/(21*a^2*d) + (5*(30*A - 21*B + 14*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a^2*d) - (7*(11*A - 8*B + 5*C)*Co
s[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + ((30*A - 21*B + 14*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a^2*d) -
((11*A - 8*B + 5*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]
^(9/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

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 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3056

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

Rule 3120

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*A - b*B + a*C)*Cos[e + f*x]*(a
+ b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x] + Dist[1/(b*(b*c - a*d)*(2*m
+ 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(
b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n -
1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^
2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]

Rule 4197

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sec[(e_.)
+ (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*
Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}
, x] &&  !IntegerQ[n] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx\\ &=-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (-\frac {3}{2} a (3 A-3 B+C)+\frac {1}{2} a (13 A-7 B+7 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \cos ^{\frac {5}{2}}(c+d x) \left (-\frac {7}{2} a^2 (11 A-8 B+5 C)+\frac {3}{2} a^2 (30 A-21 B+14 C) \cos (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(7 (11 A-8 B+5 C)) \int \cos ^{\frac {5}{2}}(c+d x) \, dx}{6 a^2}+\frac {(30 A-21 B+14 C) \int \cos ^{\frac {7}{2}}(c+d x) \, dx}{2 a^2}\\ &=-\frac {7 (11 A-8 B+5 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}+\frac {(30 A-21 B+14 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 a^2 d}-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(7 (11 A-8 B+5 C)) \int \sqrt {\cos (c+d x)} \, dx}{10 a^2}+\frac {(5 (30 A-21 B+14 C)) \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{14 a^2}\\ &=-\frac {7 (11 A-8 B+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}+\frac {5 (30 A-21 B+14 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 a^2 d}-\frac {7 (11 A-8 B+5 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}+\frac {(30 A-21 B+14 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 a^2 d}-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(5 (30 A-21 B+14 C)) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{42 a^2}\\ &=-\frac {7 (11 A-8 B+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}+\frac {5 (30 A-21 B+14 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 a^2 d}+\frac {5 (30 A-21 B+14 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 a^2 d}-\frac {7 (11 A-8 B+5 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}+\frac {(30 A-21 B+14 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 a^2 d}-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.88, size = 2174, normalized size = 8.43 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((-77*I)/5)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x
)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c]
+ (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/(
(3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -
(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])
/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] +
d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) +
(((56*I)/5)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)
*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] +
(2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((
3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(
E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/
E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d
*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (
(7*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hyper
geometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)
*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d
*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*
I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d
*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 +
E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (200*A*
Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A +
B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1
+ Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c
+ d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]
*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*
x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x -
ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*S
qrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (40*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2},
{5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]
*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin
[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[
c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(25*A - 20
*B + 15*C + 52*A*Cos[c] - 36*B*Cos[c] + 20*C*Cos[c])*Csc[c])/(5*d) + (4*(107*A - 56*B + 28*C)*Cos[d*x]*Sin[c])
/(21*d) - (8*(2*A - B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*A*Cos[3*d*x]*Sin[3*c])/(7*d) - (4*Sec[c/2]*Sec[c/2 + (d
*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(5*A*Sin[(
d*x)/2] - 4*B*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/d + (4*(107*A - 56*B + 28*C)*Cos[c]*Sin[d*x])/(21*d) - (8*(2*A
- B)*Cos[2*c]*Sin[2*d*x])/(5*d) + (4*A*Cos[3*c]*Sin[3*d*x])/(7*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c
/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)

