\(\int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx\) [404]

Optimal result

Integrand size = 21, antiderivative size = 215 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}+\frac {5 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{4 f}-\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{2 f}+\frac {3 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f} \] Output:

-1/2*(2^(1/2)-1)^(1/2)*arctan((3-2*2^(1/2)+(1-2^(1/2))*tan(f*x+e))/(-14+10 
*2^(1/2))^(1/2)/(1+tan(f*x+e))^(1/2))/f+5/4*arctanh((1+tan(f*x+e))^(1/2))/ 
f-1/2*(1+2^(1/2))^(1/2)*arctanh((3+2*2^(1/2)+(1+2^(1/2))*tan(f*x+e))/(14+1 
0*2^(1/2))^(1/2)/(1+tan(f*x+e))^(1/2))/f+3/4*cot(f*x+e)*(1+tan(f*x+e))^(1/ 
2)/f-1/2*cot(f*x+e)^2*(1+tan(f*x+e))^(1/2)/f
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.58 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {5 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )-\frac {4 \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}-\frac {4 \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i}}+3 \cot (e+f x) \sqrt {1+\tan (e+f x)}-2 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \] Input:

Integrate[Cot[e + f*x]^3/Sqrt[1 + Tan[e + f*x]],x]
 

Output:

(5*ArcTanh[Sqrt[1 + Tan[e + f*x]]] - (4*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqr 
t[1 - I]])/Sqrt[1 - I] - (4*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 + I]])/S 
qrt[1 + I] + 3*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] - 2*Cot[e + f*x]^2*Sqrt 
[1 + Tan[e + f*x]])/(4*f)
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.14, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4137, 27, 3042, 4019, 3042, 4018, 216, 220, 4117, 73, 220}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cot ^3(e+f x)}{\sqrt {\tan (e+f x)+1}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^3 \sqrt {\tan (e+f x)+1}}dx\)

\(\Big \downarrow \) 4052

\(\displaystyle -\frac {1}{2} \int \frac {\cot ^2(e+f x) \left (3 \tan ^2(e+f x)+4 \tan (e+f x)+3\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{4} \int \frac {\cot ^2(e+f x) \left (3 \tan ^2(e+f x)+4 \tan (e+f x)+3\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{4} \int \frac {3 \tan (e+f x)^2+4 \tan (e+f x)+3}{\tan (e+f x)^2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{4} \left (\int -\frac {\cot (e+f x) \left (5-3 \tan ^2(e+f x)\right )}{2 \sqrt {\tan (e+f x)+1}}dx+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}-\frac {1}{2} \int \frac {\cot (e+f x) \left (5-3 \tan ^2(e+f x)\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}-\frac {1}{2} \int \frac {5-3 \tan (e+f x)^2}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4137

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\int -\frac {8 \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx-5 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \int \frac {\tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx-5 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \int \frac {\tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4019

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \left (\frac {\int \frac {1-\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \left (\frac {\int \frac {1-\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4018

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \left (\frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3\right )^2}{\tan (e+f x)+1}-2 \left (7+5 \sqrt {2}\right )}d\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {\tan (e+f x)+1}}}{\sqrt {2} f}-\frac {\left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3\right )^2}{\tan (e+f x)+1}-2 \left (7-5 \sqrt {2}\right )}d\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {\tan (e+f x)+1}}}{\sqrt {2} f}\right )-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \left (\frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3\right )^2}{\tan (e+f x)+1}-2 \left (7+5 \sqrt {2}\right )}d\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {\tan (e+f x)+1}}}{\sqrt {2} f}-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}\right )-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}} f}\right )-5 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}} f}\right )-\frac {5 \int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}} f}\right )-\frac {10 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}} f}\right )+\frac {10 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}\right )+\frac {3 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

Input:

Int[Cot[e + f*x]^3/Sqrt[1 + Tan[e + f*x]],x]
 

Output:

-1/2*(Cot[e + f*x]^2*Sqrt[1 + Tan[e + f*x]])/f + (((10*ArcTanh[Sqrt[1 + Ta 
n[e + f*x]]])/f + 8*(-1/2*((3 - 2*Sqrt[2])*ArcTan[(3 - 2*Sqrt[2] + (1 - Sq 
rt[2])*Tan[e + f*x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/( 
Sqrt[-7 + 5*Sqrt[2]]*f) - ((3 + 2*Sqrt[2])*ArcTanh[(3 + 2*Sqrt[2] + (1 + S 
qrt[2])*Tan[e + f*x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Tan[e + f*x]])])/( 
2*Sqrt[7 + 5*Sqrt[2]]*f)))/2 + (3*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f)/ 
4
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 216

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

rule 220

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4018

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[-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 4019

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> With[{q = Rt[a^2 + b^2, 2]}, Simp[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] - Simp[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] && N 
eQ[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] && NiceSqrtQ[a^2 + b^2]
 

rule 4052

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[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] + Simp[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] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ 
erQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

rule 4117

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
 Simp[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 4132

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*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] + Simp[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)*Ta 
n[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] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

rule 4137

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) 
+ (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Sim 
p[1/(a^2 + b^2)   Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*C)*T 
an[e + f*x], x], x], x] + Simp[(A*b^2 + 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, 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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(892\) vs. \(2(167)=334\).

