\(\int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx\) [399]

Optimal result

Integrand size = 21, antiderivative size = 238 \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {25 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}-\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \]

[Out]

25/8*arctanh((1+tan(f*x+e))^(1/2))/f-arctan((3-2*2^(1/2)+(1-2^(1/2))*tan(f*x+e))/(-14+10*2^(1/2))^(1/2)/(1+tan
(f*x+e))^(1/2))*(2^(1/2)-1)^(1/2)/f-arctanh((3+2*2^(1/2)+(1+2^(1/2))*tan(f*x+e))/(14+10*2^(1/2))^(1/2)/(1+tan(
f*x+e))^(1/2))*(1+2^(1/2))^(1/2)/f+7/8*cot(f*x+e)*(1+tan(f*x+e))^(1/2)/f-7/12*cot(f*x+e)^2*(1+tan(f*x+e))^(1/2
)/f-1/3*cot(f*x+e)^3*(1+tan(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3648, 3731, 3730, 3735, 12, 3617, 3616, 209, 213, 3715, 65} \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\sqrt {\sqrt {2}-1} \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{f}+\frac {25 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{8 f}-\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{12 f}+\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{8 f} \]

[In]

Int[Cot[e + f*x]^4*(1 + Tan[e + f*x])^(3/2),x]

[Out]

-((Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + T
an[e + f*x]])])/f) + (25*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(8*f) - (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] +
(1 + Sqrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/f + (7*Cot[e + f*x]*Sqrt[1 + Ta
n[e + f*x]])/(8*f) - (7*Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/(12*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]]
)/(3*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 3648

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(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 - 2)*Simp[a*c^2*(m + 1) + a*
d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*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] && LtQ[1, n, 2] && 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 3731

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

Rule 3735

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*ta
n[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*
C)*Tan[e + f*x], x], x], x] + Dist[(A*b^2 + 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, 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 ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{3} \int \frac {\cot ^3(e+f x) \left (-\frac {7}{2}+\frac {5}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = -\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {1}{6} \int \frac {\cot ^2(e+f x) \left (-\frac {21}{4}-12 \tan (e+f x)-\frac {21}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{6} \int \frac {\cot (e+f x) \left (\frac {75}{8}-\frac {21}{8} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{6} \int -\frac {12 \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx-\frac {25}{16} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+2 \int \frac {\tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx-\frac {25 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{16 f} \\ & = \frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {\int \frac {1+\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{\sqrt {2}}+\frac {\int \frac {1+\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{\sqrt {2}}-\frac {25 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{8 f} \\ & = \frac {25 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {\left (4-3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1+\sqrt {2}\right )-4 \left (-1+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1+\sqrt {2}\right )-\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {\left (4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1-\sqrt {2}\right )-4 \left (-1-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1-\sqrt {2}\right )-\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f} \\ & = -\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {25 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}-\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.62 \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {-75 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+\frac {48 \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}+\frac {48 \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i}}-21 \cot (e+f x) \sqrt {1+\tan (e+f x)}+14 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}+8 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f} \]

[In]

Integrate[Cot[e + f*x]^4*(1 + Tan[e + f*x])^(3/2),x]

[Out]

-1/24*(-75*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (48*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (4
8*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/Sqrt[1 + I] - 21*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 14*Cot[e
+ f*x]^2*Sqrt[1 + Tan[e + f*x]] + 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/f

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4597\) vs. \(2(190)=380\).

