\(\int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx\) [1190]

Optimal result

Integrand size = 35, antiderivative size = 232 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}+\frac {5 (7 A+9 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 a d}-\frac {(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}} \]

[Out]

-(A+C)*sin(d*x+c)/d/(a+a*cos(d*x+c))/sec(d*x+c)^(7/2)+1/7*(7*A+9*C)*sin(d*x+c)/a/d/sec(d*x+c)^(5/2)-1/5*(5*A+7
*C)*sin(d*x+c)/a/d/sec(d*x+c)^(3/2)+5/21*(7*A+9*C)*sin(d*x+c)/a/d/sec(d*x+c)^(1/2)-3/5*(5*A+7*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/d
+5/21*(7*A+9*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/d

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4306, 3121, 2827, 2715, 2719, 2720} \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A+C) \sin (c+d x)}{d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac {5 (7 A+9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a d}-\frac {3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d} \]

[In]

Int[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]

[Out]

(-3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(7*A + 9*C)*Sqrt
[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Cos[c
+ d*x])*Sec[c + d*x]^(7/2)) + ((7*A + 9*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((5*A + 7*C)*Sin[c + d*
x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(7*A + 9*C)*Sin[c + d*x])/(21*a*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 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 {5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx \\ & = -\frac {(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {5}{2}}(c+d x) \left (-\frac {1}{2} a (5 A+7 C)+\frac {1}{2} a (7 A+9 C) \cos (c+d x)\right ) \, dx}{a^2} \\ & = -\frac {(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}-\frac {\left ((5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx}{2 a}+\frac {\left ((7 A+9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {7}{2}}(c+d x) \, dx}{2 a} \\ & = -\frac {(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\left (3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{10 a}+\frac {\left (5 (7 A+9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{14 a} \\ & = -\frac {3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}-\frac {(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}+\frac {\left (5 (7 A+9 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{42 a} \\ & = -\frac {3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}+\frac {5 (7 A+9 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 a d}-\frac {(A+C) \sin (c+d x)}{d (a+a \cos (c+d x)) \sec ^{\frac {7}{2}}(c+d x)}+\frac {(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (7 A+9 C) \sin (c+d x)}{21 a 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 = 7.02 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.34 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (420 \sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )+588 \sqrt {2} C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right )+1400 A \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}+1800 C \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}+\sqrt {\sec (c+d x)} \left (21 (40 A+51 C+(20 A+33 C) \cos (2 c)) \cos (d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )+20 (14 A+27 C) \cos (2 d x) \sin (2 c)-84 C \cos (3 d x) \sin (3 c)+30 C \cos (4 d x) \sin (4 c)-840 (A+C) \sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )-84 (20 A+33 C) \cos (c) \sin (d x)+20 (14 A+27 C) \cos (2 c) \sin (2 d x)-84 C \cos (3 c) \sin (3 d x)+30 C \cos (4 c) \sin (4 d x)-840 (A+C) \tan \left (\frac {c}{2}\right )\right )\right )}{420 a d (1+\cos (c+d x))} \]

[In]

Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]

[Out]

(Cos[(c + d*x)/2]^2*((420*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) + (588*Sqrt[2]*C*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) + 1400*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*S
qrt[Sec[c + d*x]] + 1800*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + Sqrt[Sec[c + d*x]
]*(21*(40*A + 51*C + (20*A + 33*C)*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2] + 20*(14*A + 27*C)*Cos[2*d*x]*Sin[2*c]
- 84*C*Cos[3*d*x]*Sin[3*c] + 30*C*Cos[4*d*x]*Sin[4*c] - 840*(A + C)*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] -
84*(20*A + 33*C)*Cos[c]*Sin[d*x] + 20*(14*A + 27*C)*Cos[2*c]*Sin[2*d*x] - 84*C*Cos[3*c]*Sin[3*d*x] + 30*C*Cos[
4*c]*Sin[4*d*x] - 840*(A + C)*Tan[c/2])))/(420*a*d*(1 + Cos[c + d*x]))

Maple [A] (verified)

Time = 3.51 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.27

method	result	size
default	\(-\frac {\sqrt {\left (-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (175 A F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+315 A E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+225 C F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+441 C E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )-480 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+864 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-280 A -888 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (630 A +930 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-245 A -321 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{105 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(295\)

[In]

int((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))/sec(d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/105*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(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)^(1/
2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(175*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+315*A*EllipticE(cos(1/2*d*x+1
/2*c),2^(1/2))+225*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+441*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-480*C*
sin(1/2*d*x+1/2*c)^10+864*C*sin(1/2*d*x+1/2*c)^8+(-280*A-888*C)*sin(1/2*d*x+1/2*c)^6+(630*A+930*C)*sin(1/2*d*x
+1/2*c)^4+(-245*A-321*C)*sin(1/2*d*x+1/2*c)^2)/a/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)/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 = 0.12 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.25 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {25 \, {\left (\sqrt {2} {\left (7 i \, A + 9 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (7 i \, A + 9 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 25 \, {\left (\sqrt {2} {\left (-7 i \, A - 9 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-7 i \, A - 9 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 63 \, {\left (\sqrt {2} {\left (5 i \, A + 7 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (5 i \, A + 7 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 63 \, {\left (\sqrt {2} {\left (-5 i \, A - 7 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-5 i \, A - 7 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (30 \, C \cos \left (d x + c\right )^{4} - 12 \, C \cos \left (d x + c\right )^{3} + 2 \, {\left (35 \, A + 39 \, C\right )} \cos \left (d x + c\right )^{2} + 25 \, {\left (7 \, A + 9 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{210 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]

[In]

integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))/sec(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

-1/210*(25*(sqrt(2)*(7*I*A + 9*I*C)*cos(d*x + c) + sqrt(2)*(7*I*A + 9*I*C))*weierstrassPInverse(-4, 0, cos(d*x
+ c) + I*sin(d*x + c)) + 25*(sqrt(2)*(-7*I*A - 9*I*C)*cos(d*x + c) + sqrt(2)*(-7*I*A - 9*I*C))*weierstrassPIn
verse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 63*(sqrt(2)*(5*I*A + 7*I*C)*cos(d*x + c) + sqrt(2)*(5*I*A + 7*I*
C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 63*(sqrt(2)*(-5*I*A -
7*I*C)*cos(d*x + c) + sqrt(2)*(-5*I*A - 7*I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c)
- I*sin(d*x + c))) - 2*(30*C*cos(d*x + c)^4 - 12*C*cos(d*x + c)^3 + 2*(35*A + 39*C)*cos(d*x + c)^2 + 25*(7*A
+ 9*C)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(a*d*cos(d*x + c) + a*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((A+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))/sec(d*x+c)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))/sec(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)/((a*cos(d*x + c) + a)*sec(d*x + c)^(5/2)), x)

Giac [F]

\[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate((A+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))/sec(d*x+c)^(5/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)/((a*cos(d*x + c) + a)*sec(d*x + c)^(5/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (a+a\,\cos \left (c+d\,x\right )\right )} \,d x \]

[In]

int((A + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(5/2)*(a + a*cos(c + d*x))),x)

[Out]

int((A + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(5/2)*(a + a*cos(c + d*x))), x)