\(\int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx\) [510]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.88 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\frac {\left (8 a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )-\sqrt {a} \left (4 a \sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+4 a \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+\cot (c+d x) (b+2 a \cot (c+d x)) \sqrt {a+b \tan (c+d x)}\right )}{4 a^{3/2} d} \] Input:

Integrate[Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]],x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.49 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4051, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4051

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]],x]
 

Output:

-1/2*(Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/d + (-1/2*(8*((a*(I*a + b)* 
ArcTan[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - (a*(I*a - b)*ArcTa 
n[Tan[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)) - (2*(8*a^2 + b^2)*ArcTa 
nh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d))/a - (b*Cot[c + d*x]*Sqr 
t[a + b*Tan[c + d*x]])/(a*d))/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 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 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 [F(-1)]

Timed out.

hanged

Input:

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2),x)
 

Output:

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (153) = 306\).

Time = 0.13 (sec) , antiderivative size = 815, normalized size of antiderivative = 4.31 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

[-1/8*(4*a^2*d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2)*log(d*sqrt((d^2*sqrt(-b^ 
2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 - 4*a^2*d*sqrt 
((d^2*sqrt(-b^2/d^4) + a)/d^2)*log(-d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + 
 sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 + 4*a^2*d*sqrt(-(d^2*sqrt(-b^2/d 
^4) - a)/d^2)*log(d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + 
 c) + a))*tan(d*x + c)^2 - 4*a^2*d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2)*log 
(-d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d* 
x + c)^2 - (8*a^2 + 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*(a*b*tan(d*x + c) 
+ 2*a^2)*sqrt(b*tan(d*x + c) + a))/(a^2*d*tan(d*x + c)^2), -1/4*(2*a^2*d*s 
qrt((d^2*sqrt(-b^2/d^4) + a)/d^2)*log(d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) 
 + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 - 2*a^2*d*sqrt((d^2*sqrt(-b^2/ 
d^4) + a)/d^2)*log(-d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + sqrt(b*tan(d*x 
+ c) + a))*tan(d*x + c)^2 + 2*a^2*d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2)*lo 
g(d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d* 
x + c)^2 - 2*a^2*d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2)*log(-d*sqrt(-(d^2*s 
qrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 + (8*a^ 
2 + b^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*tan(d*x + c) + a))*tan(d*x + c)^2 
 + (a*b*tan(d*x + c) + 2*a^2)*sqrt(b*tan(d*x + c) + a))/(a^2*d*tan(d*x + c 
)^2)]
 

Sympy [F]

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

integrate(cot(d*x+c)**3*(a+b*tan(d*x+c))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

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

integrate(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 2.14 (sec) , antiderivative size = 1910, normalized size of antiderivative = 10.11 \[ \int \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\text {Too large to display} \] Input:

int(cot(c + d*x)^3*(a + b*tan(c + d*x))^(1/2),x)
 

Output:

atan((b^14*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*2 
i)/((b^15*1i)/d + (a^2*b^13*17i)/d + (16*a^3*b^12)/d + (a^4*b^11*64i)/d + 
(16*a^5*b^10)/d + (a^6*b^9*48i)/d) - (2*b^13*(a/(4*d^2) + (b*1i)/(4*d^2))^ 
(1/2)*(a + b*tan(c + d*x))^(1/2))/((a*b^13*17i)/d + (16*a^2*b^12)/d + (a^3 
*b^11*64i)/d + (16*a^4*b^10)/d + (a^5*b^9*48i)/d + (b^15*1i)/(a*d)) + (b^1 
2*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((b^1 
3*17i)/d + (16*a*b^12)/d + (a^2*b^11*64i)/d + (16*a^3*b^10)/d + (a^4*b^9*4 
8i)/d + (b^15*1i)/(a^2*d)) + (a^2*b^10*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)* 
(a + b*tan(c + d*x))^(1/2)*128i)/((b^13*17i)/d + (16*a*b^12)/d + (a^2*b^11 
*64i)/d + (16*a^3*b^10)/d + (a^4*b^9*48i)/d + (b^15*1i)/(a^2*d)) - (96*a^3 
*b^9*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((b^13 
*17i)/d + (16*a*b^12)/d + (a^2*b^11*64i)/d + (16*a^3*b^10)/d + (a^4*b^9*48 
i)/d + (b^15*1i)/(a^2*d)))*((a + b*1i)/(4*d^2))^(1/2)*2i - atan((b^14*(a/( 
4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*2i)/((b^15*1i)/d 
 + (a^2*b^13*17i)/d - (16*a^3*b^12)/d + (a^4*b^11*64i)/d - (16*a^5*b^10)/d 
 + (a^6*b^9*48i)/d) + (2*b^13*(a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*ta 
n(c + d*x))^(1/2))/((a*b^13*17i)/d - (16*a^2*b^12)/d + (a^3*b^11*64i)/d - 
(16*a^4*b^10)/d + (a^5*b^9*48i)/d + (b^15*1i)/(a*d)) + (b^12*(a/(4*d^2) - 
(b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((b^13*17i)/d - (16* 
a*b^12)/d + (a^2*b^11*64i)/d - (16*a^3*b^10)/d + (a^4*b^9*48i)/d + (b^1...
 

Reduce [F]

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

int(cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2),x)
 

Output:

int(sqrt(tan(c + d*x)*b + a)*cot(c + d*x)**3,x)