3.233 \(\int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx\)

Optimal. Leaf size=368 \[ -\frac{d^3 \left (20 c^2+30 c d+13 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{a^3 f (c-d)^{11/2} (c+d)^{5/2}}+\frac{d \left (142 c^2 d^2-30 c^3 d+4 c^4+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 a^3 f (c-d)^5 (c+d)^2 (c+d \sec (e+f x))}+\frac{d \left (-30 c^2 d+4 c^3+146 c d^2+195 d^3\right ) \tan (e+f x)}{30 a^3 f (c-d)^4 (c+d) (c+d \sec (e+f x))^2}+\frac{\left (2 c^2-15 c d+76 d^2\right ) \tan (e+f x)}{15 f (c-d)^3 \left (a^3 \sec (e+f x)+a^3\right ) (c+d \sec (e+f x))^2}+\frac{(2 c-11 d) \tan (e+f x)}{15 a f (c-d)^2 (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2}+\frac{\tan (e+f x)}{5 f (c-d) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^2} \]

[Out]

-((d^3*(20*c^2 + 30*c*d + 13*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(a^3*(c - d)^(11/2)*(c
+ d)^(5/2)*f)) + (d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*Tan[e + f*x])/(30*a^3*(c - d)^4*(c + d)*f*(c + d*
Sec[e + f*x])^2) + Tan[e + f*x]/(5*(c - d)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2) + ((2*c - 11*d)*Ta
n[e + f*x])/(15*a*(c - d)^2*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) + ((2*c^2 - 15*c*d + 76*d^2)*Tan[
e + f*x])/(15*(c - d)^3*f*(a^3 + a^3*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) + (d*(4*c^4 - 30*c^3*d + 142*c^2*d^
2 + 525*c*d^3 + 304*d^4)*Tan[e + f*x])/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*Sec[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.706437, antiderivative size = 414, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {3987, 103, 151, 152, 12, 93, 205} \[ \frac{d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}{\sqrt{c-d} \sqrt{a-a \sec (e+f x)}}\right )}{a^2 f (c-d)^{11/2} (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\left (142 c^2 d^2-30 c^3 d+4 c^4+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 f (c-d)^5 (c+d)^2 \left (a^3 \sec (e+f x)+a^3\right )}-\frac{3 d (2 c+d) \tan (e+f x)}{2 f \left (c^2-d^2\right )^2 (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))}-\frac{d \tan (e+f x)}{2 f \left (c^2-d^2\right ) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^2}+\frac{\left (-26 c^2 d+4 c^3-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a f (c-d)^4 (c+d)^2 (a \sec (e+f x)+a)^2}+\frac{\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 f (c-d)^3 (c+d)^2 (a \sec (e+f x)+a)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c^2 + 39*c*d + 22*d^2)*Tan[e + f*x])/(10*(c - d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])^3) + ((4*c^3 - 26*c^2*
d - 184*c*d^2 - 109*d^3)*Tan[e + f*x])/(30*a*(c - d)^4*(c + d)^2*f*(a + a*Sec[e + f*x])^2) + (d^3*(20*c^2 + 30
*c*d + 13*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f
*x])/(a^2*(c - d)^(11/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((4*c^4 - 30*c^3
*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*Tan[e + f*x])/(30*(c - d)^5*(c + d)^2*f*(a^3 + a^3*Sec[e + f*x])) - (d
*Tan[e + f*x])/(2*(c^2 - d^2)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2) - (3*d*(2*c + d)*Tan[e + f*x])/
(2*(c^2 - d^2)^2*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x]))

Rule 3987

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))^(n_), x_Symbol] :> Dist[(a^2*g*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x
]]), Subst[Int[((g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^n)/Sqrt[a - b*x], x], x, Csc[e + f*x]], x] /; Free
Q[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p,
1] || IntegerQ[m - 1/2])

