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]
(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
________________________________________________________________________________________
Rubi [A] time = 0.254511, antiderivative size = 165, 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 =
{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 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 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 \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*}
Mathematica [C] time = 0.0765813, 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
________________________________________________________________________________________
Maple [C] time = 0.642, size = 2750, normalized size = 16.7 \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]
1/8/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*(1+sin(f*x+e))*
(8*I*((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)
)*2^(1/2)-8*I*((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))*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)/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)-(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)-8*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))*((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)*2^(1/2)+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)*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)-8*((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))*2^(1/2)+20*((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))*2^(1/2)-8*((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))*2^(1/2)+4*(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+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)*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+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*(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*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))*((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)-c
os(f*x+e)+sin(f*x+e)-1)/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)/co
s(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)/sin(f*x+e)^4/(cos(f*x+e)+sin(f*x+e))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )\,{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(sqrt(tan(f*x + e) + 1)*cot(f*x + e), x)
________________________________________________________________________________________
Fricas [B] time = 1.99618, size = 2857, normalized size = 17.32 \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]
-1/8*(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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\tan{\left (e + f x \right )} + 1} \cot{\left (e + f x \right )}\, 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(sqrt(tan(e + f*x) + 1)*cot(e + f*x), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\tan \left (f x + e\right ) + 1} \cot \left (f x + e\right )\,{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(sqrt(tan(f*x + e) + 1)*cot(f*x + e), x)