3.22 \(\int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx\)

Optimal. Leaf size=206 \[ \frac{2 a^3 c^5 \tan ^7(e+f x)}{7 f}+\frac{45 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac{a^3 c^5 \tan ^5(e+f x) \sec ^3(e+f x)}{8 f}+\frac{5 a^3 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{48 f}-\frac{5 a^3 c^5 \tan (e+f x) \sec ^3(e+f x)}{64 f}-\frac{a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{6 f}+\frac{5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{24 f}-\frac{35 a^3 c^5 \tan (e+f x) \sec (e+f x)}{128 f} \]

[Out]

(45*a^3*c^5*ArcTanh[Sin[e + f*x]])/(128*f) - (35*a^3*c^5*Sec[e + f*x]*Tan[e + f*x])/(128*f) - (5*a^3*c^5*Sec[e
+ f*x]^3*Tan[e + f*x])/(64*f) + (5*a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) + (5*a^3*c^5*Sec[e + f*x]^3*Ta
n[e + f*x]^3)/(48*f) - (a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) - (a^3*c^5*Sec[e + f*x]^3*Tan[e + f*x]^5)/(
8*f) + (2*a^3*c^5*Tan[e + f*x]^7)/(7*f)

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Rubi [A]  time = 0.301823, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3958, 2611, 3770, 2607, 30, 3768} \[ \frac{2 a^3 c^5 \tan ^7(e+f x)}{7 f}+\frac{45 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac{a^3 c^5 \tan ^5(e+f x) \sec ^3(e+f x)}{8 f}+\frac{5 a^3 c^5 \tan ^3(e+f x) \sec ^3(e+f x)}{48 f}-\frac{5 a^3 c^5 \tan (e+f x) \sec ^3(e+f x)}{64 f}-\frac{a^3 c^5 \tan ^5(e+f x) \sec (e+f x)}{6 f}+\frac{5 a^3 c^5 \tan ^3(e+f x) \sec (e+f x)}{24 f}-\frac{35 a^3 c^5 \tan (e+f x) \sec (e+f x)}{128 f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5,x]

[Out]

(45*a^3*c^5*ArcTanh[Sin[e + f*x]])/(128*f) - (35*a^3*c^5*Sec[e + f*x]*Tan[e + f*x])/(128*f) - (5*a^3*c^5*Sec[e
+ f*x]^3*Tan[e + f*x])/(64*f) + (5*a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^3)/(24*f) + (5*a^3*c^5*Sec[e + f*x]^3*Ta
n[e + f*x]^3)/(48*f) - (a^3*c^5*Sec[e + f*x]*Tan[e + f*x]^5)/(6*f) - (a^3*c^5*Sec[e + f*x]^3*Tan[e + f*x]^5)/(
8*f) + (2*a^3*c^5*Tan[e + f*x]^7)/(7*f)

Rule 3958

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(n_.), x_Symbol] :> Dist[(-(a*c))^m, Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n
- m), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m,
n] && GeQ[n - m, 0] && GtQ[m*n, 0]