Time = 133.75 (sec) , antiderivative size = 893, normalized size of antiderivative = 4.15

method	result	size
default	\(\text {Expression too large to display}\)	\(893\)

Input:

int(cot(f*x+e)^3/(tan(f*x+e)+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8/f*(tan(f*x+e)+1)^(1/2)/((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(1+cos(f*x 
+e))^2)^(1/2)/(1+cos(f*x+e))/(cot(f*x+e)+cot(f*x+e)^2)^(1/2)*(2*(-2+2*2^(1 
/2))^(1/2)*(1+2^(1/2))^(1/2)*(-(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*sin(f 
*x+e)^2*2^(1/2)-2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2*sin(f*x+e)^2+2*sin(f*x+e 
)*cos(f*x+e)-1))^(1/2)*arctan(1/4*(-(4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))* 
cos(f*x+e)*(3*2^(1/2)-4)/(2*sin(f*x+e)^2*2^(1/2)-2*2^(1/2)*sin(f*x+e)*cos( 
f*x+e)-2*sin(f*x+e)^2+2*sin(f*x+e)*cos(f*x+e)-1))^(1/2)*(-4*sin(f*x+e)*cos 
(f*x+e)+1+tan(f*x+e))*(-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^(1/2)*(3*cot(f*x+e)*cos(f*x+e)-7*cos(f*x+e))-4*cot(f*x 
+e)*cos(f*x+e)+10*cos(f*x+e))+5*cos(f*x+e)*(1+2^(1/2))^(1/2)*(-2*2^(1/2)+3 
)*(cot(f*x+e)+cot(f*x+e)^2)^(1/2)*ln(2*cot(f*x+e)*((cos(f*x+e)+sin(f*x+e)) 
*cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+2*csc(f*x+e)*((cos(f*x+e)+sin(f*x+e))* 
cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-2*cot(f*x+e)-1)+4*(-(cos(f*x+e)+sin(f*x 
+e))*cos(f*x+e)/(2*sin(f*x+e)^2*2^(1/2)-2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2* 
sin(f*x+e)^2+2*sin(f*x+e)*cos(f*x+e)-1))^(1/2)*arctanh((-(cos(f*x+e)+sin(f 
*x+e))*cos(f*x+e)/(2*sin(f*x+e)^2*2^(1/2)-2*2^(1/2)*sin(f*x+e)*cos(f*x+e)- 
2*sin(f*x+e)^2+2*sin(f*x+e)*cos(f*x+e)-1))^(1/2)*2^(1/2)/(1+2^(1/2))^(1/2) 
)*(2^(1/2)*(cot(f*x+e)*cos(f*x+e)-2*cos(f*x+e))-cot(f*x+e)*cos(f*x+e)+3*co 
s(f*x+e))+2*(1+2^(1/2))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(1+cos(f 
*x+e))^2)^(1/2)*(cot(f*x+e)+cot(f*x+e)^2)^(1/2)*(2*cot(f*x+e)*csc(f*x+e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (166) = 332\).

Time = 0.09 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.93 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {4 \, \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} + 4 \, \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} + 5 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{2} - 5 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{2} + 2 \, {\left (3 \, \tan \left (f x + e\right ) - 2\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{8 \, f \tan \left (f x + e\right )^{2}} \] Input:

integrate(cot(f*x+e)^3/(1+tan(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

1/8*(4*sqrt(1/2)*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(1/2)*(f^3*sqr 
t(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1))* 
tan(f*x + e)^2 - 4*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)^2 - 4*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)^2 + 4*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)^2 + 5*log(sqrt(tan( 
f*x + e) + 1) + 1)*tan(f*x + e)^2 - 5*log(sqrt(tan(f*x + e) + 1) - 1)*tan( 
f*x + e)^2 + 2*(3*tan(f*x + e) - 2)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e 
)^2)
 

Sympy [F]

\[ \int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int \frac {\cot ^{3}{\left (e + f x \right )}}{\sqrt {\tan {\left (e + f x \right )} + 1}}\, dx \] Input:

integrate(cot(f*x+e)**3/(1+tan(f*x+e))**(1/2),x)
 

Output:

Integral(cot(e + f*x)**3/sqrt(tan(e + f*x) + 1), x)
 

Maxima [F]

\[ \int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{3}}{\sqrt {\tan \left (f x + e\right ) + 1}} \,d x } \] Input:

integrate(cot(f*x+e)^3/(1+tan(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate(cot(f*x + e)^3/sqrt(tan(f*x + e) + 1), x)
 

Giac [F]

\[ \int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{3}}{\sqrt {\tan \left (f x + e\right ) + 1}} \,d x } \] Input:

integrate(cot(f*x+e)^3/(1+tan(f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate(cot(f*x + e)^3/sqrt(tan(f*x + e) + 1), x)
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.68 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{4\,f}-\frac {\frac {5\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{4}-\frac {3\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{4}}{f-2\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,2{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,2{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \] Input:

int(cot(e + f*x)^3/(tan(e + f*x) + 1)^(1/2),x)
 

Output:

atan(f*((1/8 - 1i/8)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*2i)*((1/8 - 1i/8) 
/f^2)^(1/2)*2i - ((5*(tan(e + f*x) + 1)^(1/2))/4 - (3*(tan(e + f*x) + 1)^( 
3/2))/4)/(f - 2*f*(tan(e + f*x) + 1) + f*(tan(e + f*x) + 1)^2) - (atan((ta 
n(e + f*x) + 1)^(1/2)*1i)*5i)/(4*f) + atan(f*((1/8 + 1i/8)/f^2)^(1/2)*(tan 
(e + f*x) + 1)^(1/2)*2i)*((1/8 + 1i/8)/f^2)^(1/2)*2i
 

Reduce [F]

\[ \int \frac {\cot ^3(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int \frac {\sqrt {\tan \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{3}}{\tan \left (f x +e \right )+1}d x \] Input:

int(cot(f*x+e)^3/(1+tan(f*x+e))^(1/2),x)
 

Output:

int((sqrt(tan(e + f*x) + 1)*cot(e + f*x)**3)/(tan(e + f*x) + 1),x)