3.1.56 \(\int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx\) [56]

3.1.56.1 Optimal result

Integrand size = 25, antiderivative size = 317 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx=-\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {2 \left (a^2-b^2\right ) e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a b (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 b^2 (e \cot (c+d x))^{5/2}}{5 d e}-\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d} \]

output

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

3.1.56.2 Mathematica [C] (verified)

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

Time = 2.17 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.71 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx=-\frac {(e \cot (c+d x))^{3/2} \left (\frac {2}{5} b^2 \cot ^{\frac {5}{2}}(c+d x)-\frac {4}{3} a b \cot ^{\frac {3}{2}}(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )+\frac {1}{4} \left (a^2-b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{d \cot ^{\frac {3}{2}}(c+d x)} \]

input

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

output

-(((e*Cot[c + d*x])^(3/2)*((2*b^2*Cot[c + d*x]^(5/2))/5 - (4*a*b*Cot[c + d 
*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + ((a^ 
2 - b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*Arc 
Tan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 
 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sq 
rt[Cot[c + d*x]] + Cot[c + d*x]]))/4))/(d*Cot[c + d*x]^(3/2)))
 

3.1.56.3 Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.92, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 4026, 3042, 4011, 3042, 4011, 3042, 4017, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4026

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4011

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

input

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

output

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

3.1.56.3.1 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 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 4011

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

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 4026

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

3.1.56.4 Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a e b \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+a^{2} e^{2} \sqrt {e \cot \left (d x +c \right )}-\sqrt {e \cot \left (d x +c \right )}\, b^{2} e^{2}-e^{3} \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(360\)
default	\(-\frac {2 \left (\frac {b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a e b \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+a^{2} e^{2} \sqrt {e \cot \left (d x +c \right )}-\sqrt {e \cot \left (d x +c \right )}\, b^{2} e^{2}-e^{3} \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d e}\)	\(360\)
parts	\(-\frac {2 a^{2} e \left (\sqrt {e \cot \left (d x +c \right )}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{d}-\frac {2 b^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\sqrt {e \cot \left (d x +c \right )}\, e^{2}+\frac {e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{d e}+\frac {2 a b \left (-\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\)	\(480\)

input

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

output

-2/d/e*(1/5*b^2*(e*cot(d*x+c))^(5/2)+2/3*a*e*b*(e*cot(d*x+c))^(3/2)+a^2*e^ 
2*(e*cot(d*x+c))^(1/2)-(e*cot(d*x+c))^(1/2)*b^2*e^2-e^3*(1/8*(a^2*e-b^2*e) 
*(e^2)^(1/4)/e^2*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2 
)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1 
/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*a 
rctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))+1/4*a*b/(e^2)^(1/4)*2^ 
(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/ 
2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2 
*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2 
)^(1/4)*(e*cot(d*x+c))^(1/2)+1))))
 

3.1.56.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1313 vs. \(2 (260) = 520\).

Time = 0.30 (sec) , antiderivative size = 1313, normalized size of antiderivative = 4.14 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx=\text {Too large to display} \]

input

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

output

1/30*(15*(d*cos(2*d*x + 2*c) - d)*sqrt(-(4*(a^3*b - a*b^3)*e^3 + sqrt(-(a^ 
8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^6/d^4)*d^2)/d^2)*log((a^ 
8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8)*e^4*sqrt((e*cos(2*d*x + 2*c) 
 + e)/sin(2*d*x + 2*c)) + (2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2 
*b^6 + b^8)*e^6/d^4)*a*b*d^3 + (a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*d*e^3)* 
sqrt(-(4*(a^3*b - a*b^3)*e^3 + sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a 
^2*b^6 + b^8)*e^6/d^4)*d^2)/d^2)) - 15*(d*cos(2*d*x + 2*c) - d)*sqrt(-(4*( 
a^3*b - a*b^3)*e^3 + sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b 
^8)*e^6/d^4)*d^2)/d^2)*log((a^8 - 4*a^6*b^2 - 10*a^4*b^4 - 4*a^2*b^6 + b^8 
)*e^4*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - (2*sqrt(-(a^8 - 12 
*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^6/d^4)*a*b*d^3 + (a^6 - 7*a^4* 
b^2 + 7*a^2*b^4 - b^6)*d*e^3)*sqrt(-(4*(a^3*b - a*b^3)*e^3 + sqrt(-(a^8 - 
12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^6/d^4)*d^2)/d^2)) - 15*(d*co 
s(2*d*x + 2*c) - d)*sqrt(-(4*(a^3*b - a*b^3)*e^3 - sqrt(-(a^8 - 12*a^6*b^2 
 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^6/d^4)*d^2)/d^2)*log((a^8 - 4*a^6*b^2 
- 10*a^4*b^4 - 4*a^2*b^6 + b^8)*e^4*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d* 
x + 2*c)) + (2*sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*e^ 
6/d^4)*a*b*d^3 - (a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*d*e^3)*sqrt(-(4*(a^3* 
b - a*b^3)*e^3 - sqrt(-(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)* 
e^6/d^4)*d^2)/d^2)) + 15*(d*cos(2*d*x + 2*c) - d)*sqrt(-(4*(a^3*b - a*b...
 

