∫ sec(e + fx)(a + a sec(e + fx))(c − c sec(e + fx))7∕2 dx [64]

Optimal. Leaf size=163 \[ -\frac {256 c^4 (a+a \sec (e+f x)) \tan (e+f x)}{315 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{105 f}-\frac {8 c^2 (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{21 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f} \]

-8/21*c^2*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(3/2)*tan(f*x+e)/f-2/9*c*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(5/2)*t

an(f*x+e)/f-256/315*c^4*(a+a*sec(f*x+e))*tan(f*x+e)/f/(c-c*sec(f*x+e))^(1/2)-64/105*c^3*(a+a*sec(f*x+e))*(c-c*

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Rubi [A]
time = 0.19, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4040, 4038} \begin {gather*} -\frac {256 c^4 \tan (e+f x) (a \sec (e+f x)+a)}{315 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 \tan (e+f x) (a \sec (e+f x)+a) \sqrt {c-c \sec (e+f x)}}{105 f}-\frac {8 c^2 \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{3/2}}{21 f}-\frac {2 c \tan (e+f x) (a \sec (e+f x)+a) (c-c \sec (e+f x))^{5/2}}{9 f} \end {gather*}

(-256*c^4*(a + a*Sec[e + f*x])*Tan[e + f*x])/(315*f*Sqrt[c - c*Sec[e + f*x]]) - (64*c^3*(a + a*Sec[e + f*x])*S

qrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(105*f) - (8*c^2*(a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])^(3/2)*Tan[e +

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) +

(c_)], x_Symbol] :> Simp[2*a*c*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]])),

x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -2^(-1)]

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)

)^(n_), x_Symbol] :> Simp[(-d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^(n - 1)/(f*(m + n))),

x] + Dist[c*((2*n - 1)/(m + n)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^(n - 1), x], x]

/; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 0] && !LtQ[m,

\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{7/2} \, dx &=-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f}+\frac {1}{3} (4 c) \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \, dx\\ &=-\frac {8 c^2 (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{21 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f}+\frac {1}{21} \left (32 c^2\right ) \int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \, dx\\ &=-\frac {64 c^3 (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{105 f}-\frac {8 c^2 (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{21 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f}+\frac {1}{105} \left (128 c^3\right ) \int \sec (e+f x) (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \, dx\\ &=-\frac {256 c^4 (a+a \sec (e+f x)) \tan (e+f x)}{315 f \sqrt {c-c \sec (e+f x)}}-\frac {64 c^3 (a+a \sec (e+f x)) \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{105 f}-\frac {8 c^2 (a+a \sec (e+f x)) (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{21 f}-\frac {2 c (a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{9 f}\\ \end {align*}

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Mathematica [A]
time = 0.85, size = 86, normalized size = 0.53 \begin {gather*} \frac {a c^3 \cos ^2\left (\frac {1}{2} (e+f x)\right ) (-782+1617 \cos (e+f x)-642 \cos (2 (e+f x))+319 \cos (3 (e+f x))) \cot \left (\frac {1}{2} (e+f x)\right ) \sec ^4(e+f x) \sqrt {c-c \sec (e+f x)}}{315 f} \end {gather*}

(a*c^3*Cos[(e + f*x)/2]^2*(-782 + 1617*Cos[e + f*x] - 642*Cos[2*(e + f*x)] + 319*Cos[3*(e + f*x)])*Cot[(e + f*

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method	result	size

default	\(\frac {2 a \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \left (\sin ^{3}\left (f x +e \right )\right ) \left (319 \left (\cos ^{3}\left (f x +e \right )\right )-321 \left (\cos ^{2}\left (f x +e \right )\right )+165 \cos \left (f x +e \right )-35\right )}{315 f \left (-1+\cos \left (f x +e \right )\right )^{5} \cos \left (f x +e \right )}\)	\(83\)

2/315*a/f*(c*(-1+cos(f*x+e))/cos(f*x+e))^(7/2)*sin(f*x+e)^3*(319*cos(f*x+e)^3-321*cos(f*x+e)^2+165*cos(f*x+e)-

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

-2/315*(315*(cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*(5*(a*c^3*f*cos(2*f*x + 2

*e)^4 + a*c^3*f*sin(2*f*x + 2*e)^4 + 4*a*c^3*f*cos(2*f*x + 2*e)^3 + 6*a*c^3*f*cos(2*f*x + 2*e)^2 + 4*a*c^3*f*c

os(2*f*x + 2*e) + a*c^3*f + 2*(a*c^3*f*cos(2*f*x + 2*e)^2 + 2*a*c^3*f*cos(2*f*x + 2*e) + a*c^3*f)*sin(2*f*x +

2*e)^2)*integrate((cos(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/4)*(((cos(6*f*x + 6*e)

*cos(2*f*x + 2*e) + 2*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + cos(2*f*x + 2*e)^2 + sin(6*f*x + 6*e)*sin(2*f*x + 2*

e) + 2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + sin(2*f*x + 2*e)^2)*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2

