3.4.81 \(\int \cot (e+f x) \sqrt {1+\tan (e+f x)} \, dx\) [381]
Optimal. Leaf size=165 \[ \frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f} \]
[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 = 0.18, 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 = {3653, 3617,
3616, 209, 213, 3715, 65} \begin {gather*} \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \text {ArcTan}\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} \end {gather*}
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 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 3653
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 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*} \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 {\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 \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 ) \text {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] Result contains complex when optimal does not.
time = 0.09, size = 78, normalized size = 0.47 \begin {gather*} \frac {-2 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )+\sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+\sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )}{f} \end {gather*}
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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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 18.54, size = 2750, normalized size = 16.67
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\(\text {Expression too large to display}\) |
\(2750\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(f*x+e)*(1+tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/8/f*((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*2^(1/2)
*((sin(f*x+e)-1)*(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)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*((sin(f*x+e)-
1)/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)*((sin(f*x+e)-1)*(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)/co
s(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*((sin(
f*x+e)-1)/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)*((sin(f*x+e)-1)*(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)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)*EllipticPi(1/2*
((sin(f*x+e)-1)/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))-((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)
+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e
)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticE(1/2*((sin(f*x+e)-1)/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))+((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*2^(1/2)*((2^(1/2)*cos(f*
x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x
+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticF(1/2*((sin(f*x+e)-1)/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))+2*2^(1/2)*EllipticE(1/2*((sin(f*x+e)-1)/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))*((sin(f*x+e)-1)*(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)-2^(1/2)-cos(f*
x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)+20*2^(1/2)*EllipticPi(1/2*((sin(f*x+e)-1)/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))*((sin(f*x+e)-1)*(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)-2^(1/2)-co
s(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2)-8*2^(1/2)*((sin(f*x+e)-1)*(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)-2^(1/2)-cos(f*x+e)+sin(f*x+e)
-1)/cos(f*x+e))^(1/2)*EllipticF(1/2*((sin(f*x+e)-1)/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))-8*2^(1/2)*((sin(f*x+e)-1)*(1+2^(1/2))/cos(f*x+e))^(1/2)*((2^(1/2)*sin(f*x+e)-2^(1/2)+co
s(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*((sin(f*x+e)-1)/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))-((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2
)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)
-2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticE(1/2*((sin(f*x+e)-1)/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))-6*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((2^(1/2)*
cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*s
in(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticPi(1/2*((sin(f*x+e)-1)/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*((sin(f*x+e)-1)/cos(f*x+e)
*(2+2^(1/2))*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2)
)^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)/cos(f*x+e)*2^(1/2))^(1/2)*EllipticPi(1
/2*((sin(f*x+e)-1)/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*((sin(f*x+e)-1)/cos(f*x+e)*(2+2^(1/2))*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(
1/2)+2*sin(f*x+e)-2)/cos(f*x+e)*2^(1/2))^(1/2)*((2^(1/2)*cos(f*x+e)-2^(1/2)*sin(f*x+e)+2^(1/2)-2*sin(f*x+e)+2)
/cos(f*x+e)*2^(1/2))^(1/2)*EllipticF(1/2*((sin(f*x+e)-1)/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*EllipticE(1/2*((sin(f*x+e)-1)/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))*((sin(f*x+e)-1)*(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)-2^(1/2)-cos(f*x+e)+sin(f*x+e)-1)/cos(f*x+e))^(1/2
)-4*((sin(f*x+e)-1)*(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)-2^(1/2)-cos(f...
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1016 vs.
\(2 (132) = 264\).
time = 0.97, size = 1016, normalized size = 6.16 \begin {gather*} -\frac {4 \cdot 2^{\frac {3}{4}} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - \frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - f^{2} \sqrt {\frac {1}{f^{4}}} - \sqrt {2}\right ) + 4 \cdot 2^{\frac {3}{4}} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} - \frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (f^{5} \sqrt {\frac {1}{f^{4}}} + \sqrt {2} f^{3}\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {3}{4}} + f^{2} \sqrt {\frac {1}{f^{4}}} + \sqrt {2}\right ) - 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} + 2 \, f\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{2 \, \cos \left (f x + e\right )}\right ) + 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} + 2 \, f\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 2^{\frac {1}{4}} {\left (\sqrt {2} f^{3} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, f \cos \left (f x + e\right )\right )} \sqrt {-2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \frac {1}{f^{4}}^{\frac {1}{4}} + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{2 \, \cos \left (f x + e\right )}\right ) + 8 \, \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + 1\right ) - 8 \, \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - 1\right )}{8 \, f} \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\tan {\left (e + f x \right )} + 1} \cot {\left (e + f x \right )}\, dx \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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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Mupad [B]
time = 0.16, size = 90, normalized size = 0.55 \begin {gather*} -\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}} \end {gather*}
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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