\(\int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx\) [206]

Optimal result

Integrand size = 31, antiderivative size = 161 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx=-\frac {a^3 (2 c-3 d) \text {arctanh}(\sin (e+f x))}{d^3 f}+\frac {2 a^3 (c-d)^{3/2} (2 c+3 d) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{d^3 (c+d)^{3/2} f}+\frac {2 a^3 c \tan (e+f x)}{d^2 (c+d) f}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{d (c+d) f (c+d \sec (e+f x))} \]

[Out]

-a^3*(2*c-3*d)*arctanh(sin(f*x+e))/d^3/f+2*a^3*(c-d)^(3/2)*(2*c+3*d)*arctanh((c-d)^(1/2)*tan(1/2*f*x+1/2*e)/(c
+d)^(1/2))/d^3/(c+d)^(3/2)/f+2*a^3*c*tan(f*x+e)/d^2/(c+d)/f-(c-d)*(a^3+a^3*sec(f*x+e))*tan(f*x+e)/d/(c+d)/f/(c
+d*sec(f*x+e))

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.70, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {4072, 100, 159, 163, 65, 223, 209, 95, 211} \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx=-\frac {2 a^4 (2 c-3 d) \tan (e+f x) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {2 a^4 (c-d)^{3/2} (2 c+3 d) \tan (e+f x) \arctan \left (\frac {\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}{\sqrt {c-d} \sqrt {a-a \sec (e+f x)}}\right )}{d^3 f (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 a^3 c \tan (e+f x)}{d^2 f (c+d)}-\frac {(c-d) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{d f (c+d) (c+d \sec (e+f x))} \]

[In]

Int[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^2,x]

[Out]

(2*a^3*c*Tan[e + f*x])/(d^2*(c + d)*f) - (2*a^4*(2*c - 3*d)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e
+ f*x])]]*Tan[e + f*x])/(d^3*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - (2*a^4*(c - d)^(3/2)*(2*c
+ 3*d)*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])/(d^
3*(c + d)^(3/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) - ((c - d)*(a^3 + a^3*Sec[e + f*x])*Tan[e
+ f*x])/(d*(c + d)*f*(c + d*Sec[e + f*x]))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

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 100

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

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 211

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 4072

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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{5/2}}{\sqrt {a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{d (c+d) f (c+d \sec (e+f x))}+\frac {(a \tan (e+f x)) \text {Subst}\left (\int \frac {\sqrt {a+a x} \left (a^3 (c-3 d)-2 a^3 c x\right )}{\sqrt {a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{d (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {2 a^3 c \tan (e+f x)}{d^2 (c+d) f}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{d (c+d) f (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {-a^5 (c-3 d) d-a^5 (2 c-3 d) (c+d) x}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{d^2 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {2 a^3 c \tan (e+f x)}{d^2 (c+d) f}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{d (c+d) f (c+d \sec (e+f x))}+\frac {\left (a^5 (2 c-3 d) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (a^5 (c-d)^2 (2 c+3 d) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{d^3 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {2 a^3 c \tan (e+f x)}{d^2 (c+d) f}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{d (c+d) f (c+d \sec (e+f x))}-\frac {\left (2 a^4 (2 c-3 d) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (2 a^5 (c-d)^2 (2 c+3 d) \tan (e+f x)\right ) \text {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 )}{d^3 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {2 a^3 c \tan (e+f x)}{d^2 (c+d) f}-\frac {2 a^4 (c-d)^{3/2} (2 c+3 d) \arctan \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)}{d^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{d (c+d) f (c+d \sec (e+f x))}-\frac {\left (2 a^4 (2 c-3 d) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right )}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = \frac {2 a^3 c \tan (e+f x)}{d^2 (c+d) f}-\frac {2 a^4 (2 c-3 d) \arctan \left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{d^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 a^4 (c-d)^{3/2} (2 c+3 d) \arctan \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)}{d^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-d) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{d (c+d) f (c+d \sec (e+f x))} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.18 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.83 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx=\frac {a^3 \cos (e+f x) (d+c \cos (e+f x)) \sec ^6\left (\frac {1}{2} (e+f x)\right ) (1+\sec (e+f x))^3 \left ((2 c-3 d) (d+c \cos (e+f x)) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+(-2 c+3 d) (d+c \cos (e+f x)) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-\frac {2 i (c-d)^2 (2 c+3 d) \arctan \left (\frac {(i \cos (e)+\sin (e)) \left (c \sin (e)+(-d+c \cos (e)) \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) (d+c \cos (e+f x)) (\cos (e)-i \sin (e))}{(c+d) \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {(c-d)^2 d (-d \sin (e)+c \sin (f x))}{c (c+d) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right )}+\frac {d (d+c \cos (e+f x)) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {d (d+c \cos (e+f x)) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right )}{8 d^3 f (c+d \sec (e+f x))^2} \]

