\(\int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx\) [383]

Optimal result

Integrand size = 21, antiderivative size = 273 \[ \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {139 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \]

[Out]

-139/64*arctanh((1+tan(f*x+e))^(1/2))/f+1/2*arctan(1/2*(4-3*2^(1/2)+(2-2^(1/2))*tan(f*x+e))/(-7+5*2^(1/2))^(1/
2)/(1+tan(f*x+e))^(1/2))*(-2+2*2^(1/2))^(1/2)/f+1/2*arctanh(1/2*(4+3*2^(1/2)+(2+2^(1/2))*tan(f*x+e))/(7+5*2^(1
/2))^(1/2)/(1+tan(f*x+e))^(1/2))*(2+2*2^(1/2))^(1/2)/f+11/64*cot(f*x+e)*(1+tan(f*x+e))^(1/2)/f+53/96*cot(f*x+e
)^2*(1+tan(f*x+e))^(1/2)/f-1/24*cot(f*x+e)^3*(1+tan(f*x+e))^(1/2)/f-1/4*cot(f*x+e)^4*(1+tan(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3649, 3730, 3734, 3617, 3616, 209, 213, 3715, 65} \[ \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {139 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{64 f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^4(e+f x)}{4 f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{24 f}+\frac {53 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{96 f}+\frac {11 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{64 f} \]

[In]

Int[Cot[e + f*x]^5*Sqrt[1 + Tan[e + f*x]],x]

[Out]

(Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + T
an[e + f*x]])])/f - (139*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(64*f) + (Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(4 + 3*Sqrt[
2] + (2 + Sqrt[2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/f + (11*Cot[e + f*x]*Sqrt[1
+ Tan[e + f*x]])/(64*f) + (53*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(96*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e +
f*x]])/(24*f) - (Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]])/(4*f)

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 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 213

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

Rule 3616

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2*(
d^2/f), Subst[Int[1/(2*b*c*d - 4*a*d^2 + x^2), x], x, (b*c - 2*a*d - b*d*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]
]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && EqQ[2
*a*c*d - b*(c^2 - d^2), 0]

Rule 3617

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> With[{q =
Rt[a^2 + b^2, 2]}, Dist[1/(2*q), Int[(a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*
x]], x], x] - Dist[1/(2*q), Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x
], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2
*a*c*d - b*(c^2 - d^2), 0] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(a^2
+ b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*m]

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {1}{4} \int \frac {\cot ^4(e+f x) \left (-\frac {1}{2}+4 \tan (e+f x)+\frac {7}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = -\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{12} \int \frac {\cot ^3(e+f x) \left (-\frac {53}{4}-12 \tan (e+f x)-\frac {5}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {1}{24} \int \frac {\cot ^2(e+f x) \left (\frac {33}{8}-24 \tan (e+f x)-\frac {159}{8} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{24} \int \frac {\cot (e+f x) \left (\frac {417}{16}+24 \tan (e+f x)+\frac {33}{16} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {1}{24} \int \frac {24-24 \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx+\frac {139}{128} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {\int \frac {24 \sqrt {2}+\left (48-24 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{48 \sqrt {2}}-\frac {\int \frac {-24 \sqrt {2}+\left (48+24 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{48 \sqrt {2}}+\frac {139 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{128 f} \\ & = \frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f}+\frac {139 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {\left (24 \left (4-3 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{48 \sqrt {2} \left (48-24 \sqrt {2}\right )-4 \left (48-24 \sqrt {2}\right )^2+x^2} \, dx,x,\frac {24 \sqrt {2}-2 \left (48-24 \sqrt {2}\right )-\left (48-24 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {\left (24 \left (4+3 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-48 \sqrt {2} \left (48+24 \sqrt {2}\right )-4 \left (48+24 \sqrt {2}\right )^2+x^2} \, dx,x,\frac {-24 \sqrt {2}-2 \left (48+24 \sqrt {2}\right )-\left (48+24 \sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f} \\ & = \frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {139 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{64 f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {11 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{64 f}+\frac {53 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{96 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f}-\frac {\cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.62 \[ \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\frac {-417 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+192 \sqrt {1-i} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+192 \sqrt {1+i} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+33 \cot (e+f x) \sqrt {1+\tan (e+f x)}+106 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}-8 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}-48 \cot ^4(e+f x) \sqrt {1+\tan (e+f x)}}{192 f} \]

[In]

Integrate[Cot[e + f*x]^5*Sqrt[1 + Tan[e + f*x]],x]

[Out]

(-417*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 192*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 192*Sqrt
[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 33*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 106*Cot[e + f*x
]^2*Sqrt[1 + Tan[e + f*x]] - 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]] - 48*Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]
])/(192*f)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4798\) vs. \(2(219)=438\).

Time = 100.75 (sec) , antiderivative size = 4799, normalized size of antiderivative = 17.58

method	result	size
default	\(\text {Expression too large to display}\)	\(4799\)

