\(\int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\) [410]

Optimal result

Integrand size = 21, antiderivative size = 339 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 f}+\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 f}-\frac {3 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{4 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{4 \sqrt {1+\sqrt {2}} f}+\frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \]

[Out]

-3/8*arctanh((1+tan(f*x+e))^(1/2))/f-1/4*ln(1+2^(1/2)-(2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(
1+2^(1/2))^(1/2)+1/4*ln(1+2^(1/2)+(2+2*2^(1/2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(1+2^(1/2))^(1/2)-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))*(1+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))*(1+2^(1/2))^(1/2)/f+3/8*cot(f*x+e)*(1+tan(f*x+e
))^(1/2)/f+5/12*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] (verified)

Time = 0.71 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3650, 3730, 3731, 3734, 12, 3566, 722, 1108, 648, 632, 210, 642, 3715, 65, 213} \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{2 f}+\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{2 f}-\frac {3 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{8 f}-\frac {\log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}} f}+\frac {\log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}} f}-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}+\frac {5 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{12 f}+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{8 f} \]

[In]

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

[Out]

-1/2*(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/f +
(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*(-1 + Sqrt[2])]])/(2*f) -
(3*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/(8*f) - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + T
an[e + f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Tan[e
+ f*x]]]/(4*Sqrt[1 + Sqrt[2]]*f) + (3*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/(8*f) + (5*Cot[e + f*x]^2*Sqrt[1 +
Tan[e + f*x]])/(12*f) - (Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/(3*f)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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 648

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

Rule 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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 1108

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

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 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 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]

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 3731

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*t
an[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 + a^2*C)*(a + b*Tan[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)) - a*C*(b*c*(m + 1)
+ a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 2)*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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] && ( !IntegerQ[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*} \text {integral}& = -\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 {5}{2}+3 \tan (e+f x)+\frac {5}{2} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {5 \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 {9}{4}+\frac {15}{4} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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 {9}{8}-6 \tan (e+f x)-\frac {9}{8} \tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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}{\sqrt {1+\tan (e+f x)}} \, dx+\frac {3}{16} \int \frac {\cot (e+f x) \left (1+\tan ^2(e+f x)\right )}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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 {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\tan (e+f x)\right )}{16 f}+\int \frac {1}{\sqrt {1+\tan (e+f x)}} \, dx \\ & = \frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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 {3 \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 {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {3 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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 {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{f} \\ & = -\frac {3 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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 \left (1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\tan (e+f x)}\right )}{2 \sqrt {1+\sqrt {2}} f} \\ & = -\frac {3 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}+\frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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 \sqrt {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 \sqrt {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 )}{4 \sqrt {1+\sqrt {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 )}{4 \sqrt {1+\sqrt {2}} f} \\ & = -\frac {3 \text {arctanh}\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 )}{4 \sqrt {1+\sqrt {2}} 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 )}{4 \sqrt {1+\sqrt {2}} f}+\frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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 )}{\sqrt {2} 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 )}{\sqrt {2} f} \\ & = -\frac {\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 )}{2 \sqrt {-1+\sqrt {2}} f}+\frac {\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 )}{2 \sqrt {-1+\sqrt {2}} f}-\frac {3 \text {arctanh}\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 )}{4 \sqrt {1+\sqrt {2}} 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 )}{4 \sqrt {1+\sqrt {2}} f}+\frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}+\frac {5 \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] (verified)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.43 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {-9 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+12 (1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+12 (1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+9 \cot (e+f x) \sqrt {1+\tan (e+f x)}+10 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}-8 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f} \]

[In]

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

[Out]

(-9*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 12*(1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + 12*(1 + I
)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]] + 9*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 10*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 [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4581\) vs. \(2(263)=526\).