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3768

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

Rubi steps

\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^5 \, dx &=-\left (\left (a^3 c^3\right ) \int \left (c^2 \sec (e+f x) \tan ^6(e+f x)-2 c^2 \sec ^2(e+f x) \tan ^6(e+f x)+c^2 \sec ^3(e+f x) \tan ^6(e+f x)\right ) \, dx\right )\\ &=-\left (\left (a^3 c^5\right ) \int \sec (e+f x) \tan ^6(e+f x) \, dx\right )-\left (a^3 c^5\right ) \int \sec ^3(e+f x) \tan ^6(e+f x) \, dx+\left (2 a^3 c^5\right ) \int \sec ^2(e+f x) \tan ^6(e+f x) \, dx\\ &=-\frac{a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac{a^3 c^5 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac{1}{8} \left (5 a^3 c^5\right ) \int \sec ^3(e+f x) \tan ^4(e+f x) \, dx+\frac{1}{6} \left (5 a^3 c^5\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx+\frac{\left (2 a^3 c^5\right ) \operatorname{Subst}\left (\int x^6 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac{5 a^3 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{48 f}-\frac{a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac{a^3 c^5 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac{2 a^3 c^5 \tan ^7(e+f x)}{7 f}-\frac{1}{16} \left (5 a^3 c^5\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx-\frac{1}{8} \left (5 a^3 c^5\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx\\ &=-\frac{5 a^3 c^5 \sec (e+f x) \tan (e+f x)}{16 f}-\frac{5 a^3 c^5 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac{5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac{5 a^3 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{48 f}-\frac{a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac{a^3 c^5 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac{2 a^3 c^5 \tan ^7(e+f x)}{7 f}+\frac{1}{64} \left (5 a^3 c^5\right ) \int \sec ^3(e+f x) \, dx+\frac{1}{16} \left (5 a^3 c^5\right ) \int \sec (e+f x) \, dx\\ &=\frac{5 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac{35 a^3 c^5 \sec (e+f x) \tan (e+f x)}{128 f}-\frac{5 a^3 c^5 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac{5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac{5 a^3 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{48 f}-\frac{a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac{a^3 c^5 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac{2 a^3 c^5 \tan ^7(e+f x)}{7 f}+\frac{1}{128} \left (5 a^3 c^5\right ) \int \sec (e+f x) \, dx\\ &=\frac{45 a^3 c^5 \tanh ^{-1}(\sin (e+f x))}{128 f}-\frac{35 a^3 c^5 \sec (e+f x) \tan (e+f x)}{128 f}-\frac{5 a^3 c^5 \sec ^3(e+f x) \tan (e+f x)}{64 f}+\frac{5 a^3 c^5 \sec (e+f x) \tan ^3(e+f x)}{24 f}+\frac{5 a^3 c^5 \sec ^3(e+f x) \tan ^3(e+f x)}{48 f}-\frac{a^3 c^5 \sec (e+f x) \tan ^5(e+f x)}{6 f}-\frac{a^3 c^5 \sec ^3(e+f x) \tan ^5(e+f x)}{8 f}+\frac{2 a^3 c^5 \tan ^7(e+f x)}{7 f}\\ \end{align*}

Mathematica [A]  time = 2.3447, size = 111, normalized size = 0.54 \[ -\frac{a^3 c^5 \left ((5705 \sin (e+f x)-1792 \sin (2 (e+f x))+21 \sin (3 (e+f x))+1792 \sin (4 (e+f x))+2065 \sin (5 (e+f x))-768 \sin (6 (e+f x))+581 \sin (7 (e+f x))+128 \sin (8 (e+f x))) \sec ^8(e+f x)-20160 \tanh ^{-1}(\sin (e+f x))\right )}{57344 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^3*(c - c*Sec[e + f*x])^5,x]

[Out]

-(a^3*c^5*(-20160*ArcTanh[Sin[e + f*x]] + Sec[e + f*x]^8*(5705*Sin[e + f*x] - 1792*Sin[2*(e + f*x)] + 21*Sin[3
*(e + f*x)] + 1792*Sin[4*(e + f*x)] + 2065*Sin[5*(e + f*x)] - 768*Sin[6*(e + f*x)] + 581*Sin[7*(e + f*x)] + 12
8*Sin[8*(e + f*x)])))/(57344*f)

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Maple [A]  time = 0.035, size = 217, normalized size = 1.1 \begin{align*} -{\frac{2\,{a}^{3}{c}^{5}\tan \left ( fx+e \right ) }{7\,f}}-{\frac{6\,{a}^{3}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{7\,f}}+{\frac{6\,{a}^{3}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{7\,f}}-{\frac{83\,{a}^{3}{c}^{5}\sec \left ( fx+e \right ) \tan \left ( fx+e \right ) }{128\,f}}+{\frac{45\,{a}^{3}{c}^{5}\ln \left ( \sec \left ( fx+e \right ) +\tan \left ( fx+e \right ) \right ) }{128\,f}}+{\frac{3\,{a}^{3}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{5}}{16\,f}}+{\frac{15\,{a}^{3}{c}^{5} \left ( \sec \left ( fx+e \right ) \right ) ^{3}\tan \left ( fx+e \right ) }{64\,f}}+{\frac{2\,{a}^{3}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{6}}{7\,f}}-{\frac{{a}^{3}{c}^{5}\tan \left ( fx+e \right ) \left ( \sec \left ( fx+e \right ) \right ) ^{7}}{8\,f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x)

