\(\int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx\) [1203]

Optimal result

Integrand size = 35, antiderivative size = 290 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {7 (7 A+33 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {(13 A+63 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x)}-\frac {2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}+\frac {7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}} \]

[Out]

-1/5*(A+C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^3/sec(d*x+c)^(9/2)-2/15*(A+6*C)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^2/sec
(d*x+c)^(7/2)-1/10*(13*A+63*C)*sin(d*x+c)/d/(a^3+a^3*cos(d*x+c))/sec(d*x+c)^(5/2)+7/30*(7*A+33*C)*sin(d*x+c)/a
^3/d/sec(d*x+c)^(3/2)-1/6*(13*A+63*C)*sin(d*x+c)/a^3/d/sec(d*x+c)^(1/2)+7/10*(7*A+33*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)/a^3/d-1/6*(1
3*A+63*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)/a^3/d

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4306, 3121, 3056, 2827, 2715, 2720, 2719} \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}-\frac {(13 A+63 C) \sin (c+d x)}{10 d \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}-\frac {(13 A+63 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}+\frac {7 (7 A+33 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {2 (A+6 C) \sin (c+d x)}{15 a d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac {(A+C) \sin (c+d x)}{5 d \sec ^{\frac {9}{2}}(c+d x) (a \cos (c+d x)+a)^3} \]

[In]

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

[Out]

(7*(7*A + 33*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A + 63*C)*S
qrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a
*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)) - (2*(A + 6*C)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^
(7/2)) - ((13*A + 63*C)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(5/2)) + (7*(7*A + 33*C)*Sin
[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((13*A + 63*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]])

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 3121

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[a*(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)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(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, 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 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+a \cos (c+d x))^3} \, dx \\ & = -\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (\frac {1}{2} a (A-9 C)+\frac {5}{2} a (A+3 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x)}-\frac {2 (A+6 C) \sin (c+d x)}{15 a d (a+a \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 (-7 a^2 (A+6 C)+\frac {5}{2} a^2 (5 A+21 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4} \\ & = -\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x)}-\frac {2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (-\frac {15}{4} a^3 (13 A+63 C)+\frac {35}{4} a^3 (7 A+33 C) \cos (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x)}-\frac {2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (7 (7 A+33 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx}{12 a^3}-\frac {\left ((13 A+63 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{4 a^3} \\ & = -\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x)}-\frac {2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}+\frac {7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}+\frac {\left (7 (7 A+33 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}-\frac {\left ((13 A+63 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3} \\ & = \frac {7 (7 A+33 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {(13 A+63 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(A+C) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3 \sec ^{\frac {9}{2}}(c+d x)}-\frac {2 (A+6 C) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2 \sec ^{\frac {7}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x)}+\frac {7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 12.17 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.15 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (98 \sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )+462 \sqrt {2} C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )+\frac {\left (2 (806 A+3795 C) \cos \left (\frac {1}{2} (c-d x)\right )+2 (664 A+3135 C) \cos \left (\frac {1}{2} (3 c+d x)\right )+940 A \cos \left (\frac {1}{2} (c+3 d x)\right )+4500 C \cos \left (\frac {1}{2} (c+3 d x)\right )+530 A \cos \left (\frac {1}{2} (5 c+3 d x)\right )+2430 C \cos \left (\frac {1}{2} (5 c+3 d x)\right )+234 A \cos \left (\frac {1}{2} (3 c+5 d x)\right )+1110 C \cos \left (\frac {1}{2} (3 c+5 d x)\right )+60 A \cos \left (\frac {1}{2} (7 c+5 d x)\right )+276 C \cos \left (\frac {1}{2} (7 c+5 d x)\right )+15 C \cos \left (\frac {1}{2} (5 c+7 d x)\right )-15 C \cos \left (\frac {1}{2} (9 c+7 d x)\right )-3 C \cos \left (\frac {1}{2} (7 c+9 d x)\right )+3 C \cos \left (\frac {1}{2} (11 c+9 d x)\right )\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{16 \sqrt {\sec (c+d x)}}+260 A \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}+1260 C \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right )}{15 a^3 d (1+\cos (c+d x))^3} \]

[In]

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

[Out]

-1/15*(Cos[(c + d*x)/2]^6*((98*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c
+ d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/
4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (462*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqr
t[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hyperge
ometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + ((2*(806*A + 3795*C)*Cos[(c - d*x)/2] + 2*(664*A
+ 3135*C)*Cos[(3*c + d*x)/2] + 940*A*Cos[(c + 3*d*x)/2] + 4500*C*Cos[(c + 3*d*x)/2] + 530*A*Cos[(5*c + 3*d*x)
/2] + 2430*C*Cos[(5*c + 3*d*x)/2] + 234*A*Cos[(3*c + 5*d*x)/2] + 1110*C*Cos[(3*c + 5*d*x)/2] + 60*A*Cos[(7*c +
5*d*x)/2] + 276*C*Cos[(7*c + 5*d*x)/2] + 15*C*Cos[(5*c + 7*d*x)/2] - 15*C*Cos[(9*c + 7*d*x)/2] - 3*C*Cos[(7*c
+ 9*d*x)/2] + 3*C*Cos[(11*c + 9*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(16*Sqrt[Sec[c + d*x]]) + 260*
A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 1260*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d
*x)/2, 2]*Sqrt[Sec[c + d*x]]))/(a^3*d*(1 + Cos[c + d*x])^3)

