\(\int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx\) [492]

Optimal result

Integrand size = 33, antiderivative size = 259 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {7 (7 A-17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {(13 A-33 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(13 A-33 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}+\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )} \]

[Out]

-1/6*(13*A-33*B)*sin(d*x+c)/a^3/d/sec(d*x+c)^(1/2)+1/5*(A-B)*sin(d*x+c)/d/(a+a*sec(d*x+c))^3/sec(d*x+c)^(1/2)+
1/3*(A-2*B)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^2/sec(d*x+c)^(1/2)+7/30*(7*A-17*B)*sin(d*x+c)/d/(a^3+a^3*sec(d*x+c
))/sec(d*x+c)^(1/2)+7/10*(7*A-17*B)*(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*(13*A-33*B)*(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 = 0.69 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3039, 4105, 3872, 3854, 3856, 2720, 2719} \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {(13 A-33 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}+\frac {7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}-\frac {(13 A-33 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}+\frac {7 (7 A-17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {(A-2 B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^2}+\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^3} \]

[In]

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

[Out]

(7*(7*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B)*S
qrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((13*A - 33*B)*Sin[c + d*x])/(6*a^
3*d*Sqrt[Sec[c + d*x]]) + ((A - B)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) + ((A - 2*B)*
Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) + (7*(7*A - 17*B)*Sin[c + d*x])/(30*d*Sqrt[Sec
[c + d*x]]*(a^3 + a^3*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 3039

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(
d + c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[p] && I
ntegerQ[m] && IntegerQ[n]

Rule 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4105

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(
2*m + 1))), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*B*n - a*A*
(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[
A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {B+A \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx \\ & = \frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {\int \frac {-\frac {1}{2} a (3 A-13 B)+\frac {7}{2} a (A-B) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = \frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}+\frac {\int \frac {-\frac {3}{2} a^2 (8 A-23 B)+\frac {25}{2} a^2 (A-2 B) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4} \\ & = \frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \frac {-\frac {15}{4} a^3 (13 A-33 B)+\frac {21}{4} a^3 (7 A-17 B) \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{15 a^6} \\ & = \frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(13 A-33 B) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{4 a^3}+\frac {(7 (7 A-17 B)) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3} \\ & = -\frac {(13 A-33 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}+\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(13 A-33 B) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}+\frac {\left (7 (7 A-17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3} \\ & = \frac {7 (7 A-17 B) \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-33 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}+\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left ((13 A-33 B) \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-17 B) \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-33 B) \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 {(13 A-33 B) \sin (c+d x)}{6 a^3 d \sqrt {\sec (c+d x)}}+\frac {(A-B) \sin (c+d x)}{5 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sin (c+d x)}{3 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt {\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 5.18 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.27 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (-98 \sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )+238 \sqrt {2} B e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )-\frac {\left ((806 A-1961 B) \cos \left (\frac {1}{2} (c-d x)\right )+(664 A-1609 B) \cos \left (\frac {1}{2} (3 c+d x)\right )+470 A \cos \left (\frac {1}{2} (c+3 d x)\right )-1165 B \cos \left (\frac {1}{2} (c+3 d x)\right )+265 A \cos \left (\frac {1}{2} (5 c+3 d x)\right )-620 B \cos \left (\frac {1}{2} (5 c+3 d x)\right )+117 A \cos \left (\frac {1}{2} (3 c+5 d x)\right )-292 B \cos \left (\frac {1}{2} (3 c+5 d x)\right )+30 A \cos \left (\frac {1}{2} (7 c+5 d x)\right )-65 B \cos \left (\frac {1}{2} (7 c+5 d x)\right )-5 B \cos \left (\frac {1}{2} (5 c+7 d x)\right )+5 B \cos \left (\frac {1}{2} (9 c+7 d x)\right )\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{8 \sqrt {\sec (c+d x)}}-260 A \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}+660 B \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right )}{15 a^3 d (1+\cos (c+d x))^3} \]

[In]

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

[Out]

