\(\int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx\) [9]

Optimal result

Integrand size = 25, antiderivative size = 193 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx=-\frac {\sqrt {2} a^2 e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {\sqrt {2} a^2 e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {\sqrt {2} a^2 e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}+\sqrt {e} \cot (c+d x)}\right )}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2}}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2}}{5 d e} \] Output:

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.58 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx=-\frac {2 a^2 (e \cot (c+d x))^{3/2} \left (-15 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)}+15 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)}+\cot ^{\frac {7}{4}}(c+d x) (10+3 \cot (c+d x))\right )}{15 d \cot ^{\frac {7}{4}}(c+d x)} \] Input:

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

Output:

(-2*a^2*(e*Cot[c + d*x])^(3/2)*(-15*ArcTan[(-Cot[c + d*x]^2)^(1/4)]*(-Cot[ 
c + d*x])^(1/4) + 15*ArcTanh[(-Cot[c + d*x]^2)^(1/4)]*(-Cot[c + d*x])^(1/4 
) + Cot[c + d*x]^(7/4)*(10 + 3*Cot[c + d*x])))/(15*d*Cot[c + d*x]^(7/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.66 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.20, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 4026, 27, 2030, 3042, 3954, 3042, 3957, 266, 826, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4026

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2030

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3954

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

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 2030

Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m   Int[(b*v) 
^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
 

rule 3042

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

rule 3954

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d 
*x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2   Int[(b*Tan[c + d*x])^(n - 2), x] 
, x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

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])
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(-\frac {2 a^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{3} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d e}\)	\(172\)
default	\(-\frac {2 a^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{3} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d e}\)	\(172\)
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 a^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-e^{2} \sqrt {e \cot \left (d x +c \right )}+\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^{2} \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}\)	\(481\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (153) = 306\).

Time = 0.10 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.34 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx=\frac {30 \, \sqrt {2} {\left (a^{2} e \cos \left (2 \, d x + 2 \, c\right ) - a^{2} e\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + e}{e}\right ) + 30 \, \sqrt {2} {\left (a^{2} e \cos \left (2 \, d x + 2 \, c\right ) - a^{2} e\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} - e}{e}\right ) - 15 \, \sqrt {2} {\left (a^{2} e \cos \left (2 \, d x + 2 \, c\right ) - a^{2} e\right )} \sqrt {e} \log \left (\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e \cos \left (2 \, d x + 2 \, c\right ) + e \sin \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}\right ) + 15 \, \sqrt {2} {\left (a^{2} e \cos \left (2 \, d x + 2 \, c\right ) - a^{2} e\right )} \sqrt {e} \log \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) - e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) - e}{\sin \left (2 \, d x + 2 \, c\right )}\right ) + 4 \, {\left (3 \, a^{2} e \cos \left (2 \, d x + 2 \, c\right ) + 10 \, a^{2} e \sin \left (2 \, d x + 2 \, c\right ) + 3 \, a^{2} e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{30 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}} \] Input:

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

Output:

1/30*(30*sqrt(2)*(a^2*e*cos(2*d*x + 2*c) - a^2*e)*sqrt(e)*arctan((sqrt(2)* 
sqrt(e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + e)/e) + 30*sqrt( 
2)*(a^2*e*cos(2*d*x + 2*c) - a^2*e)*sqrt(e)*arctan((sqrt(2)*sqrt(e)*sqrt(( 
e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - e)/e) - 15*sqrt(2)*(a^2*e*cos( 
2*d*x + 2*c) - a^2*e)*sqrt(e)*log((sqrt(2)*sqrt(e)*sqrt((e*cos(2*d*x + 2*c 
) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c) + e*cos(2*d*x + 2*c) + e*sin(2*d 
*x + 2*c) + e)/sin(2*d*x + 2*c)) + 15*sqrt(2)*(a^2*e*cos(2*d*x + 2*c) - a^ 
2*e)*sqrt(e)*log(-(sqrt(2)*sqrt(e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x 
 + 2*c))*sin(2*d*x + 2*c) - e*cos(2*d*x + 2*c) - e*sin(2*d*x + 2*c) - e)/s 
in(2*d*x + 2*c)) + 4*(3*a^2*e*cos(2*d*x + 2*c) + 10*a^2*e*sin(2*d*x + 2*c) 
 + 3*a^2*e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*cos(2*d*x 
+ 2*c) - d)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 10.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.54 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx=\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{d}-\frac {2\,a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e}-\frac {4\,a^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}+\frac {{\left (-1\right )}^{1/4}\,a^2\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,1{}\mathrm {i}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d} \] Input:

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

Output:

(2*(-1)^(1/4)*a^2*e^(3/2)*atan(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2) 
))/d - (2*a^2*(e*cot(c + d*x))^(5/2))/(5*d*e) - (4*a^2*(e*cot(c + d*x))^(3 
/2))/(3*d) + ((-1)^(1/4)*a^2*e^(3/2)*atan(((-1)^(1/4)*(e*cot(c + d*x))^(1/ 
2)*1i)/e^(1/2))*2i)/d
 

Reduce [F]

\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^2 \, dx=\frac {2 \sqrt {e}\, a^{2} e \left (-\sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}+5 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) d \right )}{5 d} \] Input:

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

Output:

(2*sqrt(e)*a**2*e*( - sqrt(cot(c + d*x))*cot(c + d*x)**2 + 5*int(sqrt(cot( 
c + d*x))*cot(c + d*x)**2,x)*d))/(5*d)