[Out]

-2/7/f*a^3*c^5*tan(f*x+e)-6/7/f*a^3*c^5*tan(f*x+e)*sec(f*x+e)^4+6/7/f*a^3*c^5*tan(f*x+e)*sec(f*x+e)^2-83/128*a
^3*c^5*sec(f*x+e)*tan(f*x+e)/f+45/128/f*a^3*c^5*ln(sec(f*x+e)+tan(f*x+e))+3/16/f*a^3*c^5*tan(f*x+e)*sec(f*x+e)
^5+15/64*a^3*c^5*sec(f*x+e)^3*tan(f*x+e)/f+2/7/f*a^3*c^5*tan(f*x+e)*sec(f*x+e)^6-1/8/f*a^3*c^5*tan(f*x+e)*sec(
f*x+e)^7

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Maxima [B]  time = 1.01107, size = 551, normalized size = 2.67 \begin{align*} \frac{1536 \,{\left (5 \, \tan \left (f x + e\right )^{7} + 21 \, \tan \left (f x + e\right )^{5} + 35 \, \tan \left (f x + e\right )^{3} + 35 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} - 10752 \,{\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} + 53760 \,{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{5} + 35 \, a^{3} c^{5}{\left (\frac{2 \,{\left (105 \, \sin \left (f x + e\right )^{7} - 385 \, \sin \left (f x + e\right )^{5} + 511 \, \sin \left (f x + e\right )^{3} - 279 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{8} - 4 \, \sin \left (f x + e\right )^{6} + 6 \, \sin \left (f x + e\right )^{4} - 4 \, \sin \left (f x + e\right )^{2} + 1} - 105 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 105 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 560 \, a^{3} c^{5}{\left (\frac{2 \,{\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 13440 \, a^{3} c^{5}{\left (\frac{2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 26880 \, a^{3} c^{5} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 53760 \, a^{3} c^{5} \tan \left (f x + e\right )}{26880 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x, algorithm="maxima")

[Out]

1/26880*(1536*(5*tan(f*x + e)^7 + 21*tan(f*x + e)^5 + 35*tan(f*x + e)^3 + 35*tan(f*x + e))*a^3*c^5 - 10752*(3*
tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^3*c^5 + 53760*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^3*c^
5 + 35*a^3*c^5*(2*(105*sin(f*x + e)^7 - 385*sin(f*x + e)^5 + 511*sin(f*x + e)^3 - 279*sin(f*x + e))/(sin(f*x +
e)^8 - 4*sin(f*x + e)^6 + 6*sin(f*x + e)^4 - 4*sin(f*x + e)^2 + 1) - 105*log(sin(f*x + e) + 1) + 105*log(sin(
f*x + e) - 1)) - 560*a^3*c^5*(2*(15*sin(f*x + e)^5 - 40*sin(f*x + e)^3 + 33*sin(f*x + e))/(sin(f*x + e)^6 - 3*
sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1) - 15*log(sin(f*x + e) + 1) + 15*log(sin(f*x + e) - 1)) + 13440*a^3*c^5*
(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) + 26880*a^3*c^5*log(sec(
f*x + e) + tan(f*x + e)) - 53760*a^3*c^5*tan(f*x + e))/f

