3.388 \(\int \cot ^4(e+f x) \sqrt{1+\tan (e+f x)} \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.624529, antiderivative size = 346, normalized size of antiderivative = 1., 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 = {3568, 3649, 3653, 21, 3485, 700, 1127, 1161, 618, 204, 1164, 628, 3634, 63, 207} \[ -\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )} f}-\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )} f}+\frac{7 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )}{8 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}-\frac{\sqrt{\tan (e+f x)+1} \cot ^2(e+f x)}{12 f}+\frac{9 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{8 f} \]

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 3568

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 3649

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*T
an[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 3653

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]

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 3485

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 700

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 1127

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 1161

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 618

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 204

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

Rule 1164

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 628

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 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]

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])

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 \operatorname{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 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{8 f}+\frac{\operatorname{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 \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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]  time = 0.399362, size = 151, normalized size = 0.44 \[ \frac{21 \tanh ^{-1}\left (\sqrt{\tan (e+f x)+1}\right )-24 i \sqrt{1-i} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right )+24 i \sqrt{1+i} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right )-8 \sqrt{\tan (e+f x)+1} \cot ^3(e+f x)-2 \sqrt{\tan (e+f x)+1} \cot ^2(e+f x)+27 \sqrt{\tan (e+f x)+1} \cot (e+f x)}{24 f} \]

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)

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Maple [C]  time = 0.767, size = 13941, normalized size = 40.3 \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

result too large to display

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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 )^{4}\,{d x} \end{align*}

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)

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Fricas [B]  time = 2.23811, size = 3209, normalized size = 9.27 \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)^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)

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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 ^{4}{\left (e + f x \right )}\, dx \end{align*}

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)

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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 )^{4}\,{d x} \end{align*}

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]

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