________________________________________________________________________________________

Maple [A]
time = 0.18, size = 513, normalized size = 1.99

method	result	size
default	\(-\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-2 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (750 A \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1617 A \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-525 B \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1176 B \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+350 C \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+735 C \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (750 A \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1617 A \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-525 B \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1176 B \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+350 C \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+735 C \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+960 A \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2016 A -672 B \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2608 A +896 B +560 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-5932 A +2296 B -2660 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6184 A -3682 B +2940 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1839 A +1197 B -875 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{210 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(513\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(750*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1617*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-5
25*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1176*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+350*C*EllipticF(cos(1/
2*d*x+1/2*c),2^(1/2))+735*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2*(
2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(750*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1617
*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-525*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1176*B*EllipticE(cos(1/2*
d*x+1/2*c),2^(1/2))+350*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+735*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*c
os(1/2*d*x+1/2*c)+960*A*sin(1/2*d*x+1/2*c)^12+(-2016*A-672*B)*sin(1/2*d*x+1/2*c)^10+(2608*A+896*B+560*C)*sin(1
/2*d*x+1/2*c)^8+(-5932*A+2296*B-2660*C)*sin(1/2*d*x+1/2*c)^6+(6184*A-3682*B+2940*C)*sin(1/2*d*x+1/2*c)^4+(-183
9*A+1197*B-875*C)*sin(1/2*d*x+1/2*c)^2)/a^2/cos(1/2*d*x+1/2*c)^3/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.81, size = 445, normalized size = 1.72 \begin {gather*} \frac {2 \, {\left (30 \, A \cos \left (d x + c\right )^{4} - 6 \, {\left (4 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (61 \, A - 28 \, B + 35 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (961 \, A - 658 \, B + 455 \, C\right )} \cos \left (d x + c\right ) + 750 \, A - 525 \, B + 350 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 25 \, {\left (\sqrt {2} {\left (30 i \, A - 21 i \, B + 14 i \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (30 i \, A - 21 i \, B + 14 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (30 i \, A - 21 i \, B + 14 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 25 \, {\left (\sqrt {2} {\left (-30 i \, A + 21 i \, B - 14 i \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-30 i \, A + 21 i \, B - 14 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-30 i \, A + 21 i \, B - 14 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 147 \, {\left (\sqrt {2} {\left (11 i \, A - 8 i \, B + 5 i \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (11 i \, A - 8 i \, B + 5 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (11 i \, A - 8 i \, B + 5 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 147 \, {\left (\sqrt {2} {\left (-11 i \, A + 8 i \, B - 5 i \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-11 i \, A + 8 i \, B - 5 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-11 i \, A + 8 i \, B - 5 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{210 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(2*(30*A*cos(d*x + c)^4 - 6*(4*A - 7*B)*cos(d*x + c)^3 + 2*(61*A - 28*B + 35*C)*cos(d*x + c)^2 + (961*A
- 658*B + 455*C)*cos(d*x + c) + 750*A - 525*B + 350*C)*sqrt(cos(d*x + c))*sin(d*x + c) - 25*(sqrt(2)*(30*I*A -
21*I*B + 14*I*C)*cos(d*x + c)^2 + 2*sqrt(2)*(30*I*A - 21*I*B + 14*I*C)*cos(d*x + c) + sqrt(2)*(30*I*A - 21*I*
B + 14*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 25*(sqrt(2)*(-30*I*A + 21*I*B - 14*I*
C)*cos(d*x + c)^2 + 2*sqrt(2)*(-30*I*A + 21*I*B - 14*I*C)*cos(d*x + c) + sqrt(2)*(-30*I*A + 21*I*B - 14*I*C))*
weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 147*(sqrt(2)*(11*I*A - 8*I*B + 5*I*C)*cos(d*x + c)
^2 + 2*sqrt(2)*(11*I*A - 8*I*B + 5*I*C)*cos(d*x + c) + sqrt(2)*(11*I*A - 8*I*B + 5*I*C))*weierstrassZeta(-4, 0
, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 147*(sqrt(2)*(-11*I*A + 8*I*B - 5*I*C)*cos(d*x
+ c)^2 + 2*sqrt(2)*(-11*I*A + 8*I*B - 5*I*C)*cos(d*x + c) + sqrt(2)*(-11*I*A + 8*I*B - 5*I*C))*weierstrassZeta
(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x +
c) + a^2*d)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________