3.2.9 \(\int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx\) [109]

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

[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 = 0.32, 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 = {3995, 3992, 3996, 31} \begin {gather*} \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}} \end {gather*}

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 3992

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 3995

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 3996

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_), x_Symbol] :> Dist
[(-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 ) \text {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] Result contains complex when optimal does not.
time = 6.24, size = 299, normalized size = 1.23 \begin {gather*} \frac {\sec ^{\frac {11}{2}}(e+f x) (a (1+\sec (e+f x)))^{5/2} \left (\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 )}{\left (1+e^{i (e+f x)}\right ) \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} f}-\frac {(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 ^{10}\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)}}{240 f}\right ) \sin ^{11}\left (\frac {1}{2} (e+f x)\right )}{(1+\sec (e+f x))^{5/2} (c-c \sec (e+f x))^{11/2}} \end {gather*}

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))

________________________________________________________________________________________

Maple [A]
time = 0.30, size = 415, normalized size = 1.70

method	result	size
default	\(\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (120 \left (\cos ^{5}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-240 \left (\cos ^{5}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+233 \left (\cos ^{5}\left (f x +e \right )\right )-600 \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+1200 \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-325 \left (\cos ^{4}\left (f x +e \right )\right )+1200 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-2400 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+110 \left (\cos ^{3}\left (f x +e \right )\right )-1200 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+2400 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+290 \left (\cos ^{2}\left (f x +e \right )\right )+600 \cos \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-1200 \cos \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-295 \cos \left (f x +e \right )-120 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+240 \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+83\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, a^{2}}{120 f \sin \left (f x +e \right ) \cos \left (f x +e \right )^{5} \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {11}{2}}}\)	\(415\)
risch	\(\frac {a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) x}{c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}-\frac {2 a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (f x +e \right )}{c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}+\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (105 \,{\mathrm e}^{9 i \left (f x +e \right )}-555 \,{\mathrm e}^{8 i \left (f x +e \right )}+1730 \,{\mathrm e}^{7 i \left (f x +e \right )}-3125 \,{\mathrm e}^{6 i \left (f x +e \right )}+3882 \,{\mathrm e}^{5 i \left (f x +e \right )}-3125 \,{\mathrm e}^{4 i \left (f x +e \right )}+1730 \,{\mathrm e}^{3 i \left (f x +e \right )}-555 \,{\mathrm e}^{2 i \left (f x +e \right )}+105 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15 c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}-\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\)	\(500\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 9881 vs. \(2 (234) = 468\).
time = 140.96, size = 9881, normalized size = 40.50 \begin {gather*} \text {Too large to display} \end {gather*}

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*...

________________________________________________________________________________________

Fricas [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((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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 1.78, size = 260, normalized size = 1.07 \begin {gather*} -\frac {\frac {120 \, \sqrt {-a c} a^{3} \log \left ({\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{c^{6} {\left | a \right |}} - \frac {120 \, \sqrt {-a c} a^{3} \log \left ({\left | -a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \right |}\right )}{c^{6} {\left | a \right |}} - \frac {274 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{5} \sqrt {-a c} a^{3} + 1250 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{4} \sqrt {-a c} a^{4} + 2320 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a c} a^{5} + 2165 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} \sqrt {-a c} a^{6} + 1015 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} \sqrt {-a c} a^{7} + 191 \, \sqrt {-a c} a^{8}}{a^{5} c^{6} {\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10}}}{120 \, f} \end {gather*}

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]

-1/120*(120*sqrt(-a*c)*a^3*log(abs(a)*tan(1/2*f*x + 1/2*e)^2)/(c^6*abs(a)) - 120*sqrt(-a*c)*a^3*log(abs(-a*tan
(1/2*f*x + 1/2*e)^2 - a))/(c^6*abs(a)) - (274*(a*tan(1/2*f*x + 1/2*e)^2 - a)^5*sqrt(-a*c)*a^3 + 1250*(a*tan(1/
2*f*x + 1/2*e)^2 - a)^4*sqrt(-a*c)*a^4 + 2320*(a*tan(1/2*f*x + 1/2*e)^2 - a)^3*sqrt(-a*c)*a^5 + 2165*(a*tan(1/
2*f*x + 1/2*e)^2 - a)^2*sqrt(-a*c)*a^6 + 1015*(a*tan(1/2*f*x + 1/2*e)^2 - a)*sqrt(-a*c)*a^7 + 191*sqrt(-a*c)*a
^8)/(a^5*c^6*abs(a)*tan(1/2*f*x + 1/2*e)^10))/f

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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)

________________________________________________________________________________________