\(\int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx\) [14]

Optimal result

Integrand size = 25, antiderivative size = 195 \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=-\frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{7/2}}+\frac {\sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{7/2}}+\frac {\sqrt {2} a^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}+\sqrt {e} \cot (c+d x)}\right )}{d e^{7/2}}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac {4 a^2}{3 d e^2 (e \cot (c+d x))^{3/2}} \] Output:

-2^(1/2)*a^2*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d/e^(7/2)+2^(1 
/2)*a^2*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d/e^(7/2)+2^(1/2)*a 
^2*arctanh(2^(1/2)*(e*cot(d*x+c))^(1/2)/(e^(1/2)+e^(1/2)*cot(d*x+c)))/d/e^ 
(7/2)+2/5*a^2/d/e/(e*cot(d*x+c))^(5/2)+4/3*a^2/d/e^2/(e*cot(d*x+c))^(3/2)
 

Mathematica [F(-1)]

Timed out. \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\text {\$Aborted} \] Input:

Integrate[(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2),x]
 

Output:

$Aborted
 

Rubi [A] (warning: unable to verify)

Time = 0.65 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.23, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4025, 27, 3042, 3955, 3042, 3957, 266, 755, 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.

\(\displaystyle \int \frac {(a \cot (c+d x)+a)^2}{(e \cot (c+d x))^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4025

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a^2 \int \frac {1}{(e \cot (c+d x))^{5/2}}dx}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3955

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

Int[(a + a*Cot[c + d*x])^2/(e*Cot[c + d*x])^(7/2),x]
 

Output:

(2*a^2)/(5*d*e*(e*Cot[c + d*x])^(5/2)) + (2*a^2*(2/(3*d*e*(e*Cot[c + d*x]) 
^(3/2)) + (2*((-(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*e) + ( 
-1/2*Log[e - Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + d*x]^2]/(Sqrt[2]*S 
qrt[e]) + Log[e + Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + d*x]^2]/(2*Sq 
rt[2]*Sqrt[e]))/(2*e)))/(d*e)))/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 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   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 3042

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

rule 3955

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] 
)^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2   Int[(b*Tan[c + d*x])^(n + 2), x] 
, x] /; FreeQ[{b, c, d}, x] && LtQ[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 4025

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

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(-\frac {2 a^{2} \left (-\frac {1}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{3}}\right )}{d e}\)	\(174\)
default	\(-\frac {2 a^{2} \left (-\frac {1}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{3}}\right )}{d e}\)	\(174\)
parts	\(-\frac {2 a^{2} e \left (\frac {\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^{4} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{5 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {1}{e^{4} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}-\frac {2 a^{2} \left (-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}-\frac {\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} \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d e}+\frac {2 a^{2} \left (\frac {2}{3 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\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 )}{4 e^{4}}\right )}{d}\)	\(494\)

Input:

int((a+a*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),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. 534 vs. \(2 (155) = 310\).

Time = 0.09 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.74 \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\frac {\frac {30 \, \sqrt {2} {\left (a^{2} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, a^{2} e \cos \left (2 \, d x + 2 \, c\right ) + a^{2} e\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}} + 1\right )}{\sqrt {e}} + \frac {30 \, \sqrt {2} {\left (a^{2} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, a^{2} e \cos \left (2 \, d x + 2 \, c\right ) + a^{2} e\right )} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}} - 1\right )}{\sqrt {e}} + \frac {15 \, \sqrt {2} {\left (a^{2} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, a^{2} e \cos \left (2 \, d x + 2 \, c\right ) + a^{2} e\right )} \log \left (\frac {\frac {\sqrt {2} \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 )}{\sqrt {e}} + \cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}\right )}{\sqrt {e}} - \frac {15 \, \sqrt {2} {\left (a^{2} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, a^{2} e \cos \left (2 \, d x + 2 \, c\right ) + a^{2} e\right )} \log \left (-\frac {\frac {\sqrt {2} \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 )}{\sqrt {e}} - \cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1}{\sin \left (2 \, d x + 2 \, c\right )}\right )}{\sqrt {e}} - 4 \, {\left (10 \, a^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} - 10 \, a^{2} + 3 \, {\left (a^{2} \cos \left (2 \, d x + 2 \, c\right ) - a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{30 \, {\left (d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + d e^{4}\right )}} \] Input:

integrate((a+a*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=a^{2} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {2 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx\right ) \] Input:

integrate((a+a*cot(d*x+c))**2/(e*cot(d*x+c))**(7/2),x)
 

Output:

a**2*(Integral((e*cot(c + d*x))**(-7/2), x) + Integral(2*cot(c + d*x)/(e*c 
ot(c + d*x))**(7/2), x) + Integral(cot(c + d*x)**2/(e*cot(c + d*x))**(7/2) 
, x))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a+a*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),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(-1)]

Timed out. \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 10.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.51 \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\frac {\frac {4\,a^2\,\mathrm {cot}\left (c+d\,x\right )}{3}+\frac {2\,a^2}{5}}{d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d\,e^{7/2}}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d\,e^{7/2}} \] Input:

int((a + a*cot(c + d*x))^2/(e*cot(c + d*x))^(7/2),x)
 

Output:

((4*a^2*cot(c + d*x))/3 + (2*a^2)/5)/(d*e*(e*cot(c + d*x))^(5/2)) - ((-1)^ 
(1/4)*a^2*atan(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*2i)/(d*e^(7/2) 
) - ((-1)^(1/4)*a^2*atanh(((-1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*2i) 
/(d*e^(7/2))
 

Reduce [F]

\[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\frac {\sqrt {e}\, a^{2} \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{4}}d x +2 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{3}}d x \right )+\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{2}}d x \right )}{e^{4}} \] Input:

int((a+a*cot(d*x+c))^2/(e*cot(d*x+c))^(7/2),x)
 

Output:

(sqrt(e)*a**2*(int(sqrt(cot(c + d*x))/cot(c + d*x)**4,x) + 2*int(sqrt(cot( 
c + d*x))/cot(c + d*x)**3,x) + int(sqrt(cot(c + d*x))/cot(c + d*x)**2,x))) 
/e**4