Time = 174.24 (sec) , antiderivative size = 4582, normalized size of antiderivative = 13.52

method	result	size
default	\(\text {Expression too large to display}\)	\(4582\)

[In]

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

[Out]

1/48/f*cot(f*x+e)*csc(f*x+e)^2*(1+tan(f*x+e))^(1/2)*(36*(1+2^(1/2))^(1/2)*(-2+2*2^(1/2))^(1/2)*((cos(f*x+e)+si
n(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x
+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-
2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)*(-4*sin(f*x+e)*cos(f*x+e
)+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)/(2*cos(f*x+e)^2-1))*cos(f*x+e)*sin(f*x+e)*2^(
1/2)+12*(1+2^(1/2))^(1/2)*cos(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*si
n(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(
f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e
)^2+1)*(3*2^(1/2)-4))^(1/2)*(-4*sin(f*x+e)*cos(f*x+e)+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1
/2)-4)/(2*cos(f*x+e)^2-1))*(-2+2*2^(1/2))^(1/2)*2^(1/2)+72*cos(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2
^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh((
(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*
x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))-168*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^
(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1
+2^(1/2))^(1/2))*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2
*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*sin(f*x+e)-40*2^(1/2)*(1+2^(1/2))^(1/2)*cos(f*x+e)*sin(f*x+e)*
((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)-120*2^(1/2)*cos(f*
x+e)*sin(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*
sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+
e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2
))+60*(1+2^(1/2))^(1/2)*cos(f*x+e)*sin(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot
(f*x+e)^2+cot(f*x+e))^(1/2)+120*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2
*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*((cos(f*x+e)
+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(
f*x+e)^2+1))^(1/2)*2^(1/2)*sin(f*x+e)-36*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^
2)^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)+68*2^(1/2)*(1+2^(1/2))^(1/2)*cos(f*x+e)^2*((cos(f*x+e)+sin(f*
x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)+48*2^(1/2)*cos(f*x+e)^2*((cos(f*x+e)+
sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f
*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*
2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))-102*(1+2^(1/2))^(1/2)*cos(
f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)+168*cos(f
*x+e)*sin(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2
*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x
+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/
2))-36*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2
^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(
f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^
(1/2)-4))^(1/2)*(-4*sin(f*x+e)*cos(f*x+e)+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)/(2*co
s(f*x+e)^2-1))*(-2+2*2^(1/2))^(1/2)*2^(1/2)*sin(f*x+e)+54*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e
)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)+27*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+
e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(
1/2))*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*(1+2^(1/2))^(1/2)*sin(f*x+e)-48*cos(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(
f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2
)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*
x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)-24*(1+2^(1/2))^(1/2)*cos(f*x+e)*((
cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x
+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e
)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)*(-4*sin(f*x
+e)*cos(f*x+e)+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)/(2*cos(f*x+e)^2-1))*(-2+2*2^(1/2
))^(1/2)+48*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e
)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))
*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*
(3*2^(1/2)-4))^(1/2)*(-4*sin(f*x+e)*cos(f*x+e)+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)/
(2*cos(f*x+e)^2-1))*(-2+2*2^(1/2))^(1/2)*sin(f*x+e)-12*(1+2^(1/2))^(1/2)*(-2+2*2^(1/2))^(1/2)*((cos(f*x+e)+sin
(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+
e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2
*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)*(-4*sin(f*x+e)*cos(f*x+e)
+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)/(2*cos(f*x+e)^2-1))*cos(f*x+e)^2*2^(1/2)-48*(1
+2^(1/2))^(1/2)*(-2+2*2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*si
n(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(
f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e
)^2+1)*(3*2^(1/2)-4))^(1/2)*(-4*sin(f*x+e)*cos(f*x+e)+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1
/2)-4)/(2*cos(f*x+e)^2-1))*cos(f*x+e)*sin(f*x+e)-27*(1+2^(1/2))^(1/2)*cos(f*x+e)*sin(f*x+e)*(cot(f*x+e)^2+cot(
f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)-1+2*csc
(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))-72*cos(f*x+e)^2*((cos(f*x+e)+sin(f*x+e))*
cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^
(1/2)*arctanh(((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*si
n(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2))-18*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*
x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(
f*x+e)+1)^2)^(1/2))*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)*(1+2^(1/2))^(1/2)*sin(f*x+e)+18*(1+2^(1/2))^(1/2)*
cos(f*x+e)*sin(f*x+e)*(cot(f*x+e)^2+cot(f*x+e))^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos
(f*x+e)+1)^2)^(1/2)-2*cot(f*x+e)-1+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*2
^(1/2)+24*(1+2^(1/2))^(1/2)*(-2+2*2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin
(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos
(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)
+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)*(-4*sin(f*x+e)*cos(f*x+e)+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2
)+3)*(3*2^(1/2)-4)/(2*cos(f*x+e)^2-1))*cos(f*x+e)^2)/((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/
2)/(cot(f*x+e)^2+cot(f*x+e))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2)/(3*2^(1/2)-4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.26 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {24 \, \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} + 24 \, \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 9 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{3} + 9 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{3} + 2 \, {\left (9 \, \tan \left (f x + e\right )^{2} + 10 \, \tan \left (f x + e\right ) - 8\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{48 \, f \tan \left (f x + e\right )^{3}} \]

