\(\int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [517]

Optimal result

Integrand size = 23, antiderivative size = 149 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=-\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {i (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {i (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {a \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \] Output:

-3*a^(1/2)*b*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d+I*(a-I*b)^(3/2)*arc 
tanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-I*(a+I*b)^(3/2)*arctanh((a+b* 
tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d-a*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.25 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {-3 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+\sqrt {a-i b} (i a+b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-i a \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\sqrt {a+i b} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )-a \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d} \] Input:

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

Output:

(-3*Sqrt[a]*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] + Sqrt[a - I*b]*(I 
*a + b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - I*a*Sqrt[a + I*b 
]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Sqrt[a + I*b]*b*ArcTan 
h[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] - a*Cot[c + d*x]*Sqrt[a + b*Tan[ 
c + d*x]])/d
 

Rubi [A] (warning: unable to verify)

Time = 1.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {3042, 4050, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4050

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (\frac {6 a \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{d}-2 \left (\frac {(a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}+\frac {1}{2} \left (-\frac {6 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}-2 \left (\frac {(a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}+\frac {(a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )\right )\)

Input:

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

Output:

(-2*(((a - I*b)^(3/2)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/d + ((a + I*b)^( 
3/2)*ArcTan[Tan[c + d*x]/Sqrt[a + I*b]])/d) - (6*Sqrt[a]*b*ArcTanh[Sqrt[a 
+ b*Tan[c + d*x]]/Sqrt[a]])/d)/2 - (a*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x] 
])/d
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 3042

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

rule 4020

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f)   Subst[Int[(a + (b/d)*x)^m/(d^2 + 
c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
 

rule 4022

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

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 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 [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 832 vs. \(2 (117) = 234\).

Time = 0.14 (sec) , antiderivative size = 1683, normalized size of antiderivative = 11.30 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*(d*sqrt(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^ 
4))/d^2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) + (a*d^ 
3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^2*b^2 - b^4)*d)*sqrt(-(a 
^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2))*tan(d*x 
 + c) - d*sqrt(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d 
^4))/d^2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) - (a*d 
^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^2*b^2 - b^4)*d)*sqrt(-( 
a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2))*tan(d* 
x + c) - d*sqrt(-(a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/ 
d^4))/d^2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) + (a* 
d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^2*b^2 - b^4)*d)*sqrt(- 
(a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2))*tan(d 
*x + c) + d*sqrt(-(a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6) 
/d^4))/d^2)*log(-(3*a^4*b + 2*a^2*b^3 - b^5)*sqrt(b*tan(d*x + c) + a) - (a 
*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^2*b^2 - b^4)*d)*sqrt( 
-(a^3 - 3*a*b^2 - d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4))/d^2))*tan( 
d*x + c) - 3*sqrt(a)*b*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sq 
rt(a) + 2*a)/tan(d*x + c))*tan(d*x + c) + 2*sqrt(b*tan(d*x + c) + a)*a)/(d 
*tan(d*x + c)), 1/2*(6*sqrt(-a)*b*arctan(sqrt(-a)/sqrt(b*tan(d*x + c) + a) 
)*tan(d*x + c) - d*sqrt(-(a^3 - 3*a*b^2 + d^2*sqrt(-(9*a^4*b^2 - 6*a^2*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^(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 = 1.72 (sec) , antiderivative size = 3623, normalized size of antiderivative = 24.32 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

(a*b*(a + b*tan(c + d*x))^(1/2))/(a*d - d*(a + b*tan(c + d*x))) - atan(((( 
(((16*(40*a*b^11*d^4 + 40*a^3*b^9*d^4))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^ 
8*d^4)*(a + b*tan(c + d*x))^(1/2)*((3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4* 
d^2))^(1/2))/d^4)*((3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) + (1 
6*(a + b*tan(c + d*x))^(1/2)*(44*a*b^12*d^2 + 92*a^3*b^10*d^2 - 20*a^5*b^8 
*d^2))/d^4)*((3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) - (16*(50* 
a^2*b^13*d^2 + 22*a^4*b^11*d^2 - 28*a^6*b^9*d^2))/d^5)*((3*a*b^2 + a^2*b*3 
i - a^3 - b^3*1i)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(2*b^16 
- a^2*b^14 + 66*a^4*b^12 - a^6*b^10 + 2*a^8*b^8))/d^4)*((3*a*b^2 + a^2*b*3 
i - a^3 - b^3*1i)/(4*d^2))^(1/2)*1i - (((((16*(40*a*b^11*d^4 + 40*a^3*b^9* 
d^4))/d^5 + (16*(32*b^10*d^4 + 48*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)* 
((3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2))/d^4)*((3*a*b^2 + a^2* 
b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(44*a 
*b^12*d^2 + 92*a^3*b^10*d^2 - 20*a^5*b^8*d^2))/d^4)*((3*a*b^2 + a^2*b*3i - 
 a^3 - b^3*1i)/(4*d^2))^(1/2) - (16*(50*a^2*b^13*d^2 + 22*a^4*b^11*d^2 - 2 
8*a^6*b^9*d^2))/d^5)*((3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2) + 
 (16*(a + b*tan(c + d*x))^(1/2)*(2*b^16 - a^2*b^14 + 66*a^4*b^12 - a^6*b^1 
0 + 2*a^8*b^8))/d^4)*((3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)/(4*d^2))^(1/2)*1 
i)/((32*(3*a*b^17 + 3*a^3*b^15 + 3*a^7*b^11 + 3*a^9*b^9))/d^5 + (((((16*(4 
0*a*b^11*d^4 + 40*a^3*b^9*d^4))/d^5 - (16*(32*b^10*d^4 + 48*a^2*b^8*d^4...
 

Reduce [F]

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

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

Output:

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