\(\int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx\) [399]

Optimal result

Integrand size = 21, antiderivative size = 238 \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\sqrt {-1+\sqrt {2}} \arctan \left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {25 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{8 f}-\frac {\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\tan (e+f x)}}\right )}{f}+\frac {7 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{8 f}-\frac {7 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}}{12 f}-\frac {\cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{3 f} \] Output:

-(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+25/8*arctanh((1+tan(f*x+e))^(1/2))/f-( 
1+2^(1/2))^(1/2)*arctanh((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+7/8*cot(f*x+e)*(1+tan(f*x+e))^(1/2)/f-7/ 
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
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.62 \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {-75 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+\frac {48 \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}+\frac {48 \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i}}-21 \cot (e+f x) \sqrt {1+\tan (e+f x)}+14 \cot ^2(e+f x) \sqrt {1+\tan (e+f x)}+8 \cot ^3(e+f x) \sqrt {1+\tan (e+f x)}}{24 f} \] Input:

Integrate[Cot[e + f*x]^4*(1 + Tan[e + f*x])^(3/2),x]
 

Output:

-1/24*(-75*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (48*ArcTanh[Sqrt[1 + Tan[e + 
f*x]]/Sqrt[1 - I]])/Sqrt[1 - I] + (48*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[ 
1 + I]])/Sqrt[1 + I] - 21*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]] + 14*Cot[e + 
 f*x]^2*Sqrt[1 + Tan[e + f*x]] + 8*Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/ 
f
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.17, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4050, 27, 3042, 4133, 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.

\(\displaystyle \int (\tan (e+f x)+1)^{3/2} \cot ^4(e+f x) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4050

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \int \frac {\cot ^3(e+f x) \left (7-5 \tan ^2(e+f x)\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \int \frac {7-5 \tan (e+f x)^2}{\tan (e+f x)^3 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4133

\(\displaystyle \frac {1}{6} \left (-\frac {1}{2} \int \frac {3 \cot ^2(e+f x) \left (7 \tan ^2(e+f x)+16 \tan (e+f x)+7\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \int \frac {\cot ^2(e+f x) \left (7 \tan ^2(e+f x)+16 \tan (e+f x)+7\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \int \frac {7 \tan (e+f x)^2+16 \tan (e+f x)+7}{\tan (e+f x)^2 \sqrt {\tan (e+f x)+1}}dx-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (-\int -\frac {\cot (e+f x) \left (25-7 \tan ^2(e+f x)\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {\cot (e+f x) \left (25-7 \tan ^2(e+f x)\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {25-7 \tan (e+f x)^2}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4137

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (\int -\frac {32 \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx+25 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (25 \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx-32 \int \frac {\tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (25 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-32 \int \frac {\tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4019

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (25 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-32 \left (\frac {\int \frac {1-\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (25 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-32 \left (\frac {\int \frac {1-\left (1-\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}-\frac {\int \frac {1-\left (1+\sqrt {2}\right ) \tan (e+f x)}{\sqrt {\tan (e+f x)+1}}dx}{2 \sqrt {2}}\right )\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4018

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (25 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-32 \left (\frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3\right )^2}{\tan (e+f x)+1}-2 \left (7+5 \sqrt {2}\right )}d\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {\tan (e+f x)+1}}}{\sqrt {2} f}-\frac {\left (3-2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3\right )^2}{\tan (e+f x)+1}-2 \left (7-5 \sqrt {2}\right )}d\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {\tan (e+f x)+1}}}{\sqrt {2} f}\right )\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (25 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-32 \left (\frac {\left (3+2 \sqrt {2}\right ) \int \frac {1}{\frac {\left (\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3\right )^2}{\tan (e+f x)+1}-2 \left (7+5 \sqrt {2}\right )}d\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {\tan (e+f x)+1}}}{\sqrt {2} f}-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}\right )\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (25 \int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-32 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}} f}\right )\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 4117

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (\frac {25 \int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}-32 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}} f}\right )\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (\frac {50 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}-32 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}} f}\right )\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {1}{6} \left (-\frac {3}{4} \left (\frac {1}{2} \left (-32 \left (-\frac {\left (3-2 \sqrt {2}\right ) \arctan \left (\frac {\left (1-\sqrt {2}\right ) \tan (e+f x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {5 \sqrt {2}-7} f}-\frac {\left (3+2 \sqrt {2}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {2}\right ) \tan (e+f x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\tan (e+f x)+1}}\right )}{2 \sqrt {7+5 \sqrt {2}} f}\right )-\frac {50 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\right )-\frac {7 \sqrt {\tan (e+f x)+1} \cot ^2(e+f x)}{2 f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot ^3(e+f x)}{3 f}\)

Input:

Int[Cot[e + f*x]^4*(1 + Tan[e + f*x])^(3/2),x]
 

Output:

-1/3*(Cot[e + f*x]^3*Sqrt[1 + Tan[e + f*x]])/f + ((-7*Cot[e + f*x]^2*Sqrt[ 
1 + Tan[e + f*x]])/(2*f) - (3*(((-50*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f - 
32*(-1/2*((3 - 2*Sqrt[2])*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[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 + Sqrt[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 - (7*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f))/4)/6
 

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 4050

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 
 1)*((c + d*Tan[e + f*x])^(n - 1)/(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 - 2)*Simp[a*c^2*(m + 1) + a*d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2 
*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*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] && LtQ[1, n, 2] && IntegerQ[ 
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 4133

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*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] + 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*Sim 
p[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 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. \(926\) vs. \(2(190)=380\).

