3.6.18 \(\int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx\) [518]

3.6.18.1 Optimal result

Integrand size = 23, antiderivative size = 189 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {\left (8 a^2-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 \sqrt {a} d}-\frac {(a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 d}-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d} \]

output

-(a-I*b)^(3/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-(a+I*b)^(3/ 
2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/d+1/4*(8*a^2-3*b^2)*arcta 
nh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)-5/4*b*cot(d*x+c)*(a+b*tan(d*x 
+c))^(1/2)/d-1/2*a*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/d
 

3.6.18.2 Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.89 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\frac {\left (8 a^2-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )-\sqrt {a} \left (4 (a-i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+4 (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\cot (c+d x) (5 b+2 a \cot (c+d x)) \sqrt {a+b \tan (c+d x)}\right )}{4 \sqrt {a} d} \]

input

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

output

((8*a^2 - 3*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]] - Sqrt[a]*(4*(a 
 - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + 4*(a + I*b 
)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + Cot[c + d*x]*(5* 
b + 2*a*Cot[c + d*x])*Sqrt[a + b*Tan[c + d*x]]))/(4*Sqrt[a]*d)
 

3.6.18.3 Rubi [A] (warning: unable to verify)

Time = 1.61 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4050, 27, 3042, 4132, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4050

\(\displaystyle -\frac {1}{2} \int -\frac {\cot ^2(c+d x) \left (-3 a b \tan ^2(c+d x)-4 \left (a^2-b^2\right ) \tan (c+d x)+5 a b\right )}{2 \sqrt {a+b \tan (c+d x)}}dx-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {\cot ^2(c+d x) \left (-3 a b \tan ^2(c+d x)-4 \left (a^2-b^2\right ) \tan (c+d x)+5 a b\right )}{\sqrt {a+b \tan (c+d x)}}dx-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \int \frac {-3 a b \tan (c+d x)^2-4 \left (a^2-b^2\right ) \tan (c+d x)+5 a b}{\tan (c+d x)^2 \sqrt {a+b \tan (c+d x)}}dx-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4132

\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {\cot (c+d x) \left (16 b \tan (c+d x) a^2+5 b^2 \tan ^2(c+d x) a+\left (8 a^2-3 b^2\right ) a\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a}-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {\cot (c+d x) \left (16 b \tan (c+d x) a^2+5 b^2 \tan ^2(c+d x) a+\left (8 a^2-3 b^2\right ) a\right )}{\sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {16 b \tan (c+d x) a^2+5 b^2 \tan (c+d x)^2 a+\left (8 a^2-3 b^2\right ) a}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{2 a}-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}\)

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (8 a^2-3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {1}{2} i a (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i a (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (8 a^2-3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {1}{2} i a (a-i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} i a (a+i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx\right )}{2 a}\right )\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (8 a^2-3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (-\frac {a (a-i b)^2 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {a (a+i b)^2 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (8 a^2-3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {a (a-i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {a (a+i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (8 a^2-3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {i a (a-i b)^2 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}-\frac {i a (a+i b)^2 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {a \left (8 a^2-3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx+8 \left (\frac {i a (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i a (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {\frac {a \left (8 a^2-3 b^2\right ) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}+8 \left (\frac {i a (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i a (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {\frac {2 a \left (8 a^2-3 b^2\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}+8 \left (\frac {i a (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i a (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 d}+\frac {1}{4} \left (-\frac {5 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{d}-\frac {-\frac {2 \sqrt {a} \left (8 a^2-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+8 \left (\frac {i a (a-i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d}-\frac {i a (a+i b)^{3/2} \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d}\right )}{2 a}\right )\)

input

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

output

-1/2*(a*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/d + (-1/2*(8*((I*a*(a - I 
*b)^(3/2)*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/d - (I*a*(a + I*b)^(3/2)*Arc 
Tan[Tan[c + d*x]/Sqrt[a + I*b]])/d) - (2*Sqrt[a]*(8*a^2 - 3*b^2)*ArcTanh[S 
qrt[a + b*Tan[c + d*x]]/Sqrt[a]])/d)/a - (5*b*Cot[c + d*x]*Sqrt[a + b*Tan[ 
c + d*x]])/d)/4
 

3.6.18.3.1 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 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]
 

3.6.18.4 Maple [F(-1)]

Timed out.

hanged

input

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

output

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

3.6.18.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 853 vs. \(2 (153) = 306\).

Time = 0.33 (sec) , antiderivative size = 1723, normalized size of antiderivative = 9.12 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]

input

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

output

[-1/8*(4*a*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 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) + (d^3 
*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^3 - a*b^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))*tan(d*x + c 
)^2 - 4*a*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 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) - (d^3* 
sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) - (3*a^3 - a*b^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))*tan(d*x + c) 
^2 - 4*a*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 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) + (d^3*s 
qrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^3 - a*b^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))*tan(d*x + c)^ 
2 + 4*a*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 + 2*a^2*b^2 - b^4)*sqrt(b*tan(d*x + c) + a) - (d^3*sq 
rt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/d^4) + (3*a^3 - a*b^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))*tan(d*x + c)^2 
 + (8*a^2 - 3*b^2)*sqrt(a)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a 
)*sqrt(a) + 2*a)/tan(d*x + c))*tan(d*x + c)^2 + 2*(5*a*b*tan(d*x + c) + 2* 
a^2)*sqrt(b*tan(d*x + c) + a))/(a*d*tan(d*x + c)^2), -1/4*(2*a*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...
 

3.6.18.6 Sympy [F]

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

input

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

output

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

3.6.18.7 Maxima [F]

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

input

integrate(cot(d*x+c)^3*(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)^3, x)
 

3.6.18.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.6.18.9 Mupad [B] (verification not implemented)

Time = 5.29 (sec) , antiderivative size = 4660, normalized size of antiderivative = 24.66 \[ \int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} \, dx=\text {Too large to display} \]

input

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

output

- atan(((((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8*d^4)/(2*d^5) + 
((512*b^10*d^4 + 768*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) + ((a + b*tan(c + d*x))^(1/2)*(668*a*b^12*d^2 + 1 
088*a^3*b^10*d^2 - 576*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) - (604*a*b^14*d^2 - 932*a^3*b^12*d^2 - 1344*a^5*b^10* 
d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)/(4*d 
^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(41*b^16 + 26*a^2*b^14 + 553*a^4* 
b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*(-(3*a*b^2 - a^2*b*3i - a^3 + b^3* 
1i)/(4*d^2))^(1/2)*1i - (((((384*a^2*b^10*d^4 - 384*b^12*d^4 + 768*a^4*b^8 
*d^4)/(2*d^5) - ((512*b^10*d^4 + 768*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) - ((a + b*tan(c + d*x))^(1/2)*(66 
8*a*b^12*d^2 + 1088*a^3*b^10*d^2 - 576*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) - (604*a*b^14*d^2 - 932*a^3*b^12*d^2 
- 1344*a^5*b^10*d^2 + 192*a^7*b^8*d^2)/(2*d^5))*(-(3*a*b^2 - a^2*b*3i - a^ 
3 + b^3*1i)/(4*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(41*b^16 + 26*a^2 
*b^14 + 553*a^4*b^12 - 304*a^6*b^10 + 96*a^8*b^8))/d^4)*(-(3*a*b^2 - a^2*b 
*3i - a^3 + b^3*1i)/(4*d^2))^(1/2)*1i)/((119*a^4*b^14 - 71*a^2*b^16 - 15*b 
^18 + 391*a^6*b^12 + 216*a^8*b^10)/d^5 + (((((384*a^2*b^10*d^4 - 384*b^...