[In]

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

[Out]

1/48*(24*sqrt(1/2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(1/2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1
/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 - 24*sqrt(1/2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*lo
g(-sqrt(1/2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3
- 24*sqrt(1/2)*f*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2)*log(sqrt(1/2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4)
- 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 + 24*sqrt(1/2)*f*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2)*log(-sqr
t(1/2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 - 9*lo
g(sqrt(tan(f*x + e) + 1) + 1)*tan(f*x + e)^3 + 9*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e)^3 + 2*(9*tan(f*x
+ e)^2 + 10*tan(f*x + e) - 8)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^3)

Sympy [F]

\[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\sqrt {\tan {\left (e + f x \right )} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{4}}{\sqrt {\tan \left (f x + e\right ) + 1}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.76 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {3 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right )}{16 \, f} + \frac {3 \, \log \left ({\left | \sqrt {\tan \left (f x + e\right ) + 1} - 1 \right |}\right )}{16 \, f} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{4 \, f^{3}} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{4 \, f^{3}} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{8 \, f^{3}} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{8 \, f^{3}} + \frac {9 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {5}{2}} - 8 \, {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} - 9 \, \sqrt {\tan \left (f x + e\right ) + 1}}{24 \, f \tan \left (f x + e\right )^{3}} \]

[In]

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

[Out]

-3/16*log(sqrt(tan(f*x + e) + 1) + 1)/f + 3/16*log(abs(sqrt(tan(f*x + e) + 1) - 1))/f + 1/4*(f^2*sqrt(2*sqrt(2
) - 2) + f*sqrt(2*sqrt(2) + 2)*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(2*sqrt(2) - 2) + f*sqrt(2*sqrt(2) + 2)*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 + 1/8*(f^2*sqrt(2*sqrt(2) + 2)
- f*sqrt(2*sqrt(2) - 2)*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 - 1/8*(f^2*sqrt(2*sqrt(2) + 2) - f*sqrt(2*sqrt(2) - 2)*abs(f))*log(-2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(ta
n(f*x + e) + 1) + sqrt(2) + tan(f*x + e) + 1)/f^3 + 1/24*(9*(tan(f*x + e) + 1)^(5/2) - 8*(tan(f*x + e) + 1)^(3
/2) - 9*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^3)

Mupad [B] (verification not implemented)

Time = 5.15 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.50 \[ \int \frac {\cot ^4(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,f}+\frac {\frac {3\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{8}+\frac {{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3}-\frac {3\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{8}}{f-3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3}+\mathrm {atan}\left (2\,f\,\sqrt {\frac {-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (2\,f\,\sqrt {\frac {-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]

[In]

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

[Out]

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