3.109 \(\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx\)

Optimal. Leaf size=244 \[ \frac {a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^5 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a^3 \tan (e+f x)}{c^4 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac {a^3 \tan (e+f x)}{2 c^3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{3 c^2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}} \]

[Out]

-4/5*a^3*tan(f*x+e)/f/(c-c*sec(f*x+e))^(11/2)/(a+a*sec(f*x+e))^(1/2)-1/3*a^3*tan(f*x+e)/c^2/f/(c-c*sec(f*x+e))
^(7/2)/(a+a*sec(f*x+e))^(1/2)-1/2*a^3*tan(f*x+e)/c^3/f/(c-c*sec(f*x+e))^(5/2)/(a+a*sec(f*x+e))^(1/2)-a^3*tan(f
*x+e)/c^4/f/(c-c*sec(f*x+e))^(3/2)/(a+a*sec(f*x+e))^(1/2)+a^3*ln(1-cos(f*x+e))*tan(f*x+e)/c^5/f/(a+a*sec(f*x+e
))^(1/2)/(c-c*sec(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.47, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3910, 3907, 3911, 31} \[ -\frac {a^3 \tan (e+f x)}{c^4 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac {a^3 \tan (e+f x)}{2 c^3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{3 c^2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{7/2}}+\frac {a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^5 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(11/2),x]

[Out]

(-4*a^3*Tan[e + f*x])/(5*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(11/2)) - (a^3*Tan[e + f*x])/(3*c^2*f
*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(7/2)) - (a^3*Tan[e + f*x])/(2*c^3*f*Sqrt[a + a*Sec[e + f*x]]*(
c - c*Sec[e + f*x])^(5/2)) - (a^3*Tan[e + f*x])/(c^4*f*Sqrt[a + a*Sec[e + f*x]]*(c - c*Sec[e + f*x])^(3/2)) +
(a^3*Log[1 - Cos[e + f*x]]*Tan[e + f*x])/(c^5*f*Sqrt[a + a*Sec[e + f*x]]*Sqrt[c - c*Sec[e + f*x]])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3907

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Simp[
(-2*a*Cot[e + f*x]*(c + d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[1/c, Int[Sqrt[a +
b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] &&
EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)]

Rule 3910

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(5/2)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.), x_Symbol] :> Si
mp[(-8*a^3*Cot[e + f*x]*(c + d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[a^2/c^2, Int
[Sqrt[a + b*Csc[e + f*x]]*(c + d*Csc[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*
d, 0] && EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)]

Rule 3911

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> -Dis
t[(a*c*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]]), Subst[Int[((b + a*x)^(m - 1/2)*(d
+ c*x)^(n - 1/2))/x^(m + n), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] &&
EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && EqQ[m + n, 0]

Rubi steps

\begin {align*} \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx &=-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{11/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{7/2}} \, dx}{c^2}\\ &=-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{11/2}}-\frac {a^3 \tan (e+f x)}{3 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx}{c^3}\\ &=-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{11/2}}-\frac {a^3 \tan (e+f x)}{3 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a^3 \tan (e+f x)}{2 c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx}{c^4}\\ &=-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{11/2}}-\frac {a^3 \tan (e+f x)}{3 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a^3 \tan (e+f x)}{2 c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^4 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}} \, dx}{c^5}\\ &=-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{11/2}}-\frac {a^3 \tan (e+f x)}{3 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a^3 \tan (e+f x)}{2 c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^4 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {\left (a^3 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{-c+c x} \, dx,x,\cos (e+f x)\right )}{c^4 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {4 a^3 \tan (e+f x)}{5 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{11/2}}-\frac {a^3 \tan (e+f x)}{3 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2}}-\frac {a^3 \tan (e+f x)}{2 c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^4 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\cos (e+f x)) \tan (e+f x)}{c^5 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 5.99, size = 299, normalized size = 1.23 \[ \frac {\sin ^{11}\left (\frac {1}{2} (e+f x)\right ) \sec ^{\frac {11}{2}}(e+f x) (a (\sec (e+f x)+1))^{5/2} \left (-\frac {(5612 \cos (e+f x)-5 (736 \cos (2 (e+f x))-367 \cos (3 (e+f x))+111 \cos (4 (e+f x))-21 \cos (5 (e+f x))+625)) \csc ^{10}\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \sqrt {\sec (e+f x)+1}}{240 f}+\frac {32 i \sqrt {2} e^{\frac {1}{2} i (e+f x)} \sqrt {\frac {\left (1+e^{i (e+f x)}\right )^2}{1+e^{2 i (e+f x)}}} \left (f x+2 i \log \left (1-e^{i (e+f x)}\right )\right )}{f \left (1+e^{i (e+f x)}\right ) \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}}}\right )}{(\sec (e+f x)+1)^{5/2} (c-c \sec (e+f x))^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[e + f*x])^(5/2)/(c - c*Sec[e + f*x])^(11/2),x]

