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

Optimal. Leaf size=165 \[ \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {2 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{f} \]

[Out]

-2*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

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Rubi [A]  time = 0.25, antiderivative size = 165, normalized size of antiderivative = 1.00, 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 = {3572, 3536, 3535, 203, 207, 3634, 63} \[ \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {2 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

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]

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 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 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 3572

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

Rubi steps

\begin {align*} \int \cot (e+f x) \sqrt {1+\tan (e+f x)} \, dx &=\int \frac {1-\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 {\sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx}{2 \sqrt {2}}-\frac {\int \frac {-\sqrt {2}+\left (2+\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 \sqrt {2} \left (2-\sqrt {2}\right )-4 \left (2-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {\sqrt {2}-2 \left (2-\sqrt {2}\right )-\left (2-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {\left (4+3 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-2 \sqrt {2} \left (2+\sqrt {2}\right )-4 \left (2+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {-\sqrt {2}-2 \left (2+\sqrt {2}\right )-\left (2+\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 )} \tan ^{-1}\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 {2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\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}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 78, normalized size = 0.47 \[ \frac {-2 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )+\sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {\tan (e+f x)+1}}{\sqrt {1-i}}\right )+\sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {\tan (e+f x)+1}}{\sqrt {1+i}}\right )}{f} \]

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]]] + Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + Sqrt[1 + I]*Ar
cTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/f

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fricas [B]  time = 0.58, size = 968, normalized size = 5.87 \[ \text {result too large to display} \]

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]

-1/8*(4*2^(3/4)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f*(f^(-4))^(1/4)*arctan(1/2*2^(3/4)*sqrt(1/2)*(f^5*sqrt(
f^(-4)) + sqrt(2)*f^3)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) + 2
^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4))*cos(f*x + e) + 2*f*cos(f*x + e))*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((
cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e))*(f^
(-4))^(3/4) - 1/2*2^(3/4)*(f^5*sqrt(f^(-4)) + sqrt(2)*f^3)*sqrt(-2*sqrt(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)) - sqrt(2)) + 4*2^(3/4)*sqrt(-2*sqrt(2)*f
^2*sqrt(f^(-4)) + 4)*f*(f^(-4))^(1/4)*arctan(1/2*2^(3/4)*sqrt(1/2)*(f^5*sqrt(f^(-4)) + sqrt(2)*f^3)*sqrt(-2*sq
rt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) - 2^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4))*
cos(f*x + e) + 2*f*cos(f*x + e))*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(
f*x + e))*(f^(-4))^(1/4) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4) - 1/2*2^(3/4)*(f^5*sq
rt(f^(-4)) + sqrt(2)*f^3)*sqrt(-2*sqrt(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)) + sqrt(2)) - 2^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4)) + 2*f)*sqrt(-2*sqrt(2)*f^2
*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(1/2*(2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) + 2^(1/4)*(sqrt(2)*f^3*sqrt
(f^(-4))*cos(f*x + e) + 2*f*cos(f*x + e))*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*sqrt((cos(f*x + e) + sin(f*x +
e))/cos(f*x + e))*(f^(-4))^(1/4) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e)) + 2^(1/4)*(sqrt(2)*f^3*sqrt
(f^(-4)) + 2*f)*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(1/2*(2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f
*x + e) - 2^(1/4)*(sqrt(2)*f^3*sqrt(f^(-4))*cos(f*x + e) + 2*f*cos(f*x + e))*sqrt(-2*sqrt(2)*f^2*sqrt(f^(-4))
+ 4)*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(1/4) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*
x + e)) + 8*log(sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) + 1) - 8*log(sqrt((cos(f*x + e) + sin(f*x + e
))/cos(f*x + e)) - 1))/f

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )\,{d x} \]

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

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maple [C]  time = 1.69, size = 2731, normalized size = 16.55 \[ \text {Expression too large to display} \]

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]

