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

Optimal. Leaf size=161 \[ \frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{\left (1-\sqrt{2}\right ) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left (5 \sqrt{2}-7\right )} \sqrt{\tan (e+f x)+1}}\right )}{2 f}-\frac{2 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )}{f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{\left (1+\sqrt{2}\right ) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left (7+5 \sqrt{2}\right )} \sqrt{\tan (e+f x)+1}}\right )}{2 f} \]

[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)

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Rubi [A]  time = 0.203795, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3574, 3536, 3535, 203, 207, 3634, 63} \[ \frac{\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{\left (1-\sqrt{2}\right ) \tan (e+f x)-2 \sqrt{2}+3}{\sqrt{2 \left (5 \sqrt{2}-7\right )} \sqrt{\tan (e+f x)+1}}\right )}{2 f}-\frac{2 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )}{f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{\left (1+\sqrt{2}\right ) \tan (e+f x)+2 \sqrt{2}+3}{\sqrt{2 \left (7+5 \sqrt{2}\right )} \sqrt{\tan (e+f x)+1}}\right )}{2 f} \]

Antiderivative was successfully verified.

[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 3574

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 3536

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 3535

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 203

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

Rule 207

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

Rule 3634

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 63

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]

Rubi steps

\begin{align*} \int \frac{\cot (e+f x)}{\sqrt{1+\tan (e+f x)}} \, dx &=-\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{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{2 \operatorname{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 ) \operatorname{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 ) \operatorname{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}} \tan ^{-1}\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 \tanh ^{-1}\left (\sqrt{1+\tan (e+f x)}\right )}{f}+\frac{\sqrt{1+\sqrt{2}} \tanh ^{-1}\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]  time = 0.0719469, size = 83, normalized size = 0.52 \[ -\frac{2 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )}{f}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right )}{\sqrt{1-i} f}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right )}{\sqrt{1+i} f} \]

Antiderivative was successfully verified.

[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)

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Maple [C]  time = 0.447, size = 2030, normalized size = 12.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/f/(2+2^(1/2))*((cos(f*x+e)+sin(f*x+e))/cos(f*x+e))^(1/2)*(cos(f*x+e)+1)^2*(cos(f*x+e)-1)^2*(28*2^(1/2)*((
1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))
^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*2^(1/2)*((2+2^(
1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))-16*2^(1
/2)*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f
*x+e))^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*2^(1/2)*(
(2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))
-16*EllipticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),-I*2^(1/2)/(2+2^(1/2)),I*((2
-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)
*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+
e))^(1/2)-2*EllipticE(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(
1/2)))^(1/2))*2^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f
*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)+(2^(1/
2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e))^(1/2)*(2^(1/2)*(cos(f*x+e)*2^(1/
2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(1/2)*EllipticE(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1
+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))
/cos(f*x+e))^(1/2)-(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e))^(1/2)*(
2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(1/2)*EllipticF(1/2*2^(1/2)
*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*2^(1/2)*((2+2^(1/2)
)*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)-6*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*
x+e)-2)/cos(f*x+e))^(1/2)*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^
(1/2)*EllipticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),2^(1/2)/(2+2^(1/2)),I*((2-
2^(1/2))/(2+2^(1/2)))^(1/2))*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)-4*((1+2^(1/2))*(-1+sin(f*x
+e))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((-2^(1/2)*sin(
f*x+e)+2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*EllipticE(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f
*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+4*((1+2^(1/2))*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*(
(2^(1/2)*sin(f*x+e)-2^(1/2)+cos(f*x+e)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*((-2^(1/2)*sin(f*x+e)+2^(1/2)+cos(f*x+e
)-sin(f*x+e)+1)/cos(f*x+e))^(1/2)*EllipticF(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)
,I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/co
s(f*x+e))^(1/2)*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e))^(1/2)*Elli
pticE(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((
2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)+8*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)
+2*sin(f*x+e)-2)/cos(f*x+e))^(1/2)*(2^(1/2)*(cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos
(f*x+e))^(1/2)*EllipticPi(1/2*2^(1/2)*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2),-2^(1/2)/(2+2^(1/
2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((2+2^(1/2))*2^(1/2)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2))*(1+sin(f*x+e))*
4^(1/2)/(cos(f*x+e)+sin(f*x+e))/sin(f*x+e)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )}{\sqrt{\tan \left (f x + e\right ) + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.94114, size = 2925, normalized size = 18.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*(1/2)^(3/4)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*f*(f^(-4))^(1/4)*arctan(2*(1/2)^(3/4)*(f^5*sqrt(f^(
-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos(f*x + e)
+ (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4))
+ 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + cos(f*x + e) + sin(f*x + e))/cos(f*x +
e))*(f^(-4))^(3/4) - 2*(1/2)^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*
sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) - f^2*sqrt(f^(-4)) - 2*sqrt(1/2)) + 4*(1/2)^(3
/4)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*f*(f^(-4))^(1/4)*arctan(2*(1/2)^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(1/2
)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos(f*x + e) - (1/2)^(1/4)*(
2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(
f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3
/4) - 2*(1/2)^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(1/2)*f^3)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x +
e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) + f^2*sqrt(f^(-4)) + 2*sqrt(1/2)) + (1/2)^(1/4)*(sqrt(1/2)*f^
3*sqrt(f^(-4)) + f)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log((2*sqrt(1/2)*f^2*sqrt(f^(-4))*c
os(f*x + e) + (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*s
qrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + cos(f*x + e) + sin(f*x + e)
)/cos(f*x + e)) - (1/2)^(1/4)*(sqrt(1/2)*f^3*sqrt(f^(-4)) + f)*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4)
)^(1/4)*log((2*sqrt(1/2)*f^2*sqrt(f^(-4))*cos(f*x + e) - (1/2)^(1/4)*(2*sqrt(1/2)*f^3*sqrt(f^(-4))*cos(f*x + e
) + f*cos(f*x + e))*sqrt(-4*sqrt(1/2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(
f^(-4))^(1/4) + cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) - 4*log(sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x
+ e)) + 1) + 4*log(sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) - 1))/f

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (e + f x \right )}}{\sqrt{\tan{\left (e + f x \right )} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (f x + e\right )}{\sqrt{\tan \left (f x + e\right ) + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)