Maple [A] (verified)

Time = 4.11 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.65

method	result	size
default	\(\frac {\sqrt {\left (-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-192 C \left (\cos ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+864 C \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+348 A \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+130 A \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+294 A \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+228 C \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+630 C \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1386 C \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-578 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A -1590 C \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+264 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +744 C \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-37 A \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-57 C \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 A +3 C \right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \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 {-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(479\)

[In]

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

[Out]

1/60*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-192*C*cos(1/2*d*x+1/2*c)^12+864*C*cos(1/2*d*x+
1/2*c)^10+348*A*cos(1/2*d*x+1/2*c)^8+130*A*cos(1/2*d*x+1/2*c)^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+294*A*cos(1/2*d*x+1/2*c)^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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+228*C*cos(1/2*d*x+1/2*c)^8+630*C*cos(
1/2*d*x+1/2*c)^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)*EllipticF(cos(1/2*d*x+1/2*c),2
^(1/2))+1386*C*cos(1/2*d*x+1/2*c)^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)*EllipticE(c
os(1/2*d*x+1/2*c),2^(1/2))-578*cos(1/2*d*x+1/2*c)^6*A-1590*C*cos(1/2*d*x+1/2*c)^6+264*cos(1/2*d*x+1/2*c)^4*A+7
44*C*cos(1/2*d*x+1/2*c)^4-37*A*cos(1/2*d*x+1/2*c)^2-57*C*cos(1/2*d*x+1/2*c)^2+3*A+3*C)/a^3/cos(1/2*d*x+1/2*c)^
5/(-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)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.14 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.72 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {5 \, {\left (\sqrt {2} {\left (-13 i \, A - 63 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-13 i \, A - 63 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-13 i \, A - 63 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-13 i \, A - 63 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (13 i \, A + 63 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (13 i \, A + 63 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (13 i \, A + 63 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (13 i \, A + 63 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-7 i \, A - 33 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-7 i \, A - 33 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-7 i \, A - 33 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-7 i \, A - 33 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 ) + 21 \, {\left (\sqrt {2} {\left (7 i \, A + 33 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (7 i \, A + 33 i \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (7 i \, A + 33 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (7 i \, A + 33 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 ) - \frac {2 \, {\left (12 \, C \cos \left (d x + c\right )^{5} - 24 \, C \cos \left (d x + c\right )^{4} - 3 \, {\left (29 \, A + 147 \, C\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (73 \, A + 357 \, C\right )} \cos \left (d x + c\right )^{2} - 5 \, {\left (13 \, A + 63 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

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

[Out]

-1/60*(5*(sqrt(2)*(-13*I*A - 63*I*C)*cos(d*x + c)^3 + 3*sqrt(2)*(-13*I*A - 63*I*C)*cos(d*x + c)^2 + 3*sqrt(2)*
(-13*I*A - 63*I*C)*cos(d*x + c) + sqrt(2)*(-13*I*A - 63*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(
d*x + c)) + 5*(sqrt(2)*(13*I*A + 63*I*C)*cos(d*x + c)^3 + 3*sqrt(2)*(13*I*A + 63*I*C)*cos(d*x + c)^2 + 3*sqrt(
2)*(13*I*A + 63*I*C)*cos(d*x + c) + sqrt(2)*(13*I*A + 63*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin
(d*x + c)) + 21*(sqrt(2)*(-7*I*A - 33*I*C)*cos(d*x + c)^3 + 3*sqrt(2)*(-7*I*A - 33*I*C)*cos(d*x + c)^2 + 3*sqr
t(2)*(-7*I*A - 33*I*C)*cos(d*x + c) + sqrt(2)*(-7*I*A - 33*I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4
, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(sqrt(2)*(7*I*A + 33*I*C)*cos(d*x + c)^3 + 3*sqrt(2)*(7*I*A + 33*I*C
)*cos(d*x + c)^2 + 3*sqrt(2)*(7*I*A + 33*I*C)*cos(d*x + c) + sqrt(2)*(7*I*A + 33*I*C))*weierstrassZeta(-4, 0,
weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(12*C*cos(d*x + c)^5 - 24*C*cos(d*x + c)^4 - 3*
(29*A + 147*C)*cos(d*x + c)^3 - 2*(73*A + 357*C)*cos(d*x + c)^2 - 5*(13*A + 63*C)*cos(d*x + c))*sin(d*x + c)/s
qrt(cos(d*x + c)))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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