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (a+a x)^{7/2} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{a^2 (2 c+3 d)-4 a^2 d x}{\sqrt{a-a x} (a+a x)^{7/2} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{2 \left (c^2-d^2\right ) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac{3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}-\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{a^4 \left (2 c^2+21 c d+13 d^2\right )-9 a^4 d (2 c+d) x}{\sqrt{a-a x} (a+a x)^{7/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 a^2 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac{3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{-a^6 \left (4 c^3-22 c^2 d-106 c d^2-65 d^3\right )-2 a^6 d \left (2 c^2+39 c d+22 d^2\right ) x}{\sqrt{a-a x} (a+a x)^{5/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{10 a^5 (c-d) \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac{\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac{3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}-\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{a^8 (c+d) \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right )+a^8 d \left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) x}{\sqrt{a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{30 a^8 (c-d)^2 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac{\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac{\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 (c-d)^5 (c+d)^2 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac{3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{15 a^{10} d^3 \left (20 c^2+30 c d+13 d^2\right )}{\sqrt{a-a x} \sqrt{a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{30 a^{11} (c-d)^3 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac{\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac{\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 (c-d)^5 (c+d)^2 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac{3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac{\left (d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} \sqrt{a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 a (c-d)^3 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac{\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac{\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 (c-d)^5 (c+d)^2 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac{3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}+\frac{\left (d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{a c-a d-(-a c-a d) x^2} \, dx,x,\frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{a-a \sec (e+f x)}}\right )}{a (c-d)^3 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+39 c d+22 d^2\right ) \tan (e+f x)}{10 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^3}+\frac{\left (4 c^3-26 c^2 d-184 c d^2-109 d^3\right ) \tan (e+f x)}{30 a (c-d)^4 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac{d^3 \left (20 c^2+30 c d+13 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+a \sec (e+f x)}}{\sqrt{c-d} \sqrt{a-a \sec (e+f x)}}\right ) \tan (e+f x)}{a^2 (c-d)^{11/2} (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\left (4 c^4-30 c^3 d+142 c^2 d^2+525 c d^3+304 d^4\right ) \tan (e+f x)}{30 (c-d)^5 (c+d)^2 f \left (a^3+a^3 \sec (e+f x)\right )}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2}-\frac{3 d (2 c+d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^3 (c+d \sec (e+f x))}\\ \end{align*}