[Out]

(Sec[e + f*x]^(11/2)*(a*(1 + Sec[e + f*x]))^(5/2)*(((32*I)*Sqrt[2]*E^((I/2)*(e + f*x))*Sqrt[(1 + E^(I*(e + f*x
)))^2/(1 + E^((2*I)*(e + f*x)))]*(f*x + (2*I)*Log[1 - E^(I*(e + f*x))]))/((1 + E^(I*(e + f*x)))*Sqrt[E^(I*(e +
f*x))/(1 + E^((2*I)*(e + f*x)))]*f) - ((5612*Cos[e + f*x] - 5*(625 + 736*Cos[2*(e + f*x)] - 367*Cos[3*(e + f*
x)] + 111*Cos[4*(e + f*x)] - 21*Cos[5*(e + f*x)]))*Csc[(e + f*x)/2]^10*Sec[(e + f*x)/2]*Sqrt[Sec[e + f*x]]*Sqr
t[1 + Sec[e + f*x]])/(240*f))*Sin[(e + f*x)/2]^11)/((1 + Sec[e + f*x])^(5/2)*(c - c*Sec[e + f*x])^(11/2))

________________________________________________________________________________________

fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}\right )} \sqrt {a \sec \left (f x + e\right ) + a} \sqrt {-c \sec \left (f x + e\right ) + c}}{c^{6} \sec \left (f x + e\right )^{6} - 6 \, c^{6} \sec \left (f x + e\right )^{5} + 15 \, c^{6} \sec \left (f x + e\right )^{4} - 20 \, c^{6} \sec \left (f x + e\right )^{3} + 15 \, c^{6} \sec \left (f x + e\right )^{2} - 6 \, c^{6} \sec \left (f x + e\right ) + c^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(11/2),x, algorithm="fricas")

[Out]

integral((a^2*sec(f*x + e)^2 + 2*a^2*sec(f*x + e) + a^2)*sqrt(a*sec(f*x + e) + a)*sqrt(-c*sec(f*x + e) + c)/(c
^6*sec(f*x + e)^6 - 6*c^6*sec(f*x + e)^5 + 15*c^6*sec(f*x + e)^4 - 20*c^6*sec(f*x + e)^3 + 15*c^6*sec(f*x + e)
^2 - 6*c^6*sec(f*x + e) + c^6), x)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(11/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 2.19, size = 415, normalized size = 1.70 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (240 \left (\cos ^{5}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-120 \left (\cos ^{5}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-1200 \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-233 \left (\cos ^{5}\left (f x +e \right )\right )+600 \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+2400 \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \left (\cos ^{3}\left (f x +e \right )\right )+325 \left (\cos ^{4}\left (f x +e \right )\right )-1200 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-2400 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-110 \left (\cos ^{3}\left (f x +e \right )\right )+1200 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+1200 \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \cos \left (f x +e \right )-290 \left (\cos ^{2}\left (f x +e \right )\right )-600 \cos \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-240 \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+295 \cos \left (f x +e \right )+120 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-83\right ) \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, a^{2}}{120 f \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {11}{2}} \sin \left (f x +e \right ) \cos \left (f x +e \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(11/2),x)

[Out]

-1/120/f*(-1+cos(f*x+e))*(240*cos(f*x+e)^5*ln(-(-1+cos(f*x+e))/sin(f*x+e))-120*cos(f*x+e)^5*ln(2/(1+cos(f*x+e)
))-1200*cos(f*x+e)^4*ln(-(-1+cos(f*x+e))/sin(f*x+e))-233*cos(f*x+e)^5+600*cos(f*x+e)^4*ln(2/(1+cos(f*x+e)))+24
00*ln(-(-1+cos(f*x+e))/sin(f*x+e))*cos(f*x+e)^3+325*cos(f*x+e)^4-1200*cos(f*x+e)^3*ln(2/(1+cos(f*x+e)))-2400*c
os(f*x+e)^2*ln(-(-1+cos(f*x+e))/sin(f*x+e))-110*cos(f*x+e)^3+1200*ln(2/(1+cos(f*x+e)))*cos(f*x+e)^2+1200*ln(-(
-1+cos(f*x+e))/sin(f*x+e))*cos(f*x+e)-290*cos(f*x+e)^2-600*cos(f*x+e)*ln(2/(1+cos(f*x+e)))-240*ln(-(-1+cos(f*x
+e))/sin(f*x+e))+295*cos(f*x+e)+120*ln(2/(1+cos(f*x+e)))-83)*(a*(1+cos(f*x+e))/cos(f*x+e))^(1/2)/(c*(-1+cos(f*
x+e))/cos(f*x+e))^(11/2)/sin(f*x+e)/cos(f*x+e)^5*a^2

