\(\int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx\) [382]

Optimal result

Integrand size = 21, antiderivative size = 221 \[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \arctan \left (\frac {4-3 \sqrt {2}+\left (2-\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {-7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {9 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{4 f}-\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {4+3 \sqrt {2}+\left (2+\sqrt {2}\right ) \tan (e+f x)}{2 \sqrt {7+5 \sqrt {2}} \sqrt {1+\tan (e+f x)}}\right )}{f}-\frac {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{4 f}-\frac {\cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{2 f} \] Output:

-1/2*(-2+2*2^(1/2))^(1/2)*arctan(1/2*(4-3*2^(1/2)+(2-2^(1/2))*tan(f*x+e))/ 
(-7+5*2^(1/2))^(1/2)/(1+tan(f*x+e))^(1/2))/f+9/4*arctanh((1+tan(f*x+e))^(1 
/2))/f-1/2*(2+2*2^(1/2))^(1/2)*arctanh(1/2*(4+3*2^(1/2)+(2+2^(1/2))*tan(f* 
x+e))/(7+5*2^(1/2))^(1/2)/(1+tan(f*x+e))^(1/2))/f-1/4*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 = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.56 \[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {-9 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+4 \sqrt {1-i} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+4 \sqrt {1+i} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+\cot (e+f x) \sqrt {1+\tan (e+f x)}+2 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{4 f} \] Input:

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

Output:

-1/4*(-9*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + 4*Sqrt[1 - I]*ArcTanh[Sqrt[1 + 
Tan[e + f*x]]/Sqrt[1 - I]] + 4*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Tan[e + f*x]]/ 
Sqrt[1 + I]] + Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 2*Cot[e + f*x]^2*Sqrt 
[1 + Tan[e + f*x]])/f
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 4051, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4019, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4051

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{4} \left (-\int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+8 \tan (e+f x)+9\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+8 \tan (e+f x)+9\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {\tan (e+f x)^2+8 \tan (e+f x)+9}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4136

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\int \frac {8 (1-\tan (e+f x))}{\sqrt {\tan (e+f x)+1}}dx-9 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \int \frac {1-\tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx-9 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \int \frac {1-\tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4019

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-8 \left (\frac {\int \frac {\left (2-\sqrt {2}\right ) \tan (e+f x)+\sqrt {2}}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-8 \left (\frac {\int \frac {\left (2-\sqrt {2}\right ) \tan (e+f x)+\sqrt {2}}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \left (\frac {\int \frac {\left (2-\sqrt {2}\right ) \tan (e+f x)+\sqrt {2}}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4018

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-8 \left (-\frac {\sqrt {2} \left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4\right )^2}{\tan (e+f x)+1}-4 \left (7-5 \sqrt {2}\right )}d\left (-\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{\sqrt {\tan (e+f x)+1}}\right )}{f}-\frac {\sqrt {2} \left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4\right )^2}{\tan (e+f x)+1}-4 \left (7+5 \sqrt {2}\right )}d\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{\sqrt {\tan (e+f x)+1}}}{f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}-\frac {\sqrt {2} \left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4\right )^2}{\tan (e+f x)+1}-4 \left (7+5 \sqrt {2}\right )}d\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{\sqrt {\tan (e+f x)+1}}}{f}\right )-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-9 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )} f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {9 \int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )} f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-\frac {18 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )} f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {18 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}-8 \left (\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (2-\sqrt {2}\right ) \tan (e+f x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (5 \sqrt {2}-7\right )} f}+\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (2+\sqrt {2}\right ) \tan (e+f x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\tan (e+f x)+1}}\right )}{\sqrt {2 \left (7+5 \sqrt {2}\right )} f}\right )\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\)

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 + (((18*ArcTanh[Sqrt[1 + Ta 
n[e + f*x]]])/f - 8*(((3 - 2*Sqrt[2])*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2] 
)*Tan[e + f*x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/(Sqrt[2* 
(-7 + 5*Sqrt[2])]*f) + ((3 + 2*Sqrt[2])*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt 
[2])*Tan[e + f*x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Tan[e + f*x]])])/(Sqrt[ 
2*(7 + 5*Sqrt[2])]*f)))/2 - (Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f)/4
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 4051

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

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 4136

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] :> Simp[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] + Simp[ 
(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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs. \(2(173)=346\).

Time = 122.77 (sec) , antiderivative size = 892, normalized size of antiderivative = 4.04

method	result	size
default	\(\text {Expression too large to display}\)	\(892\)

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)*(4*(-2+2*2^(1 
/2))^(1/2)*(1+2^(1/2))^(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))*(-(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)*(2*cot(f*x+e)*cos(f*x+e)-5*cos(f*x+e))-3*cot(f*x 
+e)*cos(f*x+e)+7*cos(f*x+e))+9*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))*c 
os(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*s 
in(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)+3*cos(f*x+e))+2*cot(f*x+e)*cos(f*x+e)-4* 
cos(f*x+e))+2*cot(f*x+e)*csc(f*x+e)*(1+2^(1/2))^(1/2)*(2*((-cos(f*x+e)-1)* 
sin(f*x+e)-2*cos(f*x+e)*(1+cos(f*x+e)))*2^(1/2)+3*(1+cos(f*x+e))*(sin(f...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.53 \[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {4 \, f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} + 4 \, f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 4 \, f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{2} - 9 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{2} + 9 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{2} + 2 \, {\left (\tan \left (f x + e\right ) + 2\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{8 \, f \tan \left (f x + e\right )^{2}} \] Input:

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

Output:

-1/8*(4*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(f*sqrt((f^2*sqrt(-1/f^4) + 
1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 - 4*f*sqrt((f^2*sqrt(-1/f 
^4) + 1)/f^2)*log(-f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) 
+ 1))*tan(f*x + e)^2 + 4*f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(f*sqrt(-( 
f^2*sqrt(-1/f^4) - 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 - 4*f* 
sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2)*log(-f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2) 
 + sqrt(tan(f*x + e) + 1))*tan(f*x + e)^2 - 9*log(sqrt(tan(f*x + e) + 1) + 
 1)*tan(f*x + e)^2 + 9*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e)^2 + 2* 
(tan(f*x + e) + 2)*sqrt(tan(f*x + e) + 1))/(f*tan(f*x + e)^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.67 \[ \int \cot ^3(e+f x) \sqrt {1+\tan (e+f x)} \, dx=-\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{4\,f}-\frac {\frac {\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{4}+\frac {{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{4}}{f-2\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2}-\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1-\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}-\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,\left (1+1{}\mathrm {i}\right )\right )\,\sqrt {\frac {\frac {1}{4}+\frac {1}{4}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \] Input:

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

Output:

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 - ((tan(e + f*x) + 1)^(1/2)/4 + (tan(e + f*x) + 1)^(3 
/2)/4)/(f - 2*f*(tan(e + f*x) + 1) + f*(tan(e + f*x) + 1)^2) - atan(f*((1/ 
4 - 1i/4)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*(1 - 1i))*((1/4 - 1i/4)/f^2) 
^(1/2)*2i - (atan((tan(e + f*x) + 1)^(1/2)*1i)*9i)/(4*f)
 

Reduce [F]

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