\(\int \frac {\tan ^3(d+e x)}{(a+b \cot (d+e x)+c \cot ^2(d+e x))^{3/2}} \, dx\) [15]

Optimal result

Integrand size = 33, antiderivative size = 1008 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{a^{3/2} e}+\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{7/2} e}-\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {2 \left (b^2-2 a c+b c \cot (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^3 \left (b^2-4 a c\right ) e}-\frac {2 \left (b^2-2 a c+b c \cot (d+e x)\right ) \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a^2 \left (b^2-4 a c\right ) e} \] Output:

-arctanh(1/2*(2*a+b*cot(e*x+d))/a^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1 
/2))/a^(3/2)/e+3/8*(-4*a*c+5*b^2)*arctanh(1/2*(2*a+b*cot(e*x+d))/a^(1/2)/( 
a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))/a^(7/2)/e-1/2*(2*a-2*c-(a^2-2*a*c+b^ 
2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2+(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1 
/2)*arctanh(1/2*(b^2-(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2))-b*(2*a-2*c-(a^2 
-2*a*c+b^2+c^2)^(1/2))*cot(e*x+d))*2^(1/2)/(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1 
/2))^(1/2)/(a^2-b^2-2*a*c+c^2+(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a+b* 
cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e+1/2* 
(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2* 
a*c+b^2+c^2)^(1/2))^(1/2)*arctanh(1/2*(b^2-(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^ 
(1/2))-b*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))*cot(e*x+d))*2^(1/2)/(2*a-2*c+ 
(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c 
^2)^(1/2))^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2-2*a*c 
+b^2+c^2)^(3/2)/e+2*(b^2-2*a*c+b*c*cot(e*x+d))/a/(-4*a*c+b^2)/e/(a+b*cot(e 
*x+d)+c*cot(e*x+d)^2)^(1/2)-2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(e*x+d))/(b^ 
2+(a-c)^2)/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)-1/4*b*(-52 
*a*c+15*b^2)*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)/a^3/(-4*a*c+ 
b^2)/e-2*(b^2-2*a*c+b*c*cot(e*x+d))*tan(e*x+d)^2/a/(-4*a*c+b^2)/e/(a+b*cot 
(e*x+d)+c*cot(e*x+d)^2)^(1/2)+1/2*(-12*a*c+5*b^2)*(a+b*cot(e*x+d)+c*cot(e* 
x+d)^2)^(1/2)*tan(e*x+d)^2/a^2/(-4*a*c+b^2)/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.36 (sec) , antiderivative size = 1401, normalized size of antiderivative = 1.39 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[Tan[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]
 

Output:

-((Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*((2*((-4*Sqrt[ 
a - I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) + (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTan 
[(I*b + 2*c - ((-2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b 
*Tan[d + e*x] + a*Tan[d + e*x]^2])])/(-4*a + (4*I)*b + 4*c) - (4*Sqrt[a + 
I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) - (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTan[((- 
I)*b + 2*c - ((2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + b*T 
an[d + e*x] + a*Tan[d + e*x]^2])])/(-4*a - (4*I)*b + 4*c)))/((b^2 + (a - c 
)^2)*(b^2 - 4*a*c)) - (2*Tan[d + e*x]^3*(-b^2 + 2*a*c - a*b*Tan[d + e*x])) 
/(c*(b^2 - 4*a*c)*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) - (2*(b^3 + 
 a*b*(a - 3*c) + a*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x]))/((b^2 + (a - c)^2) 
*(b^2 - 4*a*c)*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) + (4*(b^2 - 4* 
a*c)*(a^2/((b^2 - 4*a*c)*((a^2*b^2)/(b^2 - 4*a*c)^2 - (4*a^3*c)/(b^2 - 4*a 
*c)^2)))^(3/2)*(-((a*b)/(b^2 - 4*a*c)) - (2*a^2*Tan[d + e*x])/(b^2 - 4*a*c 
))*(-((a*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2))/(b^2 - 4*a*c)))^(3/2))/( 
a^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)^(3/2)*Sqrt[1 - (-((a*b)/(b^2 - 
 4*a*c)) - (2*a^2*Tan[d + e*x])/(b^2 - 4*a*c))^2/((a^2*b^2)/(b^2 - 4*a*c)^ 
2 - (4*a^3*c)/(b^2 - 4*a*c)^2)]) - (2*(b*Tan[d + e*x]^2*Sqrt[c + b*Tan[d + 
 e*x] + a*Tan[d + e*x]^2] + (((-6*a^2*b^2*c + 24*a^3*c^2)*ArcTanh[(b + 2*a 
*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])])/( 
4*a^(5/2)) + ((6*a^2*b*c - 12*a^3*c*Tan[d + e*x])*Sqrt[c + b*Tan[d + e*...
 

Rubi [A] (verified)

Time = 3.85 (sec) , antiderivative size = 985, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 4184, 7276, 2009}

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.

\(\displaystyle \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cot (d+e x)^3 \left (a+b \cot (d+e x)+c \cot (d+e x)^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\tan ^3(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {\int \left (\frac {\tan ^3(d+e x)}{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}-\frac {\tan (d+e x)}{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}+\frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}\right )d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {\left (5 b^2-12 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \tan ^2(d+e x)}{2 a^2 \left (b^2-4 a c\right )}+\frac {2 \left (b^2+c \cot (d+e x) b-2 a c\right ) \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \tan (d+e x)}{4 a^3 \left (b^2-4 a c\right )}-\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{8 a^{7/2}}+\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{a^{3/2}}+\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}-\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}-\frac {2 \left (b^2+c \cot (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}}{e}\)

Input:

Int[Tan[d + e*x]^3/(a + b*Cot[d + e*x] + c*Cot[d + e*x]^2)^(3/2),x]
 

Output:

-((ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*C 
ot[d + e*x]^2])]/a^(3/2) - (3*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x 
])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(7/2)) + 
 (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + 
 c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - 
c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c 
 + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + 
c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2] 
]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a 
*c + c^2)^(3/2)) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a 
^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b 
^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt 
[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^ 
2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + 
b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2 
]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) - (2*(b^2 - 2*a*c + b*c*Cot[d + e*x]))/ 
(a*(b^2 - 4*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 
 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a 
*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (b*(15*b^2 - 52*a*c)*Sq 
rt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(4*a^3*(b^2 - 4...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4184

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( 
f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] 
 :> Simp[-f/e   Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), 
x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ 
n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F(-1)]

Timed out.

hanged

Input:

int(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x)
 

Output:

int(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20316 vs. \(2 (923) = 1846\).

Time = 7.20 (sec) , antiderivative size = 40633, normalized size of antiderivative = 40.31 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= 
"fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate(tan(e*x+d)**3/(a+b*cot(e*x+d)+c*cot(e*x+d)**2)**(3/2),x)
 

Output:

Integral(tan(d + e*x)**3/(a + b*cot(d + e*x) + c*cot(d + e*x)**2)**(3/2), 
x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= 
"maxima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x, algorithm= 
"giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument 
Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Hanged} \] Input:

int(tan(d + e*x)^3/(a + b*cot(d + e*x) + c*cot(d + e*x)^2)^(3/2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

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

int(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x)
 

Output:

int((sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*tan(d + e*x)**3)/(cot(d 
+ e*x)**4*c**2 + 2*cot(d + e*x)**3*b*c + 2*cot(d + e*x)**2*a*c + cot(d + e 
*x)**2*b**2 + 2*cot(d + e*x)*a*b + a**2),x)