________________________________________________________________________________________

maxima [B]  time = 146.31, size = 9150, normalized size = 37.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(11/2),x, algorithm="maxima")

[Out]

-1/15*(15*(f*x + e)*a^2*cos(10*f*x + 10*e)^2 + 30375*(f*x + e)*a^2*cos(8*f*x + 8*e)^2 + 661500*(f*x + e)*a^2*c
os(6*f*x + 6*e)^2 + 661500*(f*x + e)*a^2*cos(4*f*x + 4*e)^2 + 30375*(f*x + e)*a^2*cos(2*f*x + 2*e)^2 + 1500*(f
*x + e)*a^2*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 216000*(f*x + e)*a^2*cos(7/2*arctan2(sin(
2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 952560*(f*x + e)*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
^2 + 216000*(f*x + e)*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 1500*(f*x + e)*a^2*cos(1/2*
arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 15*(f*x + e)*a^2*sin(10*f*x + 10*e)^2 + 30375*(f*x + e)*a^2*s
in(8*f*x + 8*e)^2 + 661500*(f*x + e)*a^2*sin(6*f*x + 6*e)^2 + 661500*(f*x + e)*a^2*sin(4*f*x + 4*e)^2 + 30375*
(f*x + e)*a^2*sin(2*f*x + 2*e)^2 + 1500*(f*x + e)*a^2*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 +
216000*(f*x + e)*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 952560*(f*x + e)*a^2*sin(5/2*ar
ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 216000*(f*x + e)*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e)))^2 + 1500*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 1350*(f*x + e)*a^2*
cos(2*f*x + 2*e) + 15*(f*x + e)*a^2 - 1110*a^2*sin(2*f*x + 2*e) - 30*(a^2*cos(10*f*x + 10*e)^2 + 2025*a^2*cos(
8*f*x + 8*e)^2 + 44100*a^2*cos(6*f*x + 6*e)^2 + 44100*a^2*cos(4*f*x + 4*e)^2 + 2025*a^2*cos(2*f*x + 2*e)^2 + 1
00*a^2*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 14400*a^2*cos(7/2*arctan2(sin(2*f*x + 2*e), co
s(2*f*x + 2*e)))^2 + 63504*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 14400*a^2*cos(3/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 100*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 +
a^2*sin(10*f*x + 10*e)^2 + 2025*a^2*sin(8*f*x + 8*e)^2 + 44100*a^2*sin(6*f*x + 6*e)^2 + 44100*a^2*sin(4*f*x +
4*e)^2 + 18900*a^2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 2025*a^2*sin(2*f*x + 2*e)^2 + 100*a^2*sin(9/2*arctan2(s
in(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 14400*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 635
04*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 14400*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), co
s(2*f*x + 2*e)))^2 + 100*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 90*a^2*cos(2*f*x + 2*e)
+ a^2 + 2*(45*a^2*cos(8*f*x + 8*e) + 210*a^2*cos(6*f*x + 6*e) + 210*a^2*cos(4*f*x + 4*e) + 45*a^2*cos(2*f*x +
2*e) + a^2)*cos(10*f*x + 10*e) + 90*(210*a^2*cos(6*f*x + 6*e) + 210*a^2*cos(4*f*x + 4*e) + 45*a^2*cos(2*f*x +
2*e) + a^2)*cos(8*f*x + 8*e) + 420*(210*a^2*cos(4*f*x + 4*e) + 45*a^2*cos(2*f*x + 2*e) + a^2)*cos(6*f*x + 6*e)
+ 420*(45*a^2*cos(2*f*x + 2*e) + a^2)*cos(4*f*x + 4*e) - 20*(a^2*cos(10*f*x + 10*e) + 45*a^2*cos(8*f*x + 8*e)
+ 210*a^2*cos(6*f*x + 6*e) + 210*a^2*cos(4*f*x + 4*e) + 45*a^2*cos(2*f*x + 2*e) - 120*a^2*cos(7/2*arctan2(sin
(2*f*x + 2*e), cos(2*f*x + 2*e))) - 252*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 120*a^2*cos
(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