3.1.56.6 Sympy [F]

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

input

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

output

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

3.1.56.7 Maxima [F(-2)]

Exception generated. \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.1.56.8 Giac [F]

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

input

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

output

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

3.1.56.9 Mupad [B] (verification not implemented)

Time = 15.24 (sec) , antiderivative size = 1274, normalized size of antiderivative = 4.02 \[ \int (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2 \, dx=\text {Too large to display} \]

input

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

output

atan((a^4*e^6*(e*cot(c + d*x))^(1/2)*((a*b^3*e^3)/d^2 - (b^4*e^3*1i)/(4*d^ 
2) - (a^4*e^3*1i)/(4*d^2) - (a^3*b*e^3)/d^2 + (a^2*b^2*e^3*3i)/(2*d^2))^(1 
/2)*32i)/((a^6*e^8*16i)/d - (b^6*e^8*16i)/d + (32*a*b^5*e^8)/d + (32*a^5*b 
*e^8)/d + (a^2*b^4*e^8*112i)/d - (192*a^3*b^3*e^8)/d - (a^4*b^2*e^8*112i)/ 
d) + (b^4*e^6*(e*cot(c + d*x))^(1/2)*((a*b^3*e^3)/d^2 - (b^4*e^3*1i)/(4*d^ 
2) - (a^4*e^3*1i)/(4*d^2) - (a^3*b*e^3)/d^2 + (a^2*b^2*e^3*3i)/(2*d^2))^(1 
/2)*32i)/((a^6*e^8*16i)/d - (b^6*e^8*16i)/d + (32*a*b^5*e^8)/d + (32*a^5*b 
*e^8)/d + (a^2*b^4*e^8*112i)/d - (192*a^3*b^3*e^8)/d - (a^4*b^2*e^8*112i)/ 
d) - (a^2*b^2*e^6*(e*cot(c + d*x))^(1/2)*((a*b^3*e^3)/d^2 - (b^4*e^3*1i)/( 
4*d^2) - (a^4*e^3*1i)/(4*d^2) - (a^3*b*e^3)/d^2 + (a^2*b^2*e^3*3i)/(2*d^2) 
)^(1/2)*192i)/((a^6*e^8*16i)/d - (b^6*e^8*16i)/d + (32*a*b^5*e^8)/d + (32* 
a^5*b*e^8)/d + (a^2*b^4*e^8*112i)/d - (192*a^3*b^3*e^8)/d - (a^4*b^2*e^8*1 
12i)/d))*(-(a^4*e^3*1i + b^4*e^3*1i - 4*a*b^3*e^3 + 4*a^3*b*e^3 - a^2*b^2* 
e^3*6i)/(4*d^2))^(1/2)*2i + atan((a^4*e^6*(e*cot(c + d*x))^(1/2)*((a^4*e^3 
*1i)/(4*d^2) + (b^4*e^3*1i)/(4*d^2) + (a*b^3*e^3)/d^2 - (a^3*b*e^3)/d^2 - 
(a^2*b^2*e^3*3i)/(2*d^2))^(1/2)*32i)/((b^6*e^8*16i)/d - (a^6*e^8*16i)/d + 
(32*a*b^5*e^8)/d + (32*a^5*b*e^8)/d - (a^2*b^4*e^8*112i)/d - (192*a^3*b^3* 
e^8)/d + (a^4*b^2*e^8*112i)/d) + (b^4*e^6*(e*cot(c + d*x))^(1/2)*((a^4*e^3 
*1i)/(4*d^2) + (b^4*e^3*1i)/(4*d^2) + (a*b^3*e^3)/d^2 - (a^3*b*e^3)/d^2 - 
(a^2*b^2*e^3*3i)/(2*d^2))^(1/2)*32i)/((b^6*e^8*16i)/d - (a^6*e^8*16i)/d...