-1/8/f*((cos(f*x+e)+sin(f*x+e))/cos(f*x+e))^(1/2)*(1+cos(f*x+e))^2*(-1+cos(f*x+e))^2*(1+sin(f*x+e))*(8*I*2^(1/
2)*((-1+sin(f*x+e))*(1+2^(1/2))/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)+cos(f*x+e)-sin(f*x+e)+2^(1/2)+1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*((-1+sin(f
*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),-I*2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))
-8*I*2^(1/2)*((-1+sin(f*x+e))*(1+2^(1/2))/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)+cos(f*x+e)-sin(f*x+e)+2^(1/2)+1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*
((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)
))^(1/2))-((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+
e)+2*sin(f*x+e)+2^(1/2)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)-2*sin(f*x+e)+2^(1
/2)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticF(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),
I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2)*((cos(f*x+e)
*2^(1/2)-2^(1/2)*sin(f*x+e)+2*sin(f*x+e)+2^(1/2)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin
(f*x+e)-2*sin(f*x+e)+2^(1/2)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticE(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2)
)*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))-20*2^(1/2)*((-1+sin(f*x+e))*(1+2^(1/2))/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)+cos(f*x+
e)-sin(f*x+e)+2^(1/2)+1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/
2)*2^(1/2),2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+8*2^(1/2)*EllipticF(1/2*((-1+sin(f*x+e))/cos
(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((-1+sin(f*x+e))*(1+2^(1/2))/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)+c
os(f*x+e)-sin(f*x+e)+2^(1/2)+1)/cos(f*x+e))^(1/2)-2*2^(1/2)*EllipticE(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/
2))*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((-1+sin(f*x+e))*(1+2^(1/2))/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)+cos(f*x+e)-sin(f*
x+e)+2^(1/2)+1)/cos(f*x+e))^(1/2)+8*2^(1/2)*((-1+sin(f*x+e))*(1+2^(1/2))/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)+cos(f*x+e)-sin(f*x+e)+2^(1/2)+1)/co
s(f*x+e))^(1/2)*EllipticPi(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),-I*2^(1/2)/(2+2^
(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+8*2^(1/2)*((-1+sin(f*x+e))*(1+2^(1/2))/cos(f*x+e))^(1/2)*((2^(1/2)*s
in(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)+cos(f*x+e)-sin(f*x+e)+2^(1/
2)+1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*2^(1/2
)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+6*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((co
s(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2*sin(f*x+e)+2^(1/2)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(
1/2)*sin(f*x+e)-2*sin(f*x+e)+2^(1/2)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticPi(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(
2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))-4*((-1+sin(f*x+e))/co
s(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2*sin(f*x+e)+2^(1/2)-2)/cos(f*x+e)
*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)-2*sin(f*x+e)+2^(1/2)+2)/cos(f*x+e)*2^(1/2))^(1/2)*Elli
pticF(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+((
-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)+2*sin(f*x+e)+2^(1
/2)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)-2*sin(f*x+e)+2^(1/2)+2)/cos(f*x+e)*2^
(1/2))^(1/2)*EllipticE(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^
(1/2)))^(1/2))-4*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e
)+2*sin(f*x+e)+2^(1/2)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((cos(f*x+e)*2^(1/2)-2^(1/2)*sin(f*x+e)-2*sin(f*x+e)+2^(1/
2)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticPi(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2),
-2^(1/2)/(2+2^(1/2)),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))+4*EllipticF(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2)
)*2^(1/2))^(1/2)*2^(1/2),I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((-1+sin(f*x+e))*(1+2^(1/2))/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)+cos(f*x+e)-sin(f*x+
e)+2^(1/2)+1)/cos(f*x+e))^(1/2)-4*EllipticE(1/2*((-1+sin(f*x+e))/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2)
,I*((2-2^(1/2))/(2+2^(1/2)))^(1/2))*((-1+sin(f*x+e))*(1+2^(1/2))/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)+cos(f*x+e)-sin(f*x+e)+2^(1/2)+1)/cos(f*x+e)
)^(1/2))*4^(1/2)/(cos(f*x+e)+sin(f*x+e))/sin(f*x+e)^4/(2+2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )\,{d x} \]

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

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mupad [B]  time = 0.16, size = 90, normalized size = 0.55 \[ -\frac {2\,\mathrm {atanh}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )}{f}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1+1{}\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+2\,\mathrm {atanh}\left (f\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1+1{}\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tan {\left (e + f x \right )} + 1} \cot {\left (e + f x \right )}\, dx \]

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

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