) + a^2)*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 240*(a^2*cos(10*f*x + 10*e) + 45*a^2*cos(8*f*x
+ 8*e) + 210*a^2*cos(6*f*x + 6*e) + 210*a^2*cos(4*f*x + 4*e) + 45*a^2*cos(2*f*x + 2*e) - 252*a^2*cos(5/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 120*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*a
^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a^2)*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2
*e))) - 504*(a^2*cos(10*f*x + 10*e) + 45*a^2*cos(8*f*x + 8*e) + 210*a^2*cos(6*f*x + 6*e) + 210*a^2*cos(4*f*x +
4*e) + 45*a^2*cos(2*f*x + 2*e) - 120*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*a^2*cos(1/
2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a^2)*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2
40*(a^2*cos(10*f*x + 10*e) + 45*a^2*cos(8*f*x + 8*e) + 210*a^2*cos(6*f*x + 6*e) + 210*a^2*cos(4*f*x + 4*e) + 4
5*a^2*cos(2*f*x + 2*e) - 10*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + a^2)*cos(3/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*(a^2*cos(10*f*x + 10*e) + 45*a^2*cos(8*f*x + 8*e) + 210*a^2*cos(6*f*x
+ 6*e) + 210*a^2*cos(4*f*x + 4*e) + 45*a^2*cos(2*f*x + 2*e) + a^2)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e))) + 30*(3*a^2*sin(8*f*x + 8*e) + 14*a^2*sin(6*f*x + 6*e) + 14*a^2*sin(4*f*x + 4*e) + 3*a^2*sin(2*f*x +
2*e))*sin(10*f*x + 10*e) + 1350*(14*a^2*sin(6*f*x + 6*e) + 14*a^2*sin(4*f*x + 4*e) + 3*a^2*sin(2*f*x + 2*e))*
sin(8*f*x + 8*e) + 6300*(14*a^2*sin(4*f*x + 4*e) + 3*a^2*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) - 20*(a^2*sin(10*f
*x + 10*e) + 45*a^2*sin(8*f*x + 8*e) + 210*a^2*sin(6*f*x + 6*e) + 210*a^2*sin(4*f*x + 4*e) + 45*a^2*sin(2*f*x
+ 2*e) - 120*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 252*a^2*sin(5/2*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e))) - 120*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*a^2*sin(1/2*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 240*(a^2*sin(10*
f*x + 10*e) + 45*a^2*sin(8*f*x + 8*e) + 210*a^2*sin(6*f*x + 6*e) + 210*a^2*sin(4*f*x + 4*e) + 45*a^2*sin(2*f*x
+ 2*e) - 252*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 120*a^2*sin(3/2*arctan2(sin(2*f*x + 2
*e), cos(2*f*x + 2*e))) - 10*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(7/2*arctan2(sin(2*f
*x + 2*e), cos(2*f*x + 2*e))) - 504*(a^2*sin(10*f*x + 10*e) + 45*a^2*sin(8*f*x + 8*e) + 210*a^2*sin(6*f*x + 6*
e) + 210*a^2*sin(4*f*x + 4*e) + 45*a^2*sin(2*f*x + 2*e) - 120*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e))) - 10*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))) - 240*(a^2*sin(10*f*x + 10*e) + 45*a^2*sin(8*f*x + 8*e) + 210*a^2*sin(6*f*x + 6*e) + 210*a^2*si
n(4*f*x + 4*e) + 45*a^2*sin(2*f*x + 2*e) - 10*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/
2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*(a^2*sin(10*f*x + 10*e) + 45*a^2*sin(8*f*x + 8*e) + 210*a^
2*sin(6*f*x + 6*e) + 210*a^2*sin(4*f*x + 4*e) + 45*a^2*sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos
(2*f*x + 2*e))))*arctan2(sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), cos(1/2*arctan2(sin(2*f*x + 2*e
), cos(2*f*x + 2*e))) - 1) + 10*(135*(f*x + e)*a^2*cos(8*f*x + 8*e) + 630*(f*x + e)*a^2*cos(6*f*x + 6*e) + 630
*(f*x + e)*a^2*cos(4*f*x + 4*e) + 135*(f*x + e)*a^2*cos(2*f*x + 2*e) + 3*(f*x + e)*a^2 - 111*a^2*sin(8*f*x + 8