[In]

Integrate[(Sec[e + f*x]*(a + a*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^2,x]

[Out]

(a^3*Cos[e + f*x]*(d + c*Cos[e + f*x])*Sec[(e + f*x)/2]^6*(1 + Sec[e + f*x])^3*((2*c - 3*d)*(d + c*Cos[e + f*x
])*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + (-2*c + 3*d)*(d + c*Cos[e + f*x])*Log[Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2]] - ((2*I)*(c - d)^2*(2*c + 3*d)*ArcTan[((I*Cos[e] + Sin[e])*(c*Sin[e] + (-d + c*Cos[e])*Tan[(f*x)/2]
))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(d + c*Cos[e + f*x])*(Cos[e] - I*Sin[e]))/((c + d)*Sqrt[c^2
- d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + ((c - d)^2*d*(-(d*Sin[e]) + c*Sin[f*x]))/(c*(c + d)*(Cos[e/2] - Sin[e/2]
)*(Cos[e/2] + Sin[e/2])) + (d*(d + c*Cos[e + f*x])*Sin[(f*x)/2])/((Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Si
n[(e + f*x)/2])) + (d*(d + c*Cos[e + f*x])*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f
*x)/2]))))/(8*d^3*f*(c + d*Sec[e + f*x])^2)

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {16 a^{3} \left (-\frac {\left (c^{2}-2 c d +d^{2}\right ) \left (\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {\left (2 c +3 d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 d^{3}}-\frac {1}{16 d^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {\left (2 c -3 d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 d^{3}}-\frac {1}{16 d^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {\left (-2 c +3 d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 d^{3}}\right )}{f}\)	\(216\)
default	\(\frac {16 a^{3} \left (-\frac {\left (c^{2}-2 c d +d^{2}\right ) \left (\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )}-\frac {\left (2 c +3 d \right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{2 \left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 d^{3}}-\frac {1}{16 d^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}+\frac {\left (2 c -3 d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 d^{3}}-\frac {1}{16 d^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {\left (-2 c +3 d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 d^{3}}\right )}{f}\)	\(216\)
risch	\(\frac {2 i a^{3} \left (c^{2} d \,{\mathrm e}^{3 i \left (f x +e \right )}-2 c \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+2 c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3 c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+d^{3} {\mathrm e}^{i \left (f x +e \right )}+2 c^{3}-c^{2} d +c \,d^{2}\right )}{f \,d^{2} \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \left (c +d \right ) c \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right )}+\frac {2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}+d}{c}\right ) c^{2}}{\left (c +d \right )^{2} f \,d^{3}}+\frac {\sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}+d}{c}\right ) c}{\left (c +d \right )^{2} f \,d^{2}}-\frac {3 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}+d}{c}\right )}{\left (c +d \right )^{2} f d}-\frac {2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}-d}{c}\right ) c^{2}}{\left (c +d \right )^{2} f \,d^{3}}-\frac {\sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}-d}{c}\right ) c}{\left (c +d \right )^{2} f \,d^{2}}+\frac {3 \sqrt {\left (c +d \right ) \left (c -d \right )}\, a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i \sqrt {\left (c +d \right ) \left (c -d \right )}-d}{c}\right )}{\left (c +d \right )^{2} f d}-\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c}{d^{3} f}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{d^{2} f}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c}{d^{3} f}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{d^{2} f}\)	\(654\)

[In]

int(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c+d*sec(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

16/f*a^3*(-1/4*(c^2-2*c*d+d^2)/d^3*(1/2*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)-1/2*(2*c+3*d)/(c+d)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1/2*f*x+1/2*e)/((c+d)*(c-d))^(1/2)))-1/16/d
^2/(tan(1/2*f*x+1/2*e)-1)+1/16*(2*c-3*d)/d^3*ln(tan(1/2*f*x+1/2*e)-1)-1/16/d^2/(tan(1/2*f*x+1/2*e)+1)+1/16/d^3
*(-2*c+3*d)*ln(tan(1/2*f*x+1/2*e)+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (152) = 304\).