Mathematica [C]  time = 7.69333, size = 1096, normalized size = 2.98 \[ \frac{4 \cos ^4\left (\frac{e}{2}+\frac{f x}{2}\right ) (d+c \cos (e+f x))^3 \sec \left (\frac{e}{2}\right ) \left (23 d \sin \left (\frac{e}{2}\right )-8 c \sin \left (\frac{e}{2}\right )\right ) \sec ^6(e+f x)}{15 (d-c)^4 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}+\frac{\left (20 c^2+30 d c+13 d^2\right ) \cos ^6\left (\frac{e}{2}+\frac{f x}{2}\right ) (d+c \cos (e+f x))^3 \left (-\frac{8 i \tan ^{-1}\left (\sec \left (\frac{f x}{2}\right ) \left (\frac{\cos (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{i \sin (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}\right ) \left (i c \sin \left (e+\frac{f x}{2}\right )-i d \sin \left (\frac{f x}{2}\right )\right )\right ) \cos (e) d^3}{\sqrt{c^2-d^2} f \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{8 \tan ^{-1}\left (\sec \left (\frac{f x}{2}\right ) \left (\frac{\cos (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}-\frac{i \sin (e)}{\sqrt{c^2-d^2} \sqrt{\cos (2 e)-i \sin (2 e)}}\right ) \left (i c \sin \left (e+\frac{f x}{2}\right )-i d \sin \left (\frac{f x}{2}\right )\right )\right ) \sin (e) d^3}{\sqrt{c^2-d^2} f \sqrt{\cos (2 e)-i \sin (2 e)}}\right ) \sec ^6(e+f x)}{(d-c)^5 (c+d)^2 (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}-\frac{2 \cos \left (\frac{e}{2}+\frac{f x}{2}\right ) (d+c \cos (e+f x))^3 \sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \sec ^6(e+f x)}{5 (d-c)^3 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}+\frac{4 \cos ^3\left (\frac{e}{2}+\frac{f x}{2}\right ) (d+c \cos (e+f x))^3 \sec \left (\frac{e}{2}\right ) \left (23 d \sin \left (\frac{f x}{2}\right )-8 c \sin \left (\frac{f x}{2}\right )\right ) \sec ^6(e+f x)}{15 (d-c)^4 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}-\frac{8 \cos ^5\left (\frac{e}{2}+\frac{f x}{2}\right ) (d+c \cos (e+f x))^3 \sec \left (\frac{e}{2}\right ) \left (7 \sin \left (\frac{f x}{2}\right ) c^2-44 d \sin \left (\frac{f x}{2}\right ) c+127 d^2 \sin \left (\frac{f x}{2}\right )\right ) \sec ^6(e+f x)}{15 (d-c)^5 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}+\frac{4 \cos ^6\left (\frac{e}{2}+\frac{f x}{2}\right ) (d+c \cos (e+f x)) \sec (e) \left (d^6 \sin (e)-c d^5 \sin (f x)\right ) \sec ^6(e+f x)}{c^2 (d-c)^4 (c+d) f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}-\frac{4 \cos ^6\left (\frac{e}{2}+\frac{f x}{2}\right ) (d+c \cos (e+f x))^2 \sec (e) \left (2 \sin (e) d^7-6 c \sin (e) d^6-c \sin (f x) d^6-11 c^2 \sin (e) d^5+6 c^2 \sin (f x) d^5+10 c^3 \sin (f x) d^4\right ) \sec ^6(e+f x)}{c^2 (d-c)^5 (c+d)^2 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3}-\frac{2 \cos ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) (d+c \cos (e+f x))^3 \tan \left (\frac{e}{2}\right ) \sec ^6(e+f x)}{5 (d-c)^3 f (\sec (e+f x) a+a)^3 (c+d \sec (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Cos[e/2 + (f*x)/2]^4*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*(-8*c*Sin[e/2] + 23*d*Sin[e/2]))/(15*(-
c + d)^4*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) + ((20*c^2 + 30*c*d + 13*d^2)*Cos[e/2 + (f*x)/2]^6*(
d + c*Cos[e + f*x])^3*Sec[e + f*x]^6*(((-8*I)*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] -
I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (
f*x)/2])]*Cos[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (8*d^3*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c
^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin
[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Sin[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((-c + d)^5*(c +
d)^2*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) - (2*Cos[e/2 + (f*x)/2]*(d + c*Cos[e + f*x])^3*Sec[e/2]*Se
c[e + f*x]^6*Sin[(f*x)/2])/(5*(-c + d)^3*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) + (4*Cos[e/2 + (f*x)
/2]^3*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[e + f*x]^6*(-8*c*Sin[(f*x)/2] + 23*d*Sin[(f*x)/2]))/(15*(-c + d)^4*f
*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) - (8*Cos[e/2 + (f*x)/2]^5*(d + c*Cos[e + f*x])^3*Sec[e/2]*Sec[
e + f*x]^6*(7*c^2*Sin[(f*x)/2] - 44*c*d*Sin[(f*x)/2] + 127*d^2*Sin[(f*x)/2]))/(15*(-c + d)^5*f*(a + a*Sec[e +
f*x])^3*(c + d*Sec[e + f*x])^3) + (4*Cos[e/2 + (f*x)/2]^6*(d + c*Cos[e + f*x])*Sec[e]*Sec[e + f*x]^6*(d^6*Sin[
e] - c*d^5*Sin[f*x]))/(c^2*(-c + d)^4*(c + d)*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3) - (4*Cos[e/2 +
(f*x)/2]^6*(d + c*Cos[e + f*x])^2*Sec[e]*Sec[e + f*x]^6*(-11*c^2*d^5*Sin[e] - 6*c*d^6*Sin[e] + 2*d^7*Sin[e] +
10*c^3*d^4*Sin[f*x] + 6*c^2*d^5*Sin[f*x] - c*d^6*Sin[f*x]))/(c^2*(-c + d)^5*(c + d)^2*f*(a + a*Sec[e + f*x])^3
*(c + d*Sec[e + f*x])^3) - (2*Cos[e/2 + (f*x)/2]^2*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^6*Tan[e/2])/(5*(-c + d)
^3*f*(a + a*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3)