*e) - 625*a^2*sin(6*f*x + 6*e) - 625*a^2*sin(4*f*x + 4*e) - 111*a^2*sin(2*f*x + 2*e))*cos(10*f*x + 10*e) + 450
*(630*(f*x + e)*a^2*cos(6*f*x + 6*e) + 630*(f*x + e)*a^2*cos(4*f*x + 4*e) + 135*(f*x + e)*a^2*cos(2*f*x + 2*e)
+ 3*(f*x + e)*a^2 - 107*a^2*sin(6*f*x + 6*e) - 107*a^2*sin(4*f*x + 4*e))*cos(8*f*x + 8*e) + 450*(2940*(f*x +
e)*a^2*cos(4*f*x + 4*e) + 630*(f*x + e)*a^2*cos(2*f*x + 2*e) + 14*(f*x + e)*a^2 + 107*a^2*sin(2*f*x + 2*e))*co
s(6*f*x + 6*e) + 450*(630*(f*x + e)*a^2*cos(2*f*x + 2*e) + 14*(f*x + e)*a^2 + 107*a^2*sin(2*f*x + 2*e))*cos(4*
f*x + 4*e) - 10*(30*(f*x + e)*a^2*cos(10*f*x + 10*e) + 1350*(f*x + e)*a^2*cos(8*f*x + 8*e) + 6300*(f*x + e)*a^
2*cos(6*f*x + 6*e) + 6300*(f*x + e)*a^2*cos(4*f*x + 4*e) + 1350*(f*x + e)*a^2*cos(2*f*x + 2*e) - 3600*(f*x + e
)*a^2*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 7560*(f*x + e)*a^2*cos(5/2*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e))) - 3600*(f*x + e)*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 300*(f*x +
e)*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 30*(f*x + e)*a^2 + 21*a^2*sin(10*f*x + 10*e) - 1
65*a^2*sin(8*f*x + 8*e) - 1840*a^2*sin(6*f*x + 6*e) - 1840*a^2*sin(4*f*x + 4*e) - 165*a^2*sin(2*f*x + 2*e) + 9
40*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2472*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2
*f*x + 2*e))) + 940*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(9/2*arctan2(sin(2*f*x + 2*e)
, cos(2*f*x + 2*e))) - 20*(180*(f*x + e)*a^2*cos(10*f*x + 10*e) + 8100*(f*x + e)*a^2*cos(8*f*x + 8*e) + 37800*
(f*x + e)*a^2*cos(6*f*x + 6*e) + 37800*(f*x + e)*a^2*cos(4*f*x + 4*e) + 8100*(f*x + e)*a^2*cos(2*f*x + 2*e) -
45360*(f*x + e)*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 21600*(f*x + e)*a^2*cos(3/2*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 1800*(f*x + e)*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
) + 180*(f*x + e)*a^2 + 173*a^2*sin(10*f*x + 10*e) + 1125*a^2*sin(8*f*x + 8*e) - 1170*a^2*sin(6*f*x + 6*e) - 1
170*a^2*sin(4*f*x + 4*e) + 1125*a^2*sin(2*f*x + 2*e) + 2988*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x +
2*e))) - 470*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2
*f*x + 2*e))) - 12*(630*(f*x + e)*a^2*cos(10*f*x + 10*e) + 28350*(f*x + e)*a^2*cos(8*f*x + 8*e) + 132300*(f*x
+ e)*a^2*cos(6*f*x + 6*e) + 132300*(f*x + e)*a^2*cos(4*f*x + 4*e) + 28350*(f*x + e)*a^2*cos(2*f*x + 2*e) - 756
00*(f*x + e)*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 6300*(f*x + e)*a^2*cos(1/2*arctan2(sin
(2*f*x + 2*e), cos(2*f*x + 2*e))) + 630*(f*x + e)*a^2 + 647*a^2*sin(10*f*x + 10*e) + 5805*a^2*sin(8*f*x + 8*e)
+ 4620*a^2*sin(6*f*x + 6*e) + 4620*a^2*sin(4*f*x + 4*e) + 5805*a^2*sin(2*f*x + 2*e) - 4980*a^2*sin(3/2*arctan
2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2060*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(5/
2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*(180*(f*x + e)*a^2*cos(10*f*x + 10*e) + 8100*(f*x + e)*a^2
*cos(8*f*x + 8*e) + 37800*(f*x + e)*a^2*cos(6*f*x + 6*e) + 37800*(f*x + e)*a^2*cos(4*f*x + 4*e) + 8100*(f*x +
e)*a^2*cos(2*f*x + 2*e) - 1800*(f*x + e)*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 180*(f*x +
e)*a^2 + 173*a^2*sin(10*f*x + 10*e) + 1125*a^2*sin(8*f*x + 8*e) - 1170*a^2*sin(6*f*x + 6*e) - 1170*a^2*sin(4*