Time = 61.12 (sec) , antiderivative size = 4598, normalized size of antiderivative = 19.32

method	result	size
default	\(\text {Expression too large to display}\)	\(4598\)

[In]

int(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/48/f*cot(f*x+e)*csc(f*x+e)^2*(1+tan(f*x+e))^(1/2)*(-72*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*cos(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(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)+72*2^
(1/2)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*cos(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)*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)*c
os(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))-48*cos(f*x+e)*((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)*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*si
n(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))+144*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))*((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)*sin(f*x+e)+168*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(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))*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*s
in(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)*sin(f*x+e)-56*2^(1/2
)*(1+2^(1/2))^(1/2)*cos(f*x+e)*sin(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x
+e)^2+cot(f*x+e))^(1/2)+240*(-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)+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))+96*2^(1/2)*cos(f*x+e)*sin(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))+84*(1+2^(1/2
))^(1/2)*cos(f*x+e)*sin(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^2+cot(f
*x+e))^(1/2)-96*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))*((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)*sin(f*x+e)+84*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*
x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)-116*2^(1/2)*(1+2^(1/2))^(1/2)*cos(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e
)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)-96*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*cos(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)*co
s(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))-48*2^(1/
2)*cos(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
))+174*(1+2^(1/2))^(1/2)*cos(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^
2+cot(f*x+e))^(1/2)-144*cos(f*x+e)*sin(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))-168*2^(1/2)*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*cos(f*x+e)*sin(f*x+e)*((co
s(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))-126*(1+2^(1/2))
^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)+225*ln(2*co
t(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))*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*(1+2^(1/2))^(1/2)*sin(f*x+e)+4
8*cos(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))*
2^(1/2)-225*(1+2^(1/2))^(1/2)*cos(f*x+e)*sin(f*x+e)*(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))+48*cos(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))*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))^(
1/2)*2^(1/2)/(1+2^(1/2))^(1/2))+96*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*cos(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(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))-240*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))
^(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))*((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)*sin(
f*x+e)-150*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))*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)*(1+2
^(1/2))^(1/2)*sin(f*x+e)+150*(1+2^(1/2))^(1/2)*cos(f*x+e)*sin(f*x+e)*(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)+si
n(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*2^(1/2))/((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(
1/2)/(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2)/(3*2^(1/2)-4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (189) = 378\).

Time = 0.27 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.82 \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {24 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} + 24 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} + 75 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{3} - 75 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{3} + 2 \, {\left (21 \, \tan \left (f x + e\right )^{2} - 14 \, \tan \left (f x + e\right ) - 8\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{48 \, f \tan \left (f x + e\right )^{3}} \]

[In]

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

1/48*(24*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4)
+ 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 - 24*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(-sqrt
(2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 - 24*sq
rt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(sqrt(2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)
+ 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 + 24*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(-sqrt(2)*(f^3
*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 + 75*log(sqrt(
tan(f*x + e) + 1) + 1)*tan(f*x + e)^3 - 75*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e)^3 + 2*(21*tan(f*x + e)
^2 - 14*tan(f*x + e) - 8)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^3)

Sympy [F]

\[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \cot ^{4}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**4*(1+tan(f*x+e))**(3/2),x)

[Out]

Integral((tan(e + f*x) + 1)**(3/2)*cot(e + f*x)**4, x)

Maxima [F]

\[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int { {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{4} \,d x } \]

[In]

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((tan(f*x + e) + 1)^(3/2)*cot(f*x + e)^4, x)

Giac [F(-1)]

Timed out. \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.73 \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,25{}\mathrm {i}}{8\,f}-\frac {\frac {9\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{8}-\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3}+\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{8}}{f-3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]

[In]

int(cot(e + f*x)^4*(tan(e + f*x) + 1)^(3/2),x)

[Out]

atan(f*((1/2 - 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*1i)*((1/2 - 1i/2)/f^2)^(1/2)*2i - ((9*(tan(e + f*x) +
1)^(1/2))/8 - (7*(tan(e + f*x) + 1)^(3/2))/3 + (7*(tan(e + f*x) + 1)^(5/2))/8)/(f - 3*f*(tan(e + f*x) + 1) +
3*f*(tan(e + f*x) + 1)^2 - f*(tan(e + f*x) + 1)^3) - (atan((tan(e + f*x) + 1)^(1/2)*1i)*25i)/(8*f) + atan(f*((
1/2 + 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*1i)*((1/2 + 1i/2)/f^2)^(1/2)*2i