3.4.88 \(\int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx\) [388]

Optimal. Leaf size=346 \[ -\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{f}+\frac {7 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}+\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \]

[Out]

7/8*arctanh((1+tan(f*x+e))^(1/2))/f-1/2*arctan(((2+2*2^(1/2))^(1/2)-2*(1+tan(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/
2))*(2+2*2^(1/2))^(1/2)/f+1/2*arctan(((2+2*2^(1/2))^(1/2)+2*(1+tan(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))*(2+2*2
^(1/2))^(1/2)/f+1/2*ln(1+2^(1/2)-(2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(2+2*2^(1/2))^(1/2)-1/
2*ln(1+2^(1/2)+(2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(2+2*2^(1/2))^(1/2)+9/8*cot(f*x+e)*(1+ta
n(f*x+e))^(1/2)/f-1/12*cot(f*x+e)^2*(1+tan(f*x+e))^(1/2)/f-1/3*cot(f*x+e)^3*(1+tan(f*x+e))^(1/2)/f

________________________________________________________________________________________

Rubi [A]
time = 0.43, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3649, 3730, 3734, 21, 3566, 714, 1141, 1175, 632, 210, 1178, 642, 3715, 65, 213} \begin {gather*} -\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {ArcTan}\left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{f}+\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}-\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}+\frac {7 \tanh ^{-1}\left (\sqrt {\tan (e+f x)+1}\right )}{8 f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{12 f}+\frac {9 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]],x]

[Out]

-((Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f)
+ (Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f
+ (7*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(8*f) + Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 +
Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) - Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 +
Tan[e + f*x]]]/(2*Sqrt[2*(1 + Sqrt[2])]*f) + (9*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(8*f) - (Cot[e + f*x]^2*S
qrt[1 + Tan[e + f*x]])/(12*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(3*f)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 714

Int[Sqrt[(d_) + (e_.)*(x_)]/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2 + a*e^2 - 2*c*d
*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1141

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(
a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && Lt
Q[b^2 - 4*a*c, 0] && PosQ[a*c]

Rule 1175

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !Lt
Q[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 1178

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e) - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 0]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(a^2
+ b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, 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] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*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]

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps

\begin {align*} \int \cot ^4(e+f x) \sqrt {1+\tan (e+f x)} \, dx &=-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{3} \int \frac {\cot ^3(e+f x) \left (-\frac {1}{2}+3 \tan (e+f x)+\frac {5}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {1}{6} \int \frac {\cot ^2(e+f x) \left (-\frac {27}{4}-6 \tan (e+f x)-\frac {3}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{6} \int \frac {\cot (e+f x) \left (\frac {21}{8}-6 \tan (e+f x)-\frac {27}{8} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {1}{6} \int \frac {-6-6 \tan (e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx-\frac {7}{16} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx\\ &=\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {7 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{16 f}+\int \sqrt {1+\tan (e+f x)} \, dx\\ &=\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {7 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {\text {Subst}\left (\int \frac {\sqrt {1+x}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {7 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {2 \text {Subst}\left (\int \frac {x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f}\\ &=\frac {7 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}-x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f}\\ &=\frac {7 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 f}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{-\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x-x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{-\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x-x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}\\ &=\frac {7 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}+\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}-\frac {\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}\right )}{f}-\frac {\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}\right )}{f}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {2 \left (-1+\sqrt {2}\right )} f}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {2 \left (-1+\sqrt {2}\right )} f}+\frac {7 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} f}+\frac {9 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.44, size = 151, normalized size = 0.44 \begin {gather*} \frac {21 \tanh ^{-1}\left (\sqrt {1+\tan (e+f x)}\right )-24 i \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+24 i \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+27 \cot (e+f x) \sqrt {1+\tan (e+f x)}-2 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}-8 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^4*Sqrt[1 + Tan[e + f*x]],x]

[Out]

(21*ArcTanh[Sqrt[1 + Tan[e + f*x]]] - (24*I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (24*I)*
Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 27*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] - 2*Cot[e + f
*x]^2*Sqrt[1 + Tan[e + f*x]] - 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(24*f)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.87, size = 13941, normalized size = 40.29

method	result	size
default	\(\text {Expression too large to display}\)	\(13941\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4*(1+tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