f*x + 4*e) + 1125*a^2*sin(2*f*x + 2*e) - 470*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(3/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*(30*(f*x + e)*a^2*cos(10*f*x + 10*e) + 1350*(f*x + e)*a^2*c
os(8*f*x + 8*e) + 6300*(f*x + e)*a^2*cos(6*f*x + 6*e) + 6300*(f*x + e)*a^2*cos(4*f*x + 4*e) + 1350*(f*x + e)*a
^2*cos(2*f*x + 2*e) + 30*(f*x + e)*a^2 + 21*a^2*sin(10*f*x + 10*e) - 165*a^2*sin(8*f*x + 8*e) - 1840*a^2*sin(6
*f*x + 6*e) - 1840*a^2*sin(4*f*x + 4*e) - 165*a^2*sin(2*f*x + 2*e))*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e))) + 10*(135*(f*x + e)*a^2*sin(8*f*x + 8*e) + 630*(f*x + e)*a^2*sin(6*f*x + 6*e) + 630*(f*x + e)*a^2*s
in(4*f*x + 4*e) + 135*(f*x + e)*a^2*sin(2*f*x + 2*e) + 111*a^2*cos(8*f*x + 8*e) + 625*a^2*cos(6*f*x + 6*e) + 6
25*a^2*cos(4*f*x + 4*e) + 111*a^2*cos(2*f*x + 2*e))*sin(10*f*x + 10*e) + 30*(9450*(f*x + e)*a^2*sin(6*f*x + 6*
e) + 9450*(f*x + e)*a^2*sin(4*f*x + 4*e) + 2025*(f*x + e)*a^2*sin(2*f*x + 2*e) + 1605*a^2*cos(6*f*x + 6*e) + 1
605*a^2*cos(4*f*x + 4*e) - 37*a^2)*sin(8*f*x + 8*e) + 50*(26460*(f*x + e)*a^2*sin(4*f*x + 4*e) + 5670*(f*x + e
)*a^2*sin(2*f*x + 2*e) - 963*a^2*cos(2*f*x + 2*e) - 125*a^2)*sin(6*f*x + 6*e) + 50*(5670*(f*x + e)*a^2*sin(2*f
*x + 2*e) - 963*a^2*cos(2*f*x + 2*e) - 125*a^2)*sin(4*f*x + 4*e) - 10*(30*(f*x + e)*a^2*sin(10*f*x + 10*e) + 1
350*(f*x + e)*a^2*sin(8*f*x + 8*e) + 6300*(f*x + e)*a^2*sin(6*f*x + 6*e) + 6300*(f*x + e)*a^2*sin(4*f*x + 4*e)
+ 1350*(f*x + e)*a^2*sin(2*f*x + 2*e) - 3600*(f*x + e)*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)
)) - 7560*(f*x + e)*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 3600*(f*x + e)*a^2*sin(3/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 300*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e
))) - 21*a^2*cos(10*f*x + 10*e) + 165*a^2*cos(8*f*x + 8*e) + 1840*a^2*cos(6*f*x + 6*e) + 1840*a^2*cos(4*f*x +
4*e) + 165*a^2*cos(2*f*x + 2*e) - 940*a^2*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2472*a^2*cos(
5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 940*a^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
) - 21*a^2)*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*(180*(f*x + e)*a^2*sin(10*f*x + 10*e) +
8100*(f*x + e)*a^2*sin(8*f*x + 8*e) + 37800*(f*x + e)*a^2*sin(6*f*x + 6*e) + 37800*(f*x + e)*a^2*sin(4*f*x + 4
*e) + 8100*(f*x + e)*a^2*sin(2*f*x + 2*e) - 45360*(f*x + e)*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x +
2*e))) - 21600*(f*x + e)*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 1800*(f*x + e)*a^2*sin(1/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 173*a^2*cos(10*f*x + 10*e) - 1125*a^2*cos(8*f*x + 8*e) + 1170*
a^2*cos(6*f*x + 6*e) + 1170*a^2*cos(4*f*x + 4*e) - 1125*a^2*cos(2*f*x + 2*e) - 2988*a^2*cos(5/2*arctan2(sin(2*
f*x + 2*e), cos(2*f*x + 2*e))) + 470*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 173*a^2)*sin(7
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12*(630*(f*x + e)*a^2*sin(10*f*x + 10*e) + 28350*(f*x + e)*a
^2*sin(8*f*x + 8*e) + 132300*(f*x + e)*a^2*sin(6*f*x + 6*e) + 132300*(f*x + e)*a^2*sin(4*f*x + 4*e) + 28350*(f
*x + e)*a^2*sin(2*f*x + 2*e) - 75600*(f*x + e)*a^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 6300
*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 647*a^2*cos(10*f*x + 10*e) - 5805*a^2*co