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Maple [A]  time = 0.116, size = 365, normalized size = 1. \begin{align*}{\frac{1}{4\,f{a}^{3}} \left ({\frac{1}{ \left ({c}^{3}-3\,{c}^{2}d+3\,{d}^{2}c-{d}^{3} \right ) \left ({c}^{2}-2\,cd+{d}^{2} \right ) } \left ({\frac{{c}^{2}}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{2\,cd}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{{d}^{2}}{5} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}-{\frac{2\,{c}^{2}}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+{\frac{10\,cd}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{8\,{d}^{2}}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ){c}^{2}-8\,cd\tan \left ( 1/2\,fx+e/2 \right ) +31\,\tan \left ( 1/2\,fx+e/2 \right ){d}^{2} \right ) }+16\,{\frac{{d}^{3}}{ \left ( c-d \right ) ^{5}} \left ({\frac{1}{ \left ( \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}c- \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}d-c-d \right ) ^{2}} \left ( -1/4\,{\frac{d \left ( 10\,{c}^{2}-3\,cd-7\,{d}^{2} \right ) \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{3}}{{c}^{2}+2\,cd+{d}^{2}}}+5/4\,{\frac{d \left ( 2\,c+d \right ) \tan \left ( 1/2\,fx+e/2 \right ) }{c+d}} \right ) }-1/4\,{\frac{20\,{c}^{2}+30\,cd+13\,{d}^{2}}{ \left ({c}^{2}+2\,cd+{d}^{2} \right ) \sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}{\it Artanh} \left ({\frac{\tan \left ( 1/2\,fx+e/2 \right ) \left ( c-d \right ) }{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}} \right ) } \right ) } \right ) } \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+d*sec(f*x+e))^3,x)

[Out]

1/4/f/a^3*(1/(c^3-3*c^2*d+3*c*d^2-d^3)/(c^2-2*c*d+d^2)*(1/5*tan(1/2*f*x+1/2*e)^5*c^2-2/5*tan(1/2*f*x+1/2*e)^5*
c*d+1/5*tan(1/2*f*x+1/2*e)^5*d^2-2/3*tan(1/2*f*x+1/2*e)^3*c^2+10/3*tan(1/2*f*x+1/2*e)^3*c*d-8/3*tan(1/2*f*x+1/
2*e)^3*d^2+tan(1/2*f*x+1/2*e)*c^2-8*c*d*tan(1/2*f*x+1/2*e)+31*tan(1/2*f*x+1/2*e)*d^2)+16*d^3/(c-d)^5*((-1/4*d*
(10*c^2-3*c*d-7*d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3+5/4*d*(2*c+d)/(c+d)*tan(1/2*f*x+1/2*e))/(tan(1/2*f*x
+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^2-1/4*(20*c^2+30*c*d+13*d^2)/(c^2+2*c*d+d^2)/((c+d)*(c-d))^(1/2)*arcta
nh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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+d*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.895089, size = 5948, normalized size = 16.16 \begin{align*} \text{result too large to display} \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+d*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