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)^4*(1+tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(tan(f*x + e) + 1)*cot(f*x + e)^4, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (284) = 568\).
time = 1.18, size = 1202, normalized size = 3.47 \begin {gather*} -\frac {6 \cdot 2^{\frac {1}{4}} {\left (2 \, f \cos \left (f x + e\right )^{4} - 4 \, f \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (f^{3} \cos \left (f x + e\right )^{4} - 2 \, f^{3} \cos \left (f x + e\right )^{2} + f^{3}\right )} \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^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \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}} \cos \left (f x + e\right ) + 2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 2 \, \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )}{2 \, \cos \left (f x + e\right )}\right ) - 6 \cdot 2^{\frac {1}{4}} {\left (2 \, f \cos \left (f x + e\right )^{4} - 4 \, f \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (f^{3} \cos \left (f x + e\right )^{4} - 2 \, f^{3} \cos \left (f x + e\right )^{2} + f^{3}\right )} \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^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \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}} \cos \left (f x + e\right ) - 2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 2 \, \sin \left (f x + e\right )}{2 \, \cos \left (f x + e\right )}\right ) - 21 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + 1\right ) + 21 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \log \left (\sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} - 1\right ) - 2 \, {\left (2 \, \cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} - {\left (35 \, \cos \left (f x + e\right )^{3} - 27 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {\cos \left (f x + e\right ) + \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} + \frac {24 \cdot 2^{\frac {3}{4}} {\left (f^{5} \cos \left (f x + e\right )^{4} - 2 \, f^{5} \cos \left (f x + e\right )^{2} + f^{5}\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{5} \sqrt {\frac {2^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \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}} \cos \left (f x + e\right ) + 2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) + 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 {5}{4}} - \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{5} \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 {5}{4}} - f^{2} \sqrt {\frac {1}{f^{4}}} - \sqrt {2}\right )}{f^{4}} + \frac {24 \cdot 2^{\frac {3}{4}} {\left (f^{5} \cos \left (f x + e\right )^{4} - 2 \, f^{5} \cos \left (f x + e\right )^{2} + f^{5}\right )} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} \frac {1}{f^{4}}^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{5} \sqrt {-\frac {2^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{3} \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}} \cos \left (f x + e\right ) - 2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} \cos \left (f x + e\right ) - 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 {5}{4}} - \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} f^{2} \sqrt {\frac {1}{f^{4}}} + 4} f^{5} \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 {5}{4}} + f^{2} \sqrt {\frac {1}{f^{4}}} + \sqrt {2}\right )}{f^{4}}}{48 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-1/48*(6*2^(1/4)*(2*f*cos(f*x + e)^4 - 4*f*cos(f*x + e)^2 - sqrt(2)*(f^3*cos(f*x + e)^4 - 2*f^3*cos(f*x + e)^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^(3/4)*sqrt(2*sqrt(2
)*f^2*sqrt(f^(-4)) + 4)*f^3*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4)*cos(f*x + e) + 2*s
qrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) + 2*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e)) - 6*2^(1/4)*(2*f*cos(f*x
+ e)^4 - 4*f*cos(f*x + e)^2 - sqrt(2)*(f^3*cos(f*x + e)^4 - 2*f^3*cos(f*x + e)^2 + f^3)*sqrt(f^(-4)) + 2*f)*s
qrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*log(-1/2*(2^(3/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^3*
sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4)*cos(f*x + e) - 2*sqrt(2)*f^2*sqrt(f^(-4))*cos(
f*x + e) - 2*cos(f*x + e) - 2*sin(f*x + e))/cos(f*x + e)) - 21*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*log(sqr
t((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) + 1) + 21*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*log(sqrt((cos(
f*x + e) + sin(f*x + e))/cos(f*x + e)) - 1) - 2*(2*cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (35*cos(f*x + e)^3 - 27
*cos(f*x + e))*sin(f*x + e))*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e)) + 24*2^(3/4)*(f^5*cos(f*x + e)^4
- 2*f^5*cos(f*x + e)^2 + f^5)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*arctan(1/2*2^(3/4)*sqrt(1/2
)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^5*sqrt((2^(3/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^3*sqrt((cos(f*
x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4)*cos(f*x + e) + 2*sqrt(2)*f^2*sqrt(f^(-4))*cos(f*x + e) + 2
*cos(f*x + e) + 2*sin(f*x + e))/cos(f*x + e))*(f^(-4))^(5/4) - 1/2*2^(3/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4
)*f^5*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(5/4) - f^2*sqrt(f^(-4)) - sqrt(2))/f^4 + 24*2
^(3/4)*(f^5*cos(f*x + e)^4 - 2*f^5*cos(f*x + e)^2 + f^5)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*(f^(-4))^(1/4)*a
rctan(1/2*2^(3/4)*sqrt(1/2)*sqrt(2*sqrt(2)*f^2*sqrt(f^(-4)) + 4)*f^5*sqrt(-(2^(3/4)*sqrt(2*sqrt(2)*f^2*sqrt(f^
(-4)) + 4)*f^3*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(3/4)*cos(f*x + e) - 2*sqrt(2)*f^2*sq
rt(f^(-4))*cos(f*x + e) - 2*cos(f*x + e) - 2*sin(f*x + e))/cos(f*x + e))*(f^(-4))^(5/4) - 1/2*2^(3/4)*sqrt(2*s
qrt(2)*f^2*sqrt(f^(-4)) + 4)*f^5*sqrt((cos(f*x + e) + sin(f*x + e))/cos(f*x + e))*(f^(-4))^(5/4) + f^2*sqrt(f^
(-4)) + sqrt(2))/f^4)/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\tan {\left (e + f x \right )} + 1} \cot ^{4}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4*(1+tan(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(tan(e + f*x) + 1)*cot(e + f*x)**4, x)

