\(\int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx\) [818]

Optimal result

Integrand size = 23, antiderivative size = 218 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {8 a b^2}{5 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{5 d \cot ^{\frac {5}{2}}(c+d x)} \] 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+8/5*a*b^2/d/cot(d*x+c)^(3/2)+2*b*(3*a^2-b^2)/d/cot(d*x+c)^(1/2)+2/5 
*b^2*(b+a*cot(d*x+c))/d/cot(d*x+c)^(5/2)
 

Mathematica [C] (verified)

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

Time = 0.39 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.47 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {2 \left (\left (-9 a^2 b+3 b^3\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\cot ^2(c+d x)\right )+a \left (a (9 b+5 a \cot (c+d x))-5 \left (a^2-3 b^2\right ) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\cot ^2(c+d x)\right )\right )\right )}{15 d \cot ^{\frac {5}{2}}(c+d x)} \] Input:

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

Output:

(2*((-9*a^2*b + 3*b^3)*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + 
 a*(a*(9*b + 5*a*Cot[c + d*x]) - 5*(a^2 - 3*b^2)*Cot[c + d*x]*Hypergeometr 
ic2F1[-3/4, 1, 1/4, -Cot[c + d*x]^2])))/(15*d*Cot[c + d*x]^(5/2))
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.16, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 4156, 3042, 4048, 27, 3042, 4111, 27, 3042, 4012, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4156

\(\displaystyle \int \frac {(a \cot (c+d x)+b)^3}{\cot ^{\frac {7}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\)

\(\Big \downarrow \) 4048

\(\displaystyle \frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int -\frac {12 a b^2+5 \left (3 a^2-b^2\right ) \cot (c+d x) b+a \left (5 a^2-3 b^2\right ) \cot ^2(c+d x)}{2 \cot ^{\frac {5}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \frac {12 a b^2+5 \left (3 a^2-b^2\right ) \cot (c+d x) b+a \left (5 a^2-3 b^2\right ) \cot ^2(c+d x)}{\cot ^{\frac {5}{2}}(c+d x)}dx+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{5} \int \frac {12 a b^2-5 \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) b+a \left (5 a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx+\frac {2 b^2 (a \cot (c+d x)+b)}{5 d \cot ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4111

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4012

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(2*b^2*(b + a*Cot[c + d*x]))/(5*d*Cot[c + d*x]^(5/2)) + ((8*a*b^2)/(d*Cot[ 
c + d*x]^(3/2)) + 5*((2*b*(3*a^2 - b^2))/(d*Sqrt[Cot[c + d*x]]) + (2*(-1/2 
*((a + b)*(a^2 - 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sq 
rt[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]]/Sq 
rt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2 
))/d))/5
 

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 4012

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

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 [A] (verified)

Time = 0.26 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(-\frac {-\frac {2 b^{3}}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}-\frac {2 b \left (3 a^{2}-b^{2}\right )}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 a \,b^{2}}{\cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{d}\)	\(246\)
default	\(-\frac {-\frac {2 b^{3}}{5 \cot \left (d x +c \right )^{\frac {5}{2}}}-\frac {2 b \left (3 a^{2}-b^{2}\right )}{\sqrt {\cot \left (d x +c \right )}}-\frac {2 a \,b^{2}}{\cot \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{d}\)	\(246\)

Input:

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

Output:

-1/d*(-2/5*b^3/cot(d*x+c)^(5/2)-2*b*(3*a^2-b^2)/cot(d*x+c)^(1/2)-2*a*b^2/c 
ot(d*x+c)^(3/2)+1/4*(a^3-3*a*b^2)*2^(1/2)*(ln((cot(d*x+c)+2^(1/2)*cot(d*x+ 
c)^(1/2)+1)/(cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*co 
t(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))+1/4*(-3*a^2*b+b^3)* 
2^(1/2)*(ln((cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1)/(cot(d*x+c)+2^(1/2)*co 
t(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2 
)*cot(d*x+c)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (190) = 380\).

Time = 0.10 (sec) , antiderivative size = 944, normalized size of antiderivative = 4.33 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/10*(10*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)) + 10*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)) + 5*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)) - 5*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)...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \tan (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx=\frac {8 \, {\left (b^{3} + \frac {5 \, a b^{2}}{\tan \left (d x + c\right )} + \frac {5 \, {\left (3 \, a^{2} b - b^{3}\right )}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} - 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 10 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 5 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{20 \, d} \] Input:

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

Output:

1/20*(8*(b^3 + 5*a*b^2/tan(d*x + c) + 5*(3*a^2*b - b^3)/tan(d*x + c)^2)*ta 
n(d*x + c)^(5/2) - 10*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*s 
qrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) - 10*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)))) - 5*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) + 5*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(-sqrt(2)/sq 
rt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/d
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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