\(\int \frac {\cot (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\) [403]

Optimal result

Integrand size = 19, antiderivative size = 161 \[ \int \frac {\cot (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}-\frac {2 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f} \]

[Out]

-2*arctanh((1+tan(f*x+e))^(1/2))/f+1/2*arctan((3-2*2^(1/2)+(1-2^(1/2))*tan(f*x+e))/(-14+10*2^(1/2))^(1/2)/(1+t
an(f*x+e))^(1/2))*(2^(1/2)-1)^(1/2)/f+1/2*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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3655, 3617, 3616, 209, 213, 3715, 65} \[ \int \frac {\cot (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {\sqrt {\sqrt {2}-1} \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 f}-\frac {2 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}+\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 f} \]

[In]

Int[Cot[e + f*x]/Sqrt[1 + Tan[e + f*x]],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 + Tan
[e + f*x]])])/(2*f) - (2*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/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]])])/(2*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 3655

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

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx+\int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {\int \frac {1+\left (-1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}-\frac {\int \frac {1+\left (-1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,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 )}{2 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 )}{2 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 )}{2 f}-\frac {2 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{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 )}{2 f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.52 \[ \int \frac {\cot (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {2 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i} f}+\frac {\text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i} f} \]

[In]

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

[Out]

(-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f + ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]]/(Sqrt[1 - I]*f) + ArcTanh
[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]]/(Sqrt[1 + I]*f)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2129\) vs. \(2(123)=246\).

Time = 117.36 (sec) , antiderivative size = 2130, normalized size of antiderivative = 13.23

method	result	size
default	\(\text {Expression too large to display}\)	\(2130\)

[In]

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

[Out]

-1/4/f*cot(f*x+e)*(3*(-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-ta
n(f*x+e))*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4))*2^(1/2)-7*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)*a
rctan(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)*2^(1/2)*sin(
f*x+e)-4*(-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))+10*(-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)+8*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)-2*co
s(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)+4*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*si
n(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)-12*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*cs
c(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)+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)*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))-6*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))*(1+tan(f*x+e))^(1/2)/(cos(f*x+e)+1)/(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)*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. 336 vs. \(2 (122) = 244\).

Time = 0.26 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.09 \[ \int \frac {\cot (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {\sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) - \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) - \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) + \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) + 2 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) - 2 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right )}{2 \, f} \]

[In]

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

[Out]

-1/2*(sqrt(1/2)*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(1/2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4)
+ 1)/f^2) + sqrt(tan(f*x + e) + 1)) - sqrt(1/2)*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(-sqrt(1/2)*(f^3*sqrt(-
1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1)) - sqrt(1/2)*f*sqrt(-(f^2*sqrt(-1/f^4) -
1)/f^2)*log(sqrt(1/2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2) + sqrt(tan(f*x + e) + 1)) + sq
rt(1/2)*f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(-sqrt(1/2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) - 1)
/f^2) + sqrt(tan(f*x + e) + 1)) + 2*log(sqrt(tan(f*x + e) + 1) + 1) - 2*log(sqrt(tan(f*x + e) + 1) - 1))/f

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.53 \[ \int \frac {\cot (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {2\,\mathrm {atanh}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )}{f}+2\,\mathrm {atanh}\left (2\,f\,\sqrt {\frac {\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}+2\,\mathrm {atanh}\left (2\,f\,\sqrt {\frac {\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}} \]

[In]

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

[Out]

2*atanh(2*f*((1/8 - 1i/8)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2))*((1/8 - 1i/8)/f^2)^(1/2) - (2*atanh((tan(e + f*
x) + 1)^(1/2)))/f + 2*atanh(2*f*((1/8 + 1i/8)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2))*((1/8 + 1i/8)/f^2)^(1/2)