Time = 0.56 (sec) , antiderivative size = 859, normalized size of antiderivative = 5.34 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx=\left [-\frac {{\left ({\left (2 \, a^{3} c^{3} + a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, a^{3} c^{2} d + a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {c - d}{c + d}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{2} + c d + {\left (c d + d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {c - d}{c + d}} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + {\left ({\left (2 \, a^{3} c^{3} - a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left ({\left (2 \, a^{3} c^{3} - a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (a^{3} c d^{2} + a^{3} d^{3} + {\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{2} d^{3} + c d^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (c d^{4} + d^{5}\right )} f \cos \left (f x + e\right )\right )}}, \frac {2 \, {\left ({\left (2 \, a^{3} c^{3} + a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, a^{3} c^{2} d + a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {c - d}{c + d}} \arctan \left (-\frac {{\left (d \cos \left (f x + e\right ) + c\right )} \sqrt {-\frac {c - d}{c + d}}}{{\left (c - d\right )} \sin \left (f x + e\right )}\right ) - {\left ({\left (2 \, a^{3} c^{3} - a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) + {\left ({\left (2 \, a^{3} c^{3} - a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} - 3 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (a^{3} c d^{2} + a^{3} d^{3} + {\left (2 \, a^{3} c^{2} d - a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{2} d^{3} + c d^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (c d^{4} + d^{5}\right )} f \cos \left (f x + e\right )\right )}}\right ] \]

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c+d*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

[-1/2*(((2*a^3*c^3 + a^3*c^2*d - 3*a^3*c*d^2)*cos(f*x + e)^2 + (2*a^3*c^2*d + a^3*c*d^2 - 3*a^3*d^3)*cos(f*x +
e))*sqrt((c - d)/(c + d))*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*(c^2 + c*d + (c*d + d^2)
*cos(f*x + e))*sqrt((c - d)/(c + d))*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*a^3*c^3 - a^3*c^2*d - 3*a^3*c*d^2)*cos(f*x + e)^2 + (2*a^3*c^2*d - a^3*c*d^2 - 3*a^3*d^3)*cos(f*x +
e))*log(sin(f*x + e) + 1) - ((2*a^3*c^3 - a^3*c^2*d - 3*a^3*c*d^2)*cos(f*x + e)^2 + (2*a^3*c^2*d - a^3*c*d^2 -
3*a^3*d^3)*cos(f*x + e))*log(-sin(f*x + e) + 1) - 2*(a^3*c*d^2 + a^3*d^3 + (2*a^3*c^2*d - a^3*c*d^2 + a^3*d^3
)*cos(f*x + e))*sin(f*x + e))/((c^2*d^3 + c*d^4)*f*cos(f*x + e)^2 + (c*d^4 + d^5)*f*cos(f*x + e)), 1/2*(2*((2*
a^3*c^3 + a^3*c^2*d - 3*a^3*c*d^2)*cos(f*x + e)^2 + (2*a^3*c^2*d + a^3*c*d^2 - 3*a^3*d^3)*cos(f*x + e))*sqrt(-
(c - d)/(c + d))*arctan(-(d*cos(f*x + e) + c)*sqrt(-(c - d)/(c + d))/((c - d)*sin(f*x + e))) - ((2*a^3*c^3 - a
^3*c^2*d - 3*a^3*c*d^2)*cos(f*x + e)^2 + (2*a^3*c^2*d - a^3*c*d^2 - 3*a^3*d^3)*cos(f*x + e))*log(sin(f*x + e)
+ 1) + ((2*a^3*c^3 - a^3*c^2*d - 3*a^3*c*d^2)*cos(f*x + e)^2 + (2*a^3*c^2*d - a^3*c*d^2 - 3*a^3*d^3)*cos(f*x +
e))*log(-sin(f*x + e) + 1) + 2*(a^3*c*d^2 + a^3*d^3 + (2*a^3*c^2*d - a^3*c*d^2 + a^3*d^3)*cos(f*x + e))*sin(f
*x + e))/((c^2*d^3 + c*d^4)*f*cos(f*x + e)^2 + (c*d^4 + d^5)*f*cos(f*x + e))]