[-1/60*(15*(20*c^2*d^5 + 30*c*d^6 + 13*d^7 + (20*c^4*d^3 + 30*c^3*d^4 + 13*c^2*d^5)*cos(f*x + e)^5 + (60*c^4*d
^3 + 130*c^3*d^4 + 99*c^2*d^5 + 26*c*d^6)*cos(f*x + e)^4 + (60*c^4*d^3 + 210*c^3*d^4 + 239*c^2*d^5 + 108*c*d^6
+ 13*d^7)*cos(f*x + e)^3 + (20*c^4*d^3 + 150*c^3*d^4 + 253*c^2*d^5 + 168*c*d^6 + 39*d^7)*cos(f*x + e)^2 + (40
*c^3*d^4 + 120*c^2*d^5 + 116*c*d^6 + 39*d^7)*cos(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x + e) - (c^2 - 2*
d^2)*cos(f*x + e)^2 + 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 +
2*c*d*cos(f*x + e) + d^2)) - 2*(4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7
- 304*d^8 + (14*c^8 - 60*c^7*d + 78*c^6*d^2 + 480*c^5*d^3 + 312*c^4*d^4 - 330*c^3*d^5 - 419*c^2*d^6 - 90*c*d^7
+ 15*d^8)*cos(f*x + e)^4 + (12*c^8 - 62*c^7*d + 114*c^6*d^2 + 1056*c^5*d^3 + 1626*c^4*d^4 - 81*c^3*d^5 - 1707
*c^2*d^6 - 913*c*d^7 - 45*d^8)*cos(f*x + e)^3 + (4*c^8 - 6*c^7*d - 28*c^6*d^2 + 828*c^5*d^3 + 2400*c^4*d^4 + 1
197*c^3*d^5 - 1897*c^2*d^6 - 2019*c*d^7 - 479*d^8)*cos(f*x + e)^2 + (8*c^7*d - 48*c^6*d^2 + 186*c^5*d^3 + 1224
*c^4*d^4 + 1539*c^3*d^5 - 459*c^2*d^6 - 1733*c*d^7 - 717*d^8)*cos(f*x + e))*sin(f*x + e))/((a^3*c^11 - 3*a^3*c
^10*d + 8*a^3*c^8*d^3 - 6*a^3*c^7*d^4 - 6*a^3*c^6*d^5 + 8*a^3*c^5*d^6 - 3*a^3*c^3*d^8 + a^3*c^2*d^9)*f*cos(f*x
+ e)^5 + (3*a^3*c^11 - 7*a^3*c^10*d - 6*a^3*c^9*d^2 + 24*a^3*c^8*d^3 - 2*a^3*c^7*d^4 - 30*a^3*c^6*d^5 + 12*a^
3*c^5*d^6 + 16*a^3*c^4*d^7 - 9*a^3*c^3*d^8 - 3*a^3*c^2*d^9 + 2*a^3*c*d^10)*f*cos(f*x + e)^4 + (3*a^3*c^11 - 3*
a^3*c^10*d - 17*a^3*c^9*d^2 + 21*a^3*c^8*d^3 + 30*a^3*c^7*d^4 - 46*a^3*c^6*d^5 - 18*a^3*c^5*d^6 + 42*a^3*c^4*d
^7 - a^3*c^3*d^8 - 15*a^3*c^2*d^9 + 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^3 + (a^3*c^11 + 3*a^3*c^10*d - 15*
a^3*c^9*d^2 - a^3*c^8*d^3 + 42*a^3*c^7*d^4 - 18*a^3*c^6*d^5 - 46*a^3*c^5*d^6 + 30*a^3*c^4*d^7 + 21*a^3*c^3*d^8
- 17*a^3*c^2*d^9 - 3*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^2 + (2*a^3*c^10*d - 3*a^3*c^9*d^2 - 9*a^3*c^8*d^
3 + 16*a^3*c^7*d^4 + 12*a^3*c^6*d^5 - 30*a^3*c^5*d^6 - 2*a^3*c^4*d^7 + 24*a^3*c^3*d^8 - 6*a^3*c^2*d^9 - 7*a^3*
c*d^10 + 3*a^3*d^11)*f*cos(f*x + e) + (a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4
*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f), -1/30*(15*(20*c^2*d^5 + 30*c*d^6 + 13*d^7 + (20*c^4*d^3 +