(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) + (238*Sqrt[2]*B*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))*Hypergeometr
ic2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - (((806*A - 1961*B)*Cos[(c - d*x)/2] + (664*A - 1609*B
)*Cos[(3*c + d*x)/2] + 470*A*Cos[(c + 3*d*x)/2] - 1165*B*Cos[(c + 3*d*x)/2] + 265*A*Cos[(5*c + 3*d*x)/2] - 620
*B*Cos[(5*c + 3*d*x)/2] + 117*A*Cos[(3*c + 5*d*x)/2] - 292*B*Cos[(3*c + 5*d*x)/2] + 30*A*Cos[(7*c + 5*d*x)/2]
- 65*B*Cos[(7*c + 5*d*x)/2] - 5*B*Cos[(5*c + 7*d*x)/2] + 5*B*Cos[(9*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c +
d*x)/2]^5)/(8*Sqrt[Sec[c + d*x]]) - 260*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 66
0*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]))/(15*a^3*d*(1 + Cos[c + d*x])^3)

Maple [A] (verified)

Time = 6.61 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.80

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 (-160 B \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 \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 ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\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 )-468 B \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-330 B \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 ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-714 B \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 A \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1058 B \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+264 A \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-474 B \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 )+47 B \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 A -3 B \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 {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(465\)

[In]

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

[Out]

1/60*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-160*B*cos(1/2*d*x+1/2*c)^10+348*A*cos(1/2*d*x+1
/2*c)^8+130*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1
/2))*cos(1/2*d*x+1/2*c)^5+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))-468*B*cos(1/2*d*x+1/2*c)^8-330*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^5-714*B*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))-578
*A*cos(1/2*d*x+1/2*c)^6+1058*B*cos(1/2*d*x+1/2*c)^6+264*A*cos(1/2*d*x+1/2*c)^4-474*B*cos(1/2*d*x+1/2*c)^4-37*A
*cos(1/2*d*x+1/2*c)^2+47*B*cos(1/2*d*x+1/2*c)^2+3*A-3*B)/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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.89 \[ \int \frac {A+B \cos (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 + 33 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-13 i \, A + 33 i \, B\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 - 33 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (13 i \, A - 33 i \, B\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 + 17 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-7 i \, A + 17 i \, B\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 - 17 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (7 i \, A - 17 i \, B\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 (20 \, B \cos \left (d x + c\right )^{4} - 3 \, {\left (29 \, A - 79 \, B\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (73 \, A - 188 \, B\right )} \cos \left (d x + c\right )^{2} - 5 \, {\left (13 \, A - 33 \, B\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+B*cos(d*x+c))/(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 + 33*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(-13*I*A + 33*I*B)*cos(d*x + c)^2 + 3*sqrt(2)*
(-13*I*A + 33*I*B)*cos(d*x + c) + sqrt(2)*(-13*I*A + 33*I*B))*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(
d*x + c)) + 5*(sqrt(2)*(13*I*A - 33*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(13*I*A - 33*I*B)*cos(d*x + c)^2 + 3*sqrt(
2)*(13*I*A - 33*I*B)*cos(d*x + c) + sqrt(2)*(13*I*A - 33*I*B))*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin
(d*x + c)) + 21*(sqrt(2)*(-7*I*A + 17*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(-7*I*A + 17*I*B)*cos(d*x + c)^2 + 3*sqr
t(2)*(-7*I*A + 17*I*B)*cos(d*x + c) + sqrt(2)*(-7*I*A + 17*I*B))*weierstrassZeta(-4, 0, weierstrassPInverse(-4
, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(sqrt(2)*(7*I*A - 17*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(7*I*A - 17*I*B
)*cos(d*x + c)^2 + 3*sqrt(2)*(7*I*A - 17*I*B)*cos(d*x + c) + sqrt(2)*(7*I*A - 17*I*B))*weierstrassZeta(-4, 0,
weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(20*B*cos(d*x + c)^4 - 3*(29*A - 79*B)*cos(d*x
+ c)^3 - 2*(73*A - 188*B)*cos(d*x + c)^2 - 5*(13*A - 33*B)*cos(d*x + c))*sin(d*x + c)/sqrt(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. \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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