s(8*f*x + 8*e) - 4620*a^2*cos(6*f*x + 6*e) - 4620*a^2*cos(4*f*x + 4*e) - 5805*a^2*cos(2*f*x + 2*e) + 4980*a^2*
cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2060*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x +
2*e))) - 647*a^2)*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 20*(180*(f*x + e)*a^2*sin(10*f*x + 10
*e) + 8100*(f*x + e)*a^2*sin(8*f*x + 8*e) + 37800*(f*x + e)*a^2*sin(6*f*x + 6*e) + 37800*(f*x + e)*a^2*sin(4*f
*x + 4*e) + 8100*(f*x + e)*a^2*sin(2*f*x + 2*e) - 1800*(f*x + e)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f
*x + 2*e))) - 173*a^2*cos(10*f*x + 10*e) - 1125*a^2*cos(8*f*x + 8*e) + 1170*a^2*cos(6*f*x + 6*e) + 1170*a^2*co
s(4*f*x + 4*e) - 1125*a^2*cos(2*f*x + 2*e) + 470*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 17
3*a^2)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*(30*(f*x + e)*a^2*sin(10*f*x + 10*e) + 1350*(
f*x + e)*a^2*sin(8*f*x + 8*e) + 6300*(f*x + e)*a^2*sin(6*f*x + 6*e) + 6300*(f*x + e)*a^2*sin(4*f*x + 4*e) + 13
50*(f*x + e)*a^2*sin(2*f*x + 2*e) - 21*a^2*cos(10*f*x + 10*e) + 165*a^2*cos(8*f*x + 8*e) + 1840*a^2*cos(6*f*x
+ 6*e) + 1840*a^2*cos(4*f*x + 4*e) + 165*a^2*cos(2*f*x + 2*e) - 21*a^2)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e))))*sqrt(a)*sqrt(c)/((c^6*cos(10*f*x + 10*e)^2 + 2025*c^6*cos(8*f*x + 8*e)^2 + 44100*c^6*cos(6*f*x
+ 6*e)^2 + 44100*c^6*cos(4*f*x + 4*e)^2 + 2025*c^6*cos(2*f*x + 2*e)^2 + 100*c^6*cos(9/2*arctan2(sin(2*f*x + 2
*e), cos(2*f*x + 2*e)))^2 + 14400*c^6*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 63504*c^6*cos(5
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 14400*c^6*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e)))^2 + 100*c^6*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + c^6*sin(10*f*x + 10*e)^2 + 2025*c^6*
sin(8*f*x + 8*e)^2 + 44100*c^6*sin(6*f*x + 6*e)^2 + 44100*c^6*sin(4*f*x + 4*e)^2 + 18900*c^6*sin(4*f*x + 4*e)*
sin(2*f*x + 2*e) + 2025*c^6*sin(2*f*x + 2*e)^2 + 100*c^6*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^
2 + 14400*c^6*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 63504*c^6*sin(5/2*arctan2(sin(2*f*x + 2
*e), cos(2*f*x + 2*e)))^2 + 14400*c^6*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 100*c^6*sin(1/2
*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 90*c^6*cos(2*f*x + 2*e) + c^6 + 2*(45*c^6*cos(8*f*x + 8*e) +
210*c^6*cos(6*f*x + 6*e) + 210*c^6*cos(4*f*x + 4*e) + 45*c^6*cos(2*f*x + 2*e) + c^6)*cos(10*f*x + 10*e) + 90*
(210*c^6*cos(6*f*x + 6*e) + 210*c^6*cos(4*f*x + 4*e) + 45*c^6*cos(2*f*x + 2*e) + c^6)*cos(8*f*x + 8*e) + 420*(
210*c^6*cos(4*f*x + 4*e) + 45*c^6*cos(2*f*x + 2*e) + c^6)*cos(6*f*x + 6*e) + 420*(45*c^6*cos(2*f*x + 2*e) + c^
6)*cos(4*f*x + 4*e) - 20*(c^6*cos(10*f*x + 10*e) + 45*c^6*cos(8*f*x + 8*e) + 210*c^6*cos(6*f*x + 6*e) + 210*c^
6*cos(4*f*x + 4*e) + 45*c^6*cos(2*f*x + 2*e) - 120*c^6*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) -
252*c^6*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 120*c^6*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2
*f*x + 2*e))) - 10*c^6*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + c^6)*cos(9/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) - 240*(c^6*cos(10*f*x + 10*e) + 45*c^6*cos(8*f*x + 8*e) + 210*c^6*cos(6*f*x + 6*e) +