30*c^3*d^4 + 13*c^2*d^5)*cos(f*x + e)^5 + (60*c^4*d^3 + 130*c^3*d^4 + 99*c^2*d^5 + 26*c*d^6)*cos(f*x + e)^4 +
(60*c^4*d^3 + 210*c^3*d^4 + 239*c^2*d^5 + 108*c*d^6 + 13*d^7)*cos(f*x + e)^3 + (20*c^4*d^3 + 150*c^3*d^4 + 253
*c^2*d^5 + 168*c*d^6 + 39*d^7)*cos(f*x + e)^2 + (40*c^3*d^4 + 120*c^2*d^5 + 116*c*d^6 + 39*d^7)*cos(f*x + e))*
sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) - (4*c^6*d^2 - 30*c
^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7 - 304*d^8 + (14*c^8 - 60*c^7*d + 78*c^6*d^2 + 480
*c^5*d^3 + 312*c^4*d^4 - 330*c^3*d^5 - 419*c^2*d^6 - 90*c*d^7 + 15*d^8)*cos(f*x + e)^4 + (12*c^8 - 62*c^7*d +
114*c^6*d^2 + 1056*c^5*d^3 + 1626*c^4*d^4 - 81*c^3*d^5 - 1707*c^2*d^6 - 913*c*d^7 - 45*d^8)*cos(f*x + e)^3 + (
4*c^8 - 6*c^7*d - 28*c^6*d^2 + 828*c^5*d^3 + 2400*c^4*d^4 + 1197*c^3*d^5 - 1897*c^2*d^6 - 2019*c*d^7 - 479*d^8
)*cos(f*x + e)^2 + (8*c^7*d - 48*c^6*d^2 + 186*c^5*d^3 + 1224*c^4*d^4 + 1539*c^3*d^5 - 459*c^2*d^6 - 1733*c*d^
7 - 717*d^8)*cos(f*x + e))*sin(f*x + e))/((a^3*c^11 - 3*a^3*c^10*d + 8*a^3*c^8*d^3 - 6*a^3*c^7*d^4 - 6*a^3*c^6
*d^5 + 8*a^3*c^5*d^6 - 3*a^3*c^3*d^8 + a^3*c^2*d^9)*f*cos(f*x + e)^5 + (3*a^3*c^11 - 7*a^3*c^10*d - 6*a^3*c^9*
d^2 + 24*a^3*c^8*d^3 - 2*a^3*c^7*d^4 - 30*a^3*c^6*d^5 + 12*a^3*c^5*d^6 + 16*a^3*c^4*d^7 - 9*a^3*c^3*d^8 - 3*a^
3*c^2*d^9 + 2*a^3*c*d^10)*f*cos(f*x + e)^4 + (3*a^3*c^11 - 3*a^3*c^10*d - 17*a^3*c^9*d^2 + 21*a^3*c^8*d^3 + 30
*a^3*c^7*d^4 - 46*a^3*c^6*d^5 - 18*a^3*c^5*d^6 + 42*a^3*c^4*d^7 - a^3*c^3*d^8 - 15*a^3*c^2*d^9 + 3*a^3*c*d^10
+ a^3*d^11)*f*cos(f*x + e)^3 + (a^3*c^11 + 3*a^3*c^10*d - 15*a^3*c^9*d^2 - a^3*c^8*d^3 + 42*a^3*c^7*d^4 - 18*a
^3*c^6*d^5 - 46*a^3*c^5*d^6 + 30*a^3*c^4*d^7 + 21*a^3*c^3*d^8 - 17*a^3*c^2*d^9 - 3*a^3*c*d^10 + 3*a^3*d^11)*f*
cos(f*x + e)^2 + (2*a^3*c^10*d - 3*a^3*c^9*d^2 - 9*a^3*c^8*d^3 + 16*a^3*c^7*d^4 + 12*a^3*c^6*d^5 - 30*a^3*c^5*
d^6 - 2*a^3*c^4*d^7 + 24*a^3*c^3*d^8 - 6*a^3*c^2*d^9 - 7*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e) + (a^3*c^9*d^
2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f
)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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+d*sec(f*x+e))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.53807, size = 1916, normalized size = 5.21 \begin{align*} \text{result too large to display} \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+d*sec(f*x+e))^3,x, algorithm="giac")