*e))) + (cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 2*cos(2*f*x + 2*e)*sin(4*f*x + 4*e) - cos(6*f*x + 6*e)*sin(2*f*x

+ 2*e) - 2*cos(4*f*x + 4*e)*sin(2*f*x + 2*e))*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*cos(7/2*ar

ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - ((cos(2*f*x + 2*e)*sin(6*f*x + 6*e) + 2*cos(2*f*x + 2*e)*sin(

4*f*x + 4*e) - cos(6*f*x + 6*e)*sin(2*f*x + 2*e) - 2*cos(4*f*x + 4*e)*sin(2*f*x + 2*e))*cos(9/2*arctan2(sin(2*

f*x + 2*e), cos(2*f*x + 2*e))) - (cos(6*f*x + 6*e)*cos(2*f*x + 2*e) + 2*cos(4*f*x + 4*e)*cos(2*f*x + 2*e) + co

s(2*f*x + 2*e)^2 + sin(6*f*x + 6*e)*sin(2*f*x + 2*e) + 2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + sin(2*f*x + 2*e)^

2)*sin(9/2*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) +

1)))/((cos(2*f*x + 2*e)^6 + sin(2*f*x + 2*e)^6 + 4*cos(2*f*x + 2*e)^5 + (3*cos(2*f*x + 2*e)^2 + 4*cos(2*f*x +

2*e) + 2)*sin(2*f*x + 2*e)^4 + 6*cos(2*f*x + 2*e)^4 + (cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + 4*cos(2*f*x +

2*e)^3 + 2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e)^2 + 4*cos(2*

f*x + 2*e) + 1)*cos(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + 4*cos(2*f*x + 2*e)^3 + 2*(co

s(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e)^2 + 4*cos(2*f*x + 2*e) + 1)

*cos(4*f*x + 4*e)^2 + 4*cos(2*f*x + 2*e)^3 + (cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + 4*cos(2*f*x + 2*e)^3 +

2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e)^2 + 4*cos(2*f*x + 2*e

) + 1)*sin(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + 4*cos(2*f*x + 2*e)^3 + 2*(cos(2*f*x +

2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e)^2 + 4*cos(2*f*x + 2*e) + 1)*sin(4*f*

x + 4*e)^2 + (3*cos(2*f*x + 2*e)^4 + 8*cos(2*f*x + 2*e)^3 + 8*cos(2*f*x + 2*e)^2 + 4*cos(2*f*x + 2*e) + 1)*sin

(2*f*x + 2*e)^2 + 2*(cos(2*f*x + 2*e)^5 + cos(2*f*x + 2*e)*sin(2*f*x + 2*e)^4 + 4*cos(2*f*x + 2*e)^4 + 6*cos(2

*f*x + 2*e)^3 + 2*(cos(2*f*x + 2*e)^3 + 2*cos(2*f*x + 2*e)^2 + cos(2*f*x + 2*e))*sin(2*f*x + 2*e)^2 + 2*(cos(2

*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + 4*cos(2*f*x + 2*e)^3 + 2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*si

n(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e)^2 + 4*cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e) + 4*cos(2*f*x + 2*e)^2 + co

s(2*f*x + 2*e))*cos(6*f*x + 6*e) + 4*(cos(2*f*x + 2*e)^5 + cos(2*f*x + 2*e)*sin(2*f*x + 2*e)^4 + 4*cos(2*f*x +

2*e)^4 + 6*cos(2*f*x + 2*e)^3 + 2*(cos(2*f*x + 2*e)^3 + 2*cos(2*f*x + 2*e)^2 + cos(2*f*x + 2*e))*sin(2*f*x +

2*e)^2 + 4*cos(2*f*x + 2*e)^2 + cos(2*f*x + 2*e))*cos(4*f*x + 4*e) + cos(2*f*x + 2*e)^2 + 2*(sin(2*f*x + 2*e)^

5 + 2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^3 + 2*(cos(2*f*x + 2*e)^4 + sin(2*f*x + 2

*e)^4 + 4*cos(2*f*x + 2*e)^3 + 2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*

x + 2*e)^2 + 4*cos(2*f*x + 2*e) + 1)*sin(4*f*x + 4*e) + (cos(2*f*x + 2*e)^4 + 4*cos(2*f*x + 2*e)^3 + 6*cos(2*f

*x + 2*e)^2 + 4*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e))*sin(6*f*x + 6*e) + 4*(sin(2*f*x + 2*e)^5 + 2*(cos(2*f*

x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^3 + (cos(2*f*x + 2*e)^4 + 4*cos(2*f*x + 2*e)^3 + 6*cos(2