210*c^6*cos(4*f*x + 4*e) + 45*c^6*cos(2*f*x + 2*e) - 252*c^6*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e))) - 120*c^6*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*c^6*cos(1/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) + c^6)*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 504*(c^6*cos(10*f*x + 10*e)
+ 45*c^6*cos(8*f*x + 8*e) + 210*c^6*cos(6*f*x + 6*e) + 210*c^6*cos(4*f*x + 4*e) + 45*c^6*cos(2*f*x + 2*e) - 12
0*c^6*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*c^6*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e))) + c^6)*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 240*(c^6*cos(10*f*x + 10*e) + 45*c^6*
cos(8*f*x + 8*e) + 210*c^6*cos(6*f*x + 6*e) + 210*c^6*cos(4*f*x + 4*e) + 45*c^6*cos(2*f*x + 2*e) - 10*c^6*cos(
1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + c^6)*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) -
20*(c^6*cos(10*f*x + 10*e) + 45*c^6*cos(8*f*x + 8*e) + 210*c^6*cos(6*f*x + 6*e) + 210*c^6*cos(4*f*x + 4*e) +
45*c^6*cos(2*f*x + 2*e) + c^6)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 30*(3*c^6*sin(8*f*x + 8*
e) + 14*c^6*sin(6*f*x + 6*e) + 14*c^6*sin(4*f*x + 4*e) + 3*c^6*sin(2*f*x + 2*e))*sin(10*f*x + 10*e) + 1350*(14
*c^6*sin(6*f*x + 6*e) + 14*c^6*sin(4*f*x + 4*e) + 3*c^6*sin(2*f*x + 2*e))*sin(8*f*x + 8*e) + 6300*(14*c^6*sin(
4*f*x + 4*e) + 3*c^6*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) - 20*(c^6*sin(10*f*x + 10*e) + 45*c^6*sin(8*f*x + 8*e)
+ 210*c^6*sin(6*f*x + 6*e) + 210*c^6*sin(4*f*x + 4*e) + 45*c^6*sin(2*f*x + 2*e) - 120*c^6*sin(7/2*arctan2(sin
(2*f*x + 2*e), cos(2*f*x + 2*e))) - 252*c^6*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 120*c^6*sin
(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*c^6*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))
))*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 240*(c^6*sin(10*f*x + 10*e) + 45*c^6*sin(8*f*x + 8*e
) + 210*c^6*sin(6*f*x + 6*e) + 210*c^6*sin(4*f*x + 4*e) + 45*c^6*sin(2*f*x + 2*e) - 252*c^6*sin(5/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))) - 120*c^6*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*c^6*sin
(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 504*
(c^6*sin(10*f*x + 10*e) + 45*c^6*sin(8*f*x + 8*e) + 210*c^6*sin(6*f*x + 6*e) + 210*c^6*sin(4*f*x + 4*e) + 45*c
^6*sin(2*f*x + 2*e) - 120*c^6*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 10*c^6*sin(1/2*arctan2(si
n(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 240*(c^6*sin(10*f*x
+ 10*e) + 45*c^6*sin(8*f*x + 8*e) + 210*c^6*sin(6*f*x + 6*e) + 210*c^6*sin(4*f*x + 4*e) + 45*c^6*sin(2*f*x +
2*e) - 10*c^6*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e))) - 20*(c^6*sin(10*f*x + 10*e) + 45*c^6*sin(8*f*x + 8*e) + 210*c^6*sin(6*f*x + 6*e) + 210*c^6*sin(4*f
*x + 4*e) + 45*c^6*sin(2*f*x + 2*e))*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*f)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{11/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(e + f*x))^(5/2)/(c - c/cos(e + f*x))^(11/2),x)

[Out]

int((a + a/cos(e + f*x))^(5/2)/(c - c/cos(e + f*x))^(11/2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(f*x+e))**(5/2)/(c-c*sec(f*x+e))**(11/2),x)

[Out]

Timed out

________________________________________________________________________________________