\(\int \frac {\cot ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [536]

Optimal result

Integrand size = 23, antiderivative size = 194 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\left (8 a^2-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{5/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d}+\frac {3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d}-\frac {\cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{2 a d} \] Output:

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

Mathematica [A] (verified)

Time = 2.07 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.05 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\frac {\left (8 a^2-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {4 a^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}-\frac {4 a^2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}+3 b \cot (c+d x) \sqrt {a+b \tan (c+d x)}-2 a \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 a^2 d} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.52 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4137, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4052

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4132

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4137

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

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

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 4052

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

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 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 [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. 831 vs. \(2 (158) = 316\).

Time = 0.16 (sec) , antiderivative size = 1682, normalized size of antiderivative = 8.67 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/8*(4*a^3*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4 
)) + a)/((a^2 + b^2)*d^2))*log(((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^ 
2 + b^4)*d^4)) - a*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + 
b^4)*d^4)) + a)/((a^2 + b^2)*d^2)) + sqrt(b*tan(d*x + c) + a))*tan(d*x + c 
)^2 - 4*a^3*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4 
)) + a)/((a^2 + b^2)*d^2))*log(-((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b 
^2 + b^4)*d^4)) - a*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + 
 b^4)*d^4)) + a)/((a^2 + b^2)*d^2)) + sqrt(b*tan(d*x + c) + a))*tan(d*x + 
c)^2 - 4*a^3*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d 
^4)) - a)/((a^2 + b^2)*d^2))*log(((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2* 
b^2 + b^4)*d^4)) + a*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 
 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2)) + sqrt(b*tan(d*x + c) + a))*tan(d*x 
+ c)^2 + 4*a^3*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4) 
*d^4)) - a)/((a^2 + b^2)*d^2))*log(-((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*a 
^2*b^2 + b^4)*d^4)) + a*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2* 
b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2)) + sqrt(b*tan(d*x + c) + a))*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*(3*a*b*tan(d*x 
+ c) - 2*a^2)*sqrt(b*tan(d*x + c) + a))/(a^3*d*tan(d*x + c)^2), 1/4*(2*a^3 
*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \] 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 = 0.65 (sec) , antiderivative size = 3399, normalized size of antiderivative = 17.52 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((((((((320*a^4*b^10*d^4 - 192*a^2*b^12*d^4 + 384*a^6*b^8*d^4)/(2*a^ 
4*d^5) - ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(1/(a*d^2 - b*d^2*1i))^(1/2 
)*(a + b*tan(c + d*x))^(1/2))/(4*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 
 + ((a + b*tan(c + d*x))^(1/2)*(36*a*b^12*d^2 - 192*a^3*b^10*d^2 + 576*a^5 
*b^8*d^2))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + (18*a*b^12*d^2 - 
 96*a^5*b^8*d^2)/(2*a^4*d^5))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - ((a + b*ta 
n(c + d*x))^(1/2)*(9*b^12 - 48*a^2*b^10 + 96*a^4*b^8))/(2*a^4*d^4))*(1/(a* 
d^2 - b*d^2*1i))^(1/2)*1i - ((((((((320*a^4*b^10*d^4 - 192*a^2*b^12*d^4 + 
384*a^6*b^8*d^4)/(2*a^4*d^5) + ((512*a^4*b^10*d^4 + 768*a^6*b^8*d^4)*(1/(a 
*d^2 - b*d^2*1i))^(1/2)*(a + b*tan(c + d*x))^(1/2))/(4*a^4*d^4))*(1/(a*d^2 
 - b*d^2*1i))^(1/2))/2 - ((a + b*tan(c + d*x))^(1/2)*(36*a*b^12*d^2 - 192* 
a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2) 
)/2 + (18*a*b^12*d^2 - 96*a^5*b^8*d^2)/(2*a^4*d^5))*(1/(a*d^2 - b*d^2*1i)) 
^(1/2))/2 + ((a + b*tan(c + d*x))^(1/2)*(9*b^12 - 48*a^2*b^10 + 96*a^4*b^8 
))/(2*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2)*1i)/(((((((((320*a^4*b^10*d^4 
 - 192*a^2*b^12*d^4 + 384*a^6*b^8*d^4)/(2*a^4*d^5) - ((512*a^4*b^10*d^4 + 
768*a^6*b^8*d^4)*(1/(a*d^2 - b*d^2*1i))^(1/2)*(a + b*tan(c + d*x))^(1/2))/ 
(4*a^4*d^4))*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + ((a + b*tan(c + d*x))^(1/2) 
*(36*a*b^12*d^2 - 192*a^3*b^10*d^2 + 576*a^5*b^8*d^2))/(2*a^4*d^4))*(1/(a* 
d^2 - b*d^2*1i))^(1/2))/2 + (18*a*b^12*d^2 - 96*a^5*b^8*d^2)/(2*a^4*d^5...
 

Reduce [F]

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