*f*x + 2*e)^2 + 4*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e))*sin(4*f*x + 4*e))*cos(7/2*arctan2(sin(2*f*x + 2*e),

cos(2*f*x + 2*e) + 1))^2 + (cos(2*f*x + 2*e)^6 + sin(2*f*x + 2*e)^6 + 4*cos(2*f*x + 2*e)^5 + (3*cos(2*f*x + 2*

e)^2 + 4*cos(2*f*x + 2*e) + 2)*sin(2*f*x + 2*e)^4 + 6*cos(2*f*x + 2*e)^4 + (cos(2*f*x + 2*e)^4 + sin(2*f*x + 2

*e)^4 + 4*cos(2*f*x + 2*e)^3 + 2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*

x + 2*e)^2 + 4*cos(2*f*x + 2*e) + 1)*cos(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + 4*cos(2

*f*x + 2*e)^3 + 2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e)^2 + 4*

cos(2*f*x + 2*e) + 1)*cos(4*f*x + 4*e)^2 + 4*cos(2*f*x + 2*e)^3 + (cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + 4

*cos(2*f*x + 2*e)^3 + 2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e)^

2 + 4*cos(2*f*x + 2*e) + 1)*sin(6*f*x + 6*e)^2 + 4*(cos(2*f*x + 2*e)^4 + sin(2*f*x + 2*e)^4 + 4*cos(2*f*x + 2*

e)^3 + 2*(cos(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e)^2 + 6*cos(2*f*x + 2*e)^2 + 4*cos(2*f*x

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Fricas [A]
time = 3.44, size = 128, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (319 \, a c^{3} \cos \left (f x + e\right )^{5} + 317 \, a c^{3} \cos \left (f x + e\right )^{4} - 158 \, a c^{3} \cos \left (f x + e\right )^{3} - 26 \, a c^{3} \cos \left (f x + e\right )^{2} + 95 \, a c^{3} \cos \left (f x + e\right ) - 35 \, a c^{3}\right )} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{315 \, f \cos \left (f x + e\right )^{4} \sin \left (f x + e\right )} \end {gather*}

2/315*(319*a*c^3*cos(f*x + e)^5 + 317*a*c^3*cos(f*x + e)^4 - 158*a*c^3*cos(f*x + e)^3 - 26*a*c^3*cos(f*x + e)^

2 + 95*a*c^3*cos(f*x + e) - 35*a*c^3)*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))/(f*cos(f*x + e)^4*sin(f*x + e))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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Giac [A]
time = 1.03, size = 107, normalized size = 0.66 \begin {gather*} \frac {32 \, \sqrt {2} {\left (105 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{3} c^{2} + 189 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{3} + 135 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{4} + 35 \, c^{5}\right )} a c^{3}}{315 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {9}{2}} f} \end {gather*}

32/315*sqrt(2)*(105*(c*tan(1/2*f*x + 1/2*e)^2 - c)^3*c^2 + 189*(c*tan(1/2*f*x + 1/2*e)^2 - c)^2*c^3 + 135*(c*t

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Mupad [B]
time = 9.13, size = 483, normalized size = 2.96 \begin {gather*} \frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,2{}\mathrm {i}}{f}+\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,638{}\mathrm {i}}{315\,f}\right )}{{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,32{}\mathrm {i}}{9\,f}+\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,32{}\mathrm {i}}{9\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,96{}\mathrm {i}}{7\,f}+\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,32{}\mathrm {i}}{63\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,64{}\mathrm {i}}{5\,f}-\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,736{}\mathrm {i}}{105\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2}+\frac {\sqrt {c-\frac {c}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {a\,c^3\,8{}\mathrm {i}}{3\,f}-\frac {a\,c^3\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1256{}\mathrm {i}}{315\,f}\right )}{\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )} \end {gather*}

((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^3*2i)/f + (a*c^3*exp(e*1i + f*x*1i)*638i)

/(315*f)))/(exp(e*1i + f*x*1i) - 1) - ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^3*3

2i)/(9*f) + (a*c^3*exp(e*1i + f*x*1i)*32i)/(9*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^4) + ((c

- c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^3*96i)/(7*f) + (a*c^3*exp(e*1i + f*x*1i)*32i

)/(63*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1)^3) - ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i

+ f*x*1i)/2))^(1/2)*((a*c^3*64i)/(5*f) - (a*c^3*exp(e*1i + f*x*1i)*736i)/(105*f)))/((exp(e*1i + f*x*1i) - 1)*(

exp(e*2i + f*x*2i) + 1)^2) + ((c - c/(exp(- e*1i - f*x*1i)/2 + exp(e*1i + f*x*1i)/2))^(1/2)*((a*c^3*8i)/(3*f)

- (a*c^3*exp(e*1i + f*x*1i)*1256i)/(315*f)))/((exp(e*1i + f*x*1i) - 1)*(exp(e*2i + f*x*2i) + 1))

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3.1.64 \(\int \sec (e+f x) (a+a \sec (e+f x)) (c-c \sec (e+f x))^{7/2} \, dx\) [64]