Sympy [F]

\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx=a^{3} \left (\int \frac {\sec {\left (e + f x \right )}}{c^{2} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx + \int \frac {3 \sec ^{3}{\left (e + f x \right )}}{c^{2} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx + \int \frac {\sec ^{4}{\left (e + f x \right )}}{c^{2} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{2}{\left (e + f x \right )}}\, dx\right ) \]

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))**3/(c+d*sec(f*x+e))**2,x)

[Out]

a**3*(Integral(sec(e + f*x)/(c**2 + 2*c*d*sec(e + f*x) + d**2*sec(e + f*x)**2), x) + Integral(3*sec(e + f*x)**
2/(c**2 + 2*c*d*sec(e + f*x) + d**2*sec(e + f*x)**2), x) + Integral(3*sec(e + f*x)**3/(c**2 + 2*c*d*sec(e + f*
x) + d**2*sec(e + f*x)**2), x) + Integral(sec(e + f*x)**4/(c**2 + 2*c*d*sec(e + f*x) + d**2*sec(e + f*x)**2),
x))

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c+d*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*c^2-4*d^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (152) = 304\).

Time = 0.37 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.97 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx=-\frac {\frac {2 \, {\left (2 \, a^{3} c^{3} - a^{3} c^{2} d - 4 \, a^{3} c d^{2} + 3 \, a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c d^{3} + d^{4}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {4 \, {\left (a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + c + d\right )} {\left (c d^{2} + d^{3}\right )}} + \frac {{\left (2 \, a^{3} c - 3 \, a^{3} d\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{d^{3}} - \frac {{\left (2 \, a^{3} c - 3 \, a^{3} d\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{d^{3}}}{f} \]

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c+d*sec(f*x+e))^2,x, algorithm="giac")

[Out]

-(2*(2*a^3*c^3 - a^3*c^2*d - 4*a^3*c*d^2 + 3*a^3*d^3)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(2*c - 2*d) + arcta
n((c*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2)))/((c*d^3 + d^4)*sqrt(-c^2 + d^2)) + 4*(a
^3*c^2*tan(1/2*f*x + 1/2*e)^3 - a^3*c*d*tan(1/2*f*x + 1/2*e)^3 - a^3*c^2*tan(1/2*f*x + 1/2*e) - a^3*d^2*tan(1/
2*f*x + 1/2*e))/((c*tan(1/2*f*x + 1/2*e)^4 - d*tan(1/2*f*x + 1/2*e)^4 - 2*c*tan(1/2*f*x + 1/2*e)^2 + c + d)*(c
*d^2 + d^3)) + (2*a^3*c - 3*a^3*d)*log(abs(tan(1/2*f*x + 1/2*e) + 1))/d^3 - (2*a^3*c - 3*a^3*d)*log(abs(tan(1/
2*f*x + 1/2*e) - 1))/d^3)/f

Mupad [B] (verification not implemented)

Time = 16.91 (sec) , antiderivative size = 3135, normalized size of antiderivative = 19.47 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c+d \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

int((a + a/cos(e + f*x))^3/(cos(e + f*x)*(c + d/cos(e + f*x))^2),x)

[Out]