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Fricas [A]  time = 0.522448, size = 489, normalized size = 2.37 \begin{align*} \frac{315 \, a^{3} c^{5} \cos \left (f x + e\right )^{8} \log \left (\sin \left (f x + e\right ) + 1\right ) - 315 \, a^{3} c^{5} \cos \left (f x + e\right )^{8} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \,{\left (256 \, a^{3} c^{5} \cos \left (f x + e\right )^{7} + 581 \, a^{3} c^{5} \cos \left (f x + e\right )^{6} - 768 \, a^{3} c^{5} \cos \left (f x + e\right )^{5} - 210 \, a^{3} c^{5} \cos \left (f x + e\right )^{4} + 768 \, a^{3} c^{5} \cos \left (f x + e\right )^{3} - 168 \, a^{3} c^{5} \cos \left (f x + e\right )^{2} - 256 \, a^{3} c^{5} \cos \left (f x + e\right ) + 112 \, a^{3} c^{5}\right )} \sin \left (f x + e\right )}{1792 \, f \cos \left (f x + e\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x, algorithm="fricas")

[Out]

1/1792*(315*a^3*c^5*cos(f*x + e)^8*log(sin(f*x + e) + 1) - 315*a^3*c^5*cos(f*x + e)^8*log(-sin(f*x + e) + 1) -
2*(256*a^3*c^5*cos(f*x + e)^7 + 581*a^3*c^5*cos(f*x + e)^6 - 768*a^3*c^5*cos(f*x + e)^5 - 210*a^3*c^5*cos(f*x
+ e)^4 + 768*a^3*c^5*cos(f*x + e)^3 - 168*a^3*c^5*cos(f*x + e)^2 - 256*a^3*c^5*cos(f*x + e) + 112*a^3*c^5)*si
n(f*x + e))/(f*cos(f*x + e)^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - a^{3} c^{5} \left (\int - \sec{\left (e + f x \right )}\, dx + \int 2 \sec ^{2}{\left (e + f x \right )}\, dx + \int 2 \sec ^{3}{\left (e + f x \right )}\, dx + \int - 6 \sec ^{4}{\left (e + f x \right )}\, dx + \int 6 \sec ^{6}{\left (e + f x \right )}\, dx + \int - 2 \sec ^{7}{\left (e + f x \right )}\, dx + \int - 2 \sec ^{8}{\left (e + f x \right )}\, dx + \int \sec ^{9}{\left (e + f x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))**3*(c-c*sec(f*x+e))**5,x)

[Out]

-a**3*c**5*(Integral(-sec(e + f*x), x) + Integral(2*sec(e + f*x)**2, x) + Integral(2*sec(e + f*x)**3, x) + Int
egral(-6*sec(e + f*x)**4, x) + Integral(6*sec(e + f*x)**6, x) + Integral(-2*sec(e + f*x)**7, x) + Integral(-2*
sec(e + f*x)**8, x) + Integral(sec(e + f*x)**9, x))

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Giac [A]  time = 1.42727, size = 306, normalized size = 1.49 \begin{align*} \frac{315 \, a^{3} c^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right ) - 315 \, a^{3} c^{5} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right ) - \frac{2 \,{\left (315 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{15} - 2415 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{13} + 8043 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} + 17609 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} - 15159 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 8043 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 2415 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 315 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right )}^{8}}}{896 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c-c*sec(f*x+e))^5,x, algorithm="giac")

[Out]

1/896*(315*a^3*c^5*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 315*a^3*c^5*log(abs(tan(1/2*f*x + 1/2*e) - 1)) - 2*(31
5*a^3*c^5*tan(1/2*f*x + 1/2*e)^15 - 2415*a^3*c^5*tan(1/2*f*x + 1/2*e)^13 + 8043*a^3*c^5*tan(1/2*f*x + 1/2*e)^1
1 + 17609*a^3*c^5*tan(1/2*f*x + 1/2*e)^9 - 15159*a^3*c^5*tan(1/2*f*x + 1/2*e)^7 + 8043*a^3*c^5*tan(1/2*f*x + 1
/2*e)^5 - 2415*a^3*c^5*tan(1/2*f*x + 1/2*e)^3 + 315*a^3*c^5*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)
^8)/f