________________________________________________________________________________________

Giac [A]
time = 1.20, size = 359, normalized size = 1.04 \begin {gather*} \frac {7 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right )}{16 \, f} - \frac {7 \, \log \left ({\left | \sqrt {\tan \left (f x + e\right ) + 1} - 1 \right |}\right )}{16 \, f} + \frac {{\left (f^{2} \sqrt {\sqrt {2} + 1} + f \sqrt {\sqrt {2} - 1} {\left | f \right |}\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} + \frac {{\left (f^{2} \sqrt {\sqrt {2} + 1} + f \sqrt {\sqrt {2} - 1} {\left | f \right |}\right )} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{2 \, f^{3}} + \frac {{\left (f^{2} \sqrt {\sqrt {2} - 1} - f \sqrt {\sqrt {2} + 1} {\left | f \right |}\right )} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} - \frac {{\left (f^{2} \sqrt {\sqrt {2} - 1} - f \sqrt {\sqrt {2} + 1} {\left | f \right |}\right )} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{4 \, f^{3}} + \frac {27 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {5}{2}} - 56 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} + 21 \, \sqrt {\tan \left (f x + e\right ) + 1}}{24 \, f \tan \left (f x + e\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(1/2),x, algorithm="giac")

[Out]

7/16*log(sqrt(tan(f*x + e) + 1) + 1)/f - 7/16*log(abs(sqrt(tan(f*x + e) + 1) - 1))/f + 1/2*(f^2*sqrt(sqrt(2) +
1) + f*sqrt(sqrt(2) - 1)*abs(f))*arctan(1/2*2^(3/4)*(2^(1/4)*sqrt(sqrt(2) + 2) + 2*sqrt(tan(f*x + e) + 1))/sq
rt(-sqrt(2) + 2))/f^3 + 1/2*(f^2*sqrt(sqrt(2) + 1) + f*sqrt(sqrt(2) - 1)*abs(f))*arctan(-1/2*2^(3/4)*(2^(1/4)*
sqrt(sqrt(2) + 2) - 2*sqrt(tan(f*x + e) + 1))/sqrt(-sqrt(2) + 2))/f^3 + 1/4*(f^2*sqrt(sqrt(2) - 1) - f*sqrt(sq
rt(2) + 1)*abs(f))*log(2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x + e) + 1) + sqrt(2) + tan(f*x + e) + 1)/f^3 - 1/
4*(f^2*sqrt(sqrt(2) - 1) - f*sqrt(sqrt(2) + 1)*abs(f))*log(-2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x + e) + 1) +
sqrt(2) + tan(f*x + e) + 1)/f^3 + 1/24*(27*(tan(f*x + e) + 1)^(5/2) - 56*(tan(f*x + e) + 1)^(3/2) + 21*sqrt(t
an(f*x + e) + 1))/(f*tan(f*x + e)^3)

________________________________________________________________________________________

Mupad [B]
time = 0.19, size = 175, normalized size = 0.51 \begin {gather*} -\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{8\,f}-\frac {\frac {7\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{8}-\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3}+\frac {9\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{8}}{f-3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3}-\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-\mathrm {i}\right )\right )\,\sqrt {\frac {-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^4*(tan(e + f*x) + 1)^(1/2),x)

[Out]

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 - ((7*(tan(
e + f*x) + 1)^(1/2))/8 - (7*(tan(e + f*x) + 1)^(3/2))/3 + (9*(tan(e + f*x) + 1)^(5/2))/8)/(f - 3*f*(tan(e + f*
x) + 1) + 3*f*(tan(e + f*x) + 1)^2 - f*(tan(e + f*x) + 1)^3) - 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 - (atan((tan(e + f*x) + 1)^(1/2)*1i)*7i)/(8*f)

________________________________________________________________________________________