\(\int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3 \, dx\) [817]

Optimal result

Integrand size = 23, antiderivative size = 192 \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3 \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {16 a b^2}{3 d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{3 d \cot ^{\frac {3}{2}}(c+d x)} \] Output:

-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d-1 
/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/d+1/2* 
(a+b)*(a^2-4*a*b+b^2)*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^( 
1/2)/d+16/3*a*b^2/d/cot(d*x+c)^(1/2)+2/3*b^2*(b+a*cot(d*x+c))/d/cot(d*x+c) 
^(3/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.52 \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3 \, dx=\frac {\left (-6 a^2 b+2 b^3\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\cot ^2(c+d x)\right )+6 a \left (a (b+a \cot (c+d x))-\left (a^2-3 b^2\right ) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \] Input:

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

Output:

((-6*a^2*b + 2*b^3)*Hypergeometric2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2] + 6*a 
*(a*(b + a*Cot[c + d*x]) - (a^2 - 3*b^2)*Cot[c + d*x]*Hypergeometric2F1[-1 
/4, 1, 3/4, -Cot[c + d*x]^2]))/(3*d*Cot[c + d*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.16, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 4156, 3042, 4048, 27, 3042, 4111, 27, 3042, 4017, 25, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4156

\(\displaystyle \int \frac {(a \cot (c+d x)+b)^3}{\cot ^{\frac {5}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 4048

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4017

\(\displaystyle \frac {1}{3} \left (\frac {6 \int -\frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}-\frac {6 \int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 1482

\(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{3} \left (\frac {6 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {16 a b^2}{d \sqrt {\cot (c+d x)}}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{3 d \cot ^{\frac {3}{2}}(c+d x)}\)

Input:

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

Output:

(2*b^2*(b + a*Cot[c + d*x]))/(3*d*Cot[c + d*x]^(3/2)) + ((16*a*b^2)/(d*Sqr 
t[Cot[c + d*x]]) + (6*(-1/2*((a - b)*(a^2 + 4*a*b + b^2)*(-(ArcTan[1 - Sqr 
t[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] 
/Sqrt[2])) + ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Cot[c 
 + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Co 
t[c + d*x]]/(2*Sqrt[2])))/2))/d)/3
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

rule 3042

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

rule 4017

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[2/f   Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq 
rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & 
& NeQ[c^2 + d^2, 0]
 

rule 4048

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m 
 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 
/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + 
f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c 
*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) 
*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( 
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] && GtQ[m, 2] && LtQ 
[n, -1] && IntegerQ[2*m]
 

rule 4111

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + 
 (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - 
 a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x 
] + Simp[1/(a^2 + b^2)   Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - 
 C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B 
, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0 
]
 

rule 4156

Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x 
_)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p)   Int[(d*Cot[e + f*x])^(m - n*p 
)*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && 
  !IntegerQ[m] && IntegersQ[n, p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(167)=334\).

Time = 0.22 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.41

method	result	size
derivativedivides	\(\frac {\sqrt {\frac {1}{\tan \left (d x +c \right )}}\, \sqrt {\tan \left (d x +c \right )}\, \left (8 b^{3} \tan \left (d x +c \right )^{\frac {3}{2}}+3 \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{3}-9 \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a \,b^{2}+6 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+18 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -18 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-6 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}+6 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+18 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -18 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-6 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}+9 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a^{2} b -3 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) b^{3}+72 a \,b^{2} \sqrt {\tan \left (d x +c \right )}\right )}{12 d}\)	\(462\)
default	\(\frac {\sqrt {\frac {1}{\tan \left (d x +c \right )}}\, \sqrt {\tan \left (d x +c \right )}\, \left (8 b^{3} \tan \left (d x +c \right )^{\frac {3}{2}}+3 \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{3}-9 \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a \,b^{2}+6 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+18 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -18 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-6 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}+6 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}+18 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b -18 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}-6 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}+9 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a^{2} b -3 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) b^{3}+72 a \,b^{2} \sqrt {\tan \left (d x +c \right )}\right )}{12 d}\)	\(462\)

Input:

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

Output:

1/12/d*(1/tan(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*(8*b^3*tan(d*x+c)^(3/2)+3*2^( 
1/2)*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2) 
-tan(d*x+c)-1))*a^3-9*2^(1/2)*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/ 
(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a*b^2+6*2^(1/2)*arctan(1+2^(1/2)* 
tan(d*x+c)^(1/2))*a^3+18*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b- 
18*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-6*2^(1/2)*arctan(1+2^( 
1/2)*tan(d*x+c)^(1/2))*b^3+6*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a 
^3+18*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b-18*2^(1/2)*arctan( 
-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b^2-6*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^ 
(1/2))*b^3+9*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+ 
c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*a^2*b-3*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1 
/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*b^3+72*a*b^2*ta 
n(d*x+c)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (167) = 334\).

Time = 0.09 (sec) , antiderivative size = 910, normalized size of antiderivative = 4.74 \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

-1/6*(6*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b 
^4 + 6*a*b^5 + b^6)/d^2)*arctan(-(2*sqrt(1/2)*(a^3 - 3*a^2*b - 3*a*b^2 + b 
^3)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + 
 b^6)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a 
^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 
 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b^2 + 15*a^2* 
b^4 - b^6)) + 6*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 
 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*arctan(-(2*sqrt(1/2)*(a^3 - 3*a^2*b - 3*a 
*b^2 + b^3)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6 
*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - d^2*sqrt((a^6 + 6*a^5*b + 3*a^4*b^ 
2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6*a^5*b + 3*a 
^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b^2 + 
 15*a^2*b^4 - b^6)) - 3*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a 
^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*log(a^3 - 3*a^2*b - 3*a*b^2 + b^3 
 + 2*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 
- 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) + (a^3 - 3*a^2*b - 3*a*b^2 + b^3) 
*tan(d*x + c)) + 3*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^ 
3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*log(a^3 - 3*a^2*b - 3*a*b^2 + b^3 - 2* 
sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a 
*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) + (a^3 - 3*a^2*b - 3*a*b^2 + b^3)*t...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.15 \[ \int \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3 \, dx=-\frac {6 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 3 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 3 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 8 \, {\left (b^{3} + \frac {9 \, a b^{2}}{\tan \left (d x + c\right )}\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}{12 \, d} \] Input:

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

Output:

-1/12*(6*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt( 
2) + 2/sqrt(tan(d*x + c)))) + 6*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*ar 
ctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 3*sqrt(2)*(a^3 - 3*a 
^2*b - 3*a*b^2 + b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) 
 + 3*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)/sqrt(tan(d*x + c 
)) + 1/tan(d*x + c) + 1) - 8*(b^3 + 9*a*b^2/tan(d*x + c))*tan(d*x + c)^(3/ 
2))/d
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(cot(c + d*x)),x)*a**3 + int(sqrt(cot(c + d*x))*tan(c + d*x)**3,x) 
*b**3 + 3*int(sqrt(cot(c + d*x))*tan(c + d*x)**2,x)*a*b**2 + 3*int(sqrt(co 
t(c + d*x))*tan(c + d*x),x)*a**2*b