Time = 126.22 (sec) , antiderivative size = 927, normalized size of antiderivative = 3.89

method	result	size
default	\(\text {Expression too large to display}\)	\(927\)

Input:

int(cot(f*x+e)^4*(tan(f*x+e)+1)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/48/f*cot(f*x+e)*csc(f*x+e)^2*(24*(-2+2*2^(1/2))^(1/2)*(1+2^(1/2))^(1/2)* 
((cos(f*x+e)-1)*(7*sin(f*x+e)-3*cos(f*x+e))*2^(1/2)+10*(1-cos(f*x+e))*sin( 
f*x+e)+4*cos(f*x+e)*(cos(f*x+e)-1))*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*2^(1/2)*sin(f*x+e)*cos(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)/(2*cos 
(f*x+e)^2-1)*(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))*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2) 
*sin(f*x+e)*cos(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*si 
n(f*x+e)^2+1))^(1/2)+75*sin(f*x+e)*(1+2^(1/2))^(1/2)*(2*2^(1/2)*(1-cos(f*x 
+e))-3+3*cos(f*x+e))*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)*(cot(f*x+e)+cot(f*x+e)^2)^(1/2)+48*( 
(2*(1-cos(f*x+e))*sin(f*x+e)+cos(f*x+e)*(cos(f*x+e)-1))*2^(1/2)+3*(cos(f*x 
+e)-1)*sin(f*x+e)-cos(f*x+e)*(cos(f*x+e)-1))*arctanh(((cos(f*x+e)+sin(f*x+ 
e))*cos(f*x+e)/(2*2^(1/2)*sin(f*x+e)*cos(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*s 
in(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)*sin(f*x+e)*cos(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)*(2*(-21+29*cos(f*x+e)^2+14*sin(f*x+e)*cos(f*x+e))*2^(1/2)+63-8 
7*cos(f*x+e)^2-42*sin(f*x+e)*cos(f*x+e))*(cot(f*x+e)+cot(f*x+e)^2)^(1/2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (189) = 378\).

Time = 0.09 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.82 \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\frac {24 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, \sqrt {2} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} - 24 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} + 24 \, \sqrt {2} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {2} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right )^{3} + 75 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right )^{3} - 75 \, \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right )^{3} + 2 \, {\left (21 \, \tan \left (f x + e\right )^{2} - 14 \, \tan \left (f x + e\right ) - 8\right )} \sqrt {\tan \left (f x + e\right ) + 1}}{48 \, f \tan \left (f x + e\right )^{3}} \] Input:

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

1/48*(24*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(2)*(f^3*sqrt( 
-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))* 
tan(f*x + e)^3 - 24*sqrt(2)*f*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2)*log(-sqrt(2 
)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) + 1)/f^2) + 2*sqrt(tan(f*x 
 + e) + 1))*tan(f*x + e)^3 - 24*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2 
)*log(sqrt(2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) - 1)/f^2) + 2 
*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 + 24*sqrt(2)*f*sqrt(-(f^2*sqrt(-1/ 
f^4) - 1)/f^2)*log(-sqrt(2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) 
 - 1)/f^2) + 2*sqrt(tan(f*x + e) + 1))*tan(f*x + e)^3 + 75*log(sqrt(tan(f* 
x + e) + 1) + 1)*tan(f*x + e)^3 - 75*log(sqrt(tan(f*x + e) + 1) - 1)*tan(f 
*x + e)^3 + 2*(21*tan(f*x + e)^2 - 14*tan(f*x + e) - 8)*sqrt(tan(f*x + e) 
+ 1))/(f*tan(f*x + e)^3)
 

Sympy [F]

\[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \left (\tan {\left (e + f x \right )} + 1\right )^{\frac {3}{2}} \cot ^{4}{\left (e + f x \right )}\, dx \] Input:

integrate(cot(f*x+e)**4*(1+tan(f*x+e))**(3/2),x)
 

Output:

Integral((tan(e + f*x) + 1)**(3/2)*cot(e + f*x)**4, x)
 

Maxima [F]

\[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int { {\left (\tan \left (f x + e\right ) + 1\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{4} \,d x } \] Input:

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

integrate((tan(f*x + e) + 1)^(3/2)*cot(f*x + e)^4, x)
 

Giac [F(-2)]

Exception generated. \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.73 \[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=-\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,25{}\mathrm {i}}{8\,f}-\frac {\frac {9\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{8}-\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{3/2}}{3}+\frac {7\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^{5/2}}{8}}{f-3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )+3\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^2-f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}^3}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}-\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (f\,\sqrt {\frac {\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,\sqrt {\frac {\frac {1}{2}+\frac {1}{2}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \] Input:

int(cot(e + f*x)^4*(tan(e + f*x) + 1)^(3/2),x)
 

Output:

atan(f*((1/2 - 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1)^(1/2)*1i)*((1/2 - 1i/2) 
/f^2)^(1/2)*2i - ((9*(tan(e + f*x) + 1)^(1/2))/8 - (7*(tan(e + f*x) + 1)^( 
3/2))/3 + (7*(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((tan(e + f*x) + 1 
)^(1/2)*1i)*25i)/(8*f) + atan(f*((1/2 + 1i/2)/f^2)^(1/2)*(tan(e + f*x) + 1 
)^(1/2)*1i)*((1/2 + 1i/2)/f^2)^(1/2)*2i
 

Reduce [F]

\[ \int \cot ^4(e+f x) (1+\tan (e+f x))^{3/2} \, dx=\int \sqrt {\tan \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{4} \tan \left (f x +e \right )d x +\int \sqrt {\tan \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{4}d x \] Input:

int(cot(f*x+e)^4*(1+tan(f*x+e))^(3/2),x)
 

Output:

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