[Out]

-1/60*(60*(20*c^2*d^3 + 30*c*d^4 + 13*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) + arctan(-(c*tan(
1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2)))/((a^3*c^7 - 3*a^3*c^6*d + a^3*c^5*d^2 + 5*a^3*c^
4*d^3 - 5*a^3*c^3*d^4 - a^3*c^2*d^5 + 3*a^3*c*d^6 - a^3*d^7)*sqrt(-c^2 + d^2)) - (3*a^12*c^12*tan(1/2*f*x + 1/
2*e)^5 - 36*a^12*c^11*d*tan(1/2*f*x + 1/2*e)^5 + 198*a^12*c^10*d^2*tan(1/2*f*x + 1/2*e)^5 - 660*a^12*c^9*d^3*t
an(1/2*f*x + 1/2*e)^5 + 1485*a^12*c^8*d^4*tan(1/2*f*x + 1/2*e)^5 - 2376*a^12*c^7*d^5*tan(1/2*f*x + 1/2*e)^5 +
2772*a^12*c^6*d^6*tan(1/2*f*x + 1/2*e)^5 - 2376*a^12*c^5*d^7*tan(1/2*f*x + 1/2*e)^5 + 1485*a^12*c^4*d^8*tan(1/
2*f*x + 1/2*e)^5 - 660*a^12*c^3*d^9*tan(1/2*f*x + 1/2*e)^5 + 198*a^12*c^2*d^10*tan(1/2*f*x + 1/2*e)^5 - 36*a^1
2*c*d^11*tan(1/2*f*x + 1/2*e)^5 + 3*a^12*d^12*tan(1/2*f*x + 1/2*e)^5 - 10*a^12*c^12*tan(1/2*f*x + 1/2*e)^3 + 1
50*a^12*c^11*d*tan(1/2*f*x + 1/2*e)^3 - 990*a^12*c^10*d^2*tan(1/2*f*x + 1/2*e)^3 + 3850*a^12*c^9*d^3*tan(1/2*f
*x + 1/2*e)^3 - 9900*a^12*c^8*d^4*tan(1/2*f*x + 1/2*e)^3 + 17820*a^12*c^7*d^5*tan(1/2*f*x + 1/2*e)^3 - 23100*a
^12*c^6*d^6*tan(1/2*f*x + 1/2*e)^3 + 21780*a^12*c^5*d^7*tan(1/2*f*x + 1/2*e)^3 - 14850*a^12*c^4*d^8*tan(1/2*f*
x + 1/2*e)^3 + 7150*a^12*c^3*d^9*tan(1/2*f*x + 1/2*e)^3 - 2310*a^12*c^2*d^10*tan(1/2*f*x + 1/2*e)^3 + 450*a^12
*c*d^11*tan(1/2*f*x + 1/2*e)^3 - 40*a^12*d^12*tan(1/2*f*x + 1/2*e)^3 + 15*a^12*c^12*tan(1/2*f*x + 1/2*e) - 270
*a^12*c^11*d*tan(1/2*f*x + 1/2*e) + 2340*a^12*c^10*d^2*tan(1/2*f*x + 1/2*e) - 11850*a^12*c^9*d^3*tan(1/2*f*x +
1/2*e) + 38475*a^12*c^8*d^4*tan(1/2*f*x + 1/2*e) - 84780*a^12*c^7*d^5*tan(1/2*f*x + 1/2*e) + 131040*a^12*c^6*
d^6*tan(1/2*f*x + 1/2*e) - 144180*a^12*c^5*d^7*tan(1/2*f*x + 1/2*e) + 112725*a^12*c^4*d^8*tan(1/2*f*x + 1/2*e)
- 61350*a^12*c^3*d^9*tan(1/2*f*x + 1/2*e) + 22140*a^12*c^2*d^10*tan(1/2*f*x + 1/2*e) - 4770*a^12*c*d^11*tan(1
/2*f*x + 1/2*e) + 465*a^12*d^12*tan(1/2*f*x + 1/2*e))/(a^15*c^15 - 15*a^15*c^14*d + 105*a^15*c^13*d^2 - 455*a^
15*c^12*d^3 + 1365*a^15*c^11*d^4 - 3003*a^15*c^10*d^5 + 5005*a^15*c^9*d^6 - 6435*a^15*c^8*d^7 + 6435*a^15*c^7*
d^8 - 5005*a^15*c^6*d^9 + 3003*a^15*c^5*d^10 - 1365*a^15*c^4*d^11 + 455*a^15*c^3*d^12 - 105*a^15*c^2*d^13 + 15
*a^15*c*d^14 - a^15*d^15) + 60*(10*c^2*d^4*tan(1/2*f*x + 1/2*e)^3 - 3*c*d^5*tan(1/2*f*x + 1/2*e)^3 - 7*d^6*tan
(1/2*f*x + 1/2*e)^3 - 10*c^2*d^4*tan(1/2*f*x + 1/2*e) - 15*c*d^5*tan(1/2*f*x + 1/2*e) - 5*d^6*tan(1/2*f*x + 1/
2*e))/((a^3*c^7 - 3*a^3*c^6*d + a^3*c^5*d^2 + 5*a^3*c^4*d^3 - 5*a^3*c^3*d^4 - a^3*c^2*d^5 + 3*a^3*c*d^6 - a^3*
d^7)*(c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)^2))/f