(a^3*atan(((a^3*((64*tan(e/2 + (f*x)/2)*(4*a^6*c^7 - 9*a^6*d^7 + 27*a^6*c*d^6 - 12*a^6*c^6*d - 16*a^6*c^2*d^5
- 24*a^6*c^3*d^4 + 29*a^6*c^4*d^3 + a^6*c^5*d^2))/(2*c*d^5 + d^6 + c^2*d^4) + (a^3*((64*(3*a^3*d^11 - 3*a^3*c*
d^10 - 4*a^3*c^2*d^9 + 4*a^3*c^3*d^8 + a^3*c^4*d^7 - a^3*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6) - (64*a^3*tan(e/2
+ (f*x)/2)*(2*c - 3*d)*(c*d^10 - 2*c^3*d^8 + c^5*d^6))/(d^3*(2*c*d^5 + d^6 + c^2*d^4)))*(2*c - 3*d))/d^3)*(2*
c - 3*d)*1i)/d^3 + (a^3*((64*tan(e/2 + (f*x)/2)*(4*a^6*c^7 - 9*a^6*d^7 + 27*a^6*c*d^6 - 12*a^6*c^6*d - 16*a^6*
c^2*d^5 - 24*a^6*c^3*d^4 + 29*a^6*c^4*d^3 + a^6*c^5*d^2))/(2*c*d^5 + d^6 + c^2*d^4) - (a^3*((64*(3*a^3*d^11 -
3*a^3*c*d^10 - 4*a^3*c^2*d^9 + 4*a^3*c^3*d^8 + a^3*c^4*d^7 - a^3*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6) + (64*a^3
*tan(e/2 + (f*x)/2)*(2*c - 3*d)*(c*d^10 - 2*c^3*d^8 + c^5*d^6))/(d^3*(2*c*d^5 + d^6 + c^2*d^4)))*(2*c - 3*d))/
d^3)*(2*c - 3*d)*1i)/d^3)/((128*(4*a^9*c^7 - 9*a^9*c*d^6 - 16*a^9*c^6*d + 36*a^9*c^2*d^5 - 50*a^9*c^3*d^4 + 20
*a^9*c^4*d^3 + 15*a^9*c^5*d^2))/(2*c*d^7 + d^8 + c^2*d^6) + (a^3*((64*tan(e/2 + (f*x)/2)*(4*a^6*c^7 - 9*a^6*d^
7 + 27*a^6*c*d^6 - 12*a^6*c^6*d - 16*a^6*c^2*d^5 - 24*a^6*c^3*d^4 + 29*a^6*c^4*d^3 + a^6*c^5*d^2))/(2*c*d^5 +
d^6 + c^2*d^4) + (a^3*((64*(3*a^3*d^11 - 3*a^3*c*d^10 - 4*a^3*c^2*d^9 + 4*a^3*c^3*d^8 + a^3*c^4*d^7 - a^3*c^5*
d^6))/(2*c*d^7 + d^8 + c^2*d^6) - (64*a^3*tan(e/2 + (f*x)/2)*(2*c - 3*d)*(c*d^10 - 2*c^3*d^8 + c^5*d^6))/(d^3*
(2*c*d^5 + d^6 + c^2*d^4)))*(2*c - 3*d))/d^3)*(2*c - 3*d))/d^3 - (a^3*((64*tan(e/2 + (f*x)/2)*(4*a^6*c^7 - 9*a
^6*d^7 + 27*a^6*c*d^6 - 12*a^6*c^6*d - 16*a^6*c^2*d^5 - 24*a^6*c^3*d^4 + 29*a^6*c^4*d^3 + a^6*c^5*d^2))/(2*c*d
^5 + d^6 + c^2*d^4) - (a^3*((64*(3*a^3*d^11 - 3*a^3*c*d^10 - 4*a^3*c^2*d^9 + 4*a^3*c^3*d^8 + a^3*c^4*d^7 - a^3
*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6) + (64*a^3*tan(e/2 + (f*x)/2)*(2*c - 3*d)*(c*d^10 - 2*c^3*d^8 + c^5*d^6))/
(d^3*(2*c*d^5 + d^6 + c^2*d^4)))*(2*c - 3*d))/d^3)*(2*c - 3*d))/d^3))*(2*c - 3*d)*2i)/(d^3*f) - ((4*tan(e/2 +
(f*x)/2)^3*(a^3*c^2 - a^3*c*d))/(d^2*(c + d)) - (4*a^3*tan(e/2 + (f*x)/2)*(c^2 + d^2))/(d^2*(c + d)))/(f*(c +
d + tan(e/2 + (f*x)/2)^4*(c - d) - 2*c*tan(e/2 + (f*x)/2)^2)) + (a^3*atan(((a^3*((64*tan(e/2 + (f*x)/2)*(4*a^6
*c^7 - 9*a^6*d^7 + 27*a^6*c*d^6 - 12*a^6*c^6*d - 16*a^6*c^2*d^5 - 24*a^6*c^3*d^4 + 29*a^6*c^4*d^3 + a^6*c^5*d^
2))/(2*c*d^5 + d^6 + c^2*d^4) + (a^3*((64*(3*a^3*d^11 - 3*a^3*c*d^10 - 4*a^3*c^2*d^9 + 4*a^3*c^3*d^8 + a^3*c^4
*d^7 - a^3*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6) - (64*a^3*tan(e/2 + (f*x)/2)*((c + d)^3*(c - d)^3)^(1/2)*(2*c +