[In]

int(cot(f*x+e)^5*(1+tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/384/f*(1+tan(f*x+e))^(1/2)/(cot(f*x+e)^2+cot(f*x+e))^(1/2)/((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1
)^2)^(1/2)*(576*cot(f*x+e)^2*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*
x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*
arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2
^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x
+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))-132*cot(f*x+e)*csc(f*x+e)^2*((cos(f*x+e)+s
in(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)-384*co
t(f*x+e)^2*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*co
s(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2
^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e
)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f*x+e))*
(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))*2^(1/2)-288*cot(f*x+e)^2*csc(f*x+e)^2*((cos(f*x+e)+sin(f*x+e
))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)+198*cot(f*x+e)*csc(f*x
+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^
(1/2)-384*cot(f*x+e)^3*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(
1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*c
os(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2
))^(1/2))+768*cot(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2
*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/
2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^
(1/2))^(1/2))-576*cot(f*x+e)^3*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2
^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/
4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*
sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan
(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))+1344*cot(f*x+e)^2*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^
(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)
*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*c
os(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*
cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))+164*c
ot(f*x+e)^3*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*
x+e))^(1/2)*2^(1/2)-616*cot(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(
1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)-192*cot(f*x+e)^2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(
2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan
h(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos
(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)+834*cot(f*x+e)^2*(1+2^(1/2))^(1/2)*(cot(f*
x+e)^2+cot(f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x
+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*2^(1/2)+576*cot(f*x+e)*csc(f*x
+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*c
os(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e
)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)+1
251*cot(f*x+e)*csc(f*x+e)*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f
*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos
(f*x+e)+1)^2)^(1/2))+960*cot(f*x+e)*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))
*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))
^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x
+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*
cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))*2^(1/2)+384*cot(f*x+e)^2*csc(f*x+e)
*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(
f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2
*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))+192*cot(f*x+
e)^3*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*
cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+
e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)-
576*cot(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2
*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x
+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/
2))*2^(1/2)-246*cot(f*x+e)^3*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(co
t(f*x+e)^2+cot(f*x+e))^(1/2)+924*cot(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2
^(1/2))^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)-768*cot(f*x+e)*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2
*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh
(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(
f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))-1251*cot(f*x+e)^2*(1+2^(1/2))^(1/2)*(cot(f*x+e)^2+c
ot(f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)-1+2*
csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))-834*cot(f*x+e)*csc(f*x+e)*(1+2^(1/2))^
(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1
/2)-2*cot(f*x+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*2^(1/2)+192*cot(f
*x+e)^2*csc(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1+2^(1/2))^(1/2)*(cot(f*x+e)
^2+cot(f*x+e))^(1/2)*2^(1/2)+384*cot(f*x+e)^3*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*
cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^
(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+
e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*c
os(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))*2^(1/2)-960*cot(f*x+e)^2*(-2+2*2^(1/
2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^
2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*c
os(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3
*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*x+e)*cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3
)*(3*2^(1/2)-4))*2^(1/2)-1344*cot(f*x+e)*csc(f*x+e)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*
x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^
2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*si
n(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)/(2*cos(f*x+e)^2-1)*(4*sin(f*
x+e)*cos(f*x+e)-1-tan(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)))*2^(1/2)/(1+2^(1/2))^(1/2)/(3*
2^(1/2)-4)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.32 \[ \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\frac {192 \, f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 192 \, f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} + 192 \, f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 192 \, f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{4} - 417 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{4} + 417 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{4} + 2 \, {\left (33 \, \tan \left (f x + e\right )^{3} + 106 \, \tan \left (f x + e\right )^{2} - 8 \, \tan \left (f x + e\right ) - 48\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{384 \, f \tan \left (f x + e\right )^{4}} \]

[In]

integrate(cot(f*x+e)^5*(1+tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/384*(192*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1))
*tan(f*x + e)^4 - 192*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(-f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*
x + e) + 1))*tan(f*x + e)^4 + 192*f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)
+ sqrt(tan(f*x + e) + 1))*tan(f*x + e)^4 - 192*f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(-f*sqrt(-(f^2*sqrt(-1/f
^4) - 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^4 - 417*log(sqrt(tan(f*x + e) + 1) + 1)*tan(f*x + e)^4 +
417*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e)^4 + 2*(33*tan(f*x + e)^3 + 106*tan(f*x + e)^2 - 8*tan(f*x + e
) - 48)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^4)

Sympy [F]

\[ \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\int \sqrt {\tan {\left (e + f x \right )} + 1} \cot ^{5}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**5*(1+tan(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(tan(e + f*x) + 1)*cot(e + f*x)**5, x)

Maxima [F]

\[ \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\int { \sqrt {\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(cot(f*x+e)^5*(1+tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(tan(f*x + e) + 1)*cot(f*x + e)^5, x)

Giac [F]

\[ \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\int { \sqrt {\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )^{5} \,d x } \]

[In]

integrate(cot(f*x+e)^5*(1+tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(tan(f*x + e) + 1)*cot(f*x + e)^5, x)

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.73 \[ \int \cot ^5(e+f x) \sqrt {1+\tan (e+f x)} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,139{}\mathrm {i}}{64\,f}+\frac {\frac {11\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{64}-\frac {121\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{192}+\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{192}+\frac {11\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{7/2}}{64}}{f-4\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+6\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-4\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^4}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1-\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1+1{}\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]

[In]

int(cot(e + f*x)^5*(tan(e + f*x) + 1)^(1/2),x)

[Out]

(atan((tan(e + f*x) + 1)^(1/2)*1i)*139i)/(64*f) + ((11*(tan(e + f*x) + 1)^(1/2))/64 - (121*(tan(e + f*x) + 1)^
(3/2))/192 + (7*(tan(e + f*x) + 1)^(5/2))/192 + (11*(tan(e + f*x) + 1)^(7/2))/64)/(f - 4*f*(tan(e + f*x) + 1)
+ 6*f*(tan(e + f*x) + 1)^2 - 4*f*(tan(e + f*x) + 1)^3 + f*(tan(e + f*x) + 1)^4) + atan(f*((1/4 - 1i/4)/f^2)^(1
/2)*(tan(e + f*x) + 1)^(1/2)*(1 - 1i))*((1/4 - 1i/4)/f^2)^(1/2)*2i - atan(f*((1/4 + 1i/4)/f^2)^(1/2)*(tan(e +
f*x) + 1)^(1/2)*(1 + 1i))*((1/4 + 1i/4)/f^2)^(1/2)*2i