3*d)*(c*d^10 - 2*c^3*d^8 + c^5*d^6))/((2*c*d^5 + d^6 + c^2*d^4)*(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3)))*((c +
d)^3*(c - d)^3)^(1/2)*(2*c + 3*d))/(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3))*((c + d)^3*(c - d)^3)^(1/2)*(2*c +
3*d)*1i)/(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3) + (a^3*((64*tan(e/2 + (f*x)/2)*(4*a^6*c^7 - 9*a^6*d^7 + 27*a^6*
c*d^6 - 12*a^6*c^6*d - 16*a^6*c^2*d^5 - 24*a^6*c^3*d^4 + 29*a^6*c^4*d^3 + a^6*c^5*d^2))/(2*c*d^5 + d^6 + c^2*d
^4) - (a^3*((64*(3*a^3*d^11 - 3*a^3*c*d^10 - 4*a^3*c^2*d^9 + 4*a^3*c^3*d^8 + a^3*c^4*d^7 - a^3*c^5*d^6))/(2*c*
d^7 + d^8 + c^2*d^6) + (64*a^3*tan(e/2 + (f*x)/2)*((c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d)*(c*d^10 - 2*c^3*d^8
+ c^5*d^6))/((2*c*d^5 + d^6 + c^2*d^4)*(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3)))*((c + d)^3*(c - d)^3)^(1/2)*(2*
c + 3*d))/(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3))*((c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d)*1i)/(3*c*d^5 + d^6 +
3*c^2*d^4 + c^3*d^3))/((128*(4*a^9*c^7 - 9*a^9*c*d^6 - 16*a^9*c^6*d + 36*a^9*c^2*d^5 - 50*a^9*c^3*d^4 + 20*a^9
*c^4*d^3 + 15*a^9*c^5*d^2))/(2*c*d^7 + d^8 + c^2*d^6) + (a^3*((64*tan(e/2 + (f*x)/2)*(4*a^6*c^7 - 9*a^6*d^7 +
27*a^6*c*d^6 - 12*a^6*c^6*d - 16*a^6*c^2*d^5 - 24*a^6*c^3*d^4 + 29*a^6*c^4*d^3 + a^6*c^5*d^2))/(2*c*d^5 + d^6
+ c^2*d^4) + (a^3*((64*(3*a^3*d^11 - 3*a^3*c*d^10 - 4*a^3*c^2*d^9 + 4*a^3*c^3*d^8 + a^3*c^4*d^7 - a^3*c^5*d^6)
)/(2*c*d^7 + d^8 + c^2*d^6) - (64*a^3*tan(e/2 + (f*x)/2)*((c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d)*(c*d^10 - 2*c
^3*d^8 + c^5*d^6))/((2*c*d^5 + d^6 + c^2*d^4)*(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3)))*((c + d)^3*(c - d)^3)^(1
/2)*(2*c + 3*d))/(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3))*((c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d))/(3*c*d^5 + d^
6 + 3*c^2*d^4 + c^3*d^3) - (a^3*((64*tan(e/2 + (f*x)/2)*(4*a^6*c^7 - 9*a^6*d^7 + 27*a^6*c*d^6 - 12*a^6*c^6*d -
16*a^6*c^2*d^5 - 24*a^6*c^3*d^4 + 29*a^6*c^4*d^3 + a^6*c^5*d^2))/(2*c*d^5 + d^6 + c^2*d^4) - (a^3*((64*(3*a^3
*d^11 - 3*a^3*c*d^10 - 4*a^3*c^2*d^9 + 4*a^3*c^3*d^8 + a^3*c^4*d^7 - a^3*c^5*d^6))/(2*c*d^7 + d^8 + c^2*d^6) +
(64*a^3*tan(e/2 + (f*x)/2)*((c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d)*(c*d^10 - 2*c^3*d^8 + c^5*d^6))/((2*c*d^5
+ d^6 + c^2*d^4)*(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3)))*((c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d))/(3*c*d^5 + d
^6 + 3*c^2*d^4 + c^3*d^3))*((c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d))/(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3)))*((
c + d)^3*(c - d)^3)^(1/2)*(2*c + 3*d)*2i)/(f*(3*c*d^5 + d^6 + 3*c^2*d^4 + c^3*d^3))