\(\int \frac {\cot ^{-1}(x)}{(a+a x^2)^{7/2}} \, dx\) [12]

Optimal result

Integrand size = 14, antiderivative size = 118 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=-\frac {1}{25 a \left (a+a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a+a x^2\right )^{3/2}}-\frac {8}{15 a^3 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a+a x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(x)}{15 a^3 \sqrt {a+a x^2}} \] Output:

-1/25/a/(a*x^2+a)^(5/2)-4/45/a^2/(a*x^2+a)^(3/2)-8/15/a^3/(a*x^2+a)^(1/2)+ 
1/5*x*arccot(x)/a/(a*x^2+a)^(5/2)+4/15*x*arccot(x)/a^2/(a*x^2+a)^(3/2)+8/1 
5*x*arccot(x)/a^3/(a*x^2+a)^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {-149-260 x^2-120 x^4+15 x \left (15+20 x^2+8 x^4\right ) \cot ^{-1}(x)}{225 a \left (a \left (1+x^2\right )\right )^{5/2}} \] Input:

Integrate[ArcCot[x]/(a + a*x^2)^(7/2),x]
 

Output:

(-149 - 260*x^2 - 120*x^4 + 15*x*(15 + 20*x^2 + 8*x^4)*ArcCot[x])/(225*a*( 
a*(1 + x^2))^(5/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5432, 5432, 5430}

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.

\(\displaystyle \int \frac {\cot ^{-1}(x)}{\left (a x^2+a\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 5432

\(\displaystyle \frac {4 \int \frac {\cot ^{-1}(x)}{\left (a x^2+a\right )^{5/2}}dx}{5 a}-\frac {1}{25 a \left (a x^2+a\right )^{5/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a x^2+a\right )^{5/2}}\)

\(\Big \downarrow \) 5432

\(\displaystyle \frac {4 \left (\frac {2 \int \frac {\cot ^{-1}(x)}{\left (a x^2+a\right )^{3/2}}dx}{3 a}-\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}}\right )}{5 a}-\frac {1}{25 a \left (a x^2+a\right )^{5/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a x^2+a\right )^{5/2}}\)

\(\Big \downarrow \) 5430

\(\displaystyle -\frac {1}{25 a \left (a x^2+a\right )^{5/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a x^2+a\right )^{5/2}}+\frac {4 \left (-\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}}+\frac {2 \left (\frac {x \cot ^{-1}(x)}{a \sqrt {a x^2+a}}-\frac {1}{a \sqrt {a x^2+a}}\right )}{3 a}\right )}{5 a}\)

Input:

Int[ArcCot[x]/(a + a*x^2)^(7/2),x]
 

Output:

-1/25*1/(a*(a + a*x^2)^(5/2)) + (x*ArcCot[x])/(5*a*(a + a*x^2)^(5/2)) + (4 
*(-1/9*1/(a*(a + a*x^2)^(3/2)) + (x*ArcCot[x])/(3*a*(a + a*x^2)^(3/2)) + ( 
2*(-(1/(a*Sqrt[a + a*x^2])) + (x*ArcCot[x])/(a*Sqrt[a + a*x^2])))/(3*a)))/ 
(5*a)
 

Defintions of rubi rules used

rule 5430

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcCot[c*x])/(d*Sq 
rt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
 

rule 5432

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol 
] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + 
e*x^2)^(q + 1)*((a + b*ArcCot[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2* 
d*(q + 1))   Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x], x]) /; FreeQ[ 
{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
 

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75

method	result	size
orering	\(\frac {\left (\frac {64}{15} x^{7}+\frac {616}{45} x^{5}+\frac {3388}{225} x^{3}+\frac {1268}{225} x \right ) \operatorname {arccot}\left (x \right )}{\left (a \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (120 x^{4}+260 x^{2}+149\right ) \left (x^{2}+1\right )^{2} \left (-\frac {1}{\left (x^{2}+1\right ) \left (a \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {7 \,\operatorname {arccot}\left (x \right ) a x}{\left (a \,x^{2}+a \right )^{\frac {9}{2}}}\right )}{225}\)	\(88\)
risch	\(\frac {i x \left (8 x^{4}+20 x^{2}+15\right ) \ln \left (i x +1\right )}{30 a^{3} \left (x^{2}+1\right )^{2} \sqrt {a \left (x^{2}+1\right )}}+\frac {-120 i x^{5} \ln \left (-i x +1\right )+120 x^{5} \pi -300 i x^{3} \ln \left (-i x +1\right )+300 \pi \,x^{3}-240 x^{4}-225 i \ln \left (-i x +1\right ) x +225 \pi x -520 x^{2}-298}{450 a^{3} \left (x^{2}+1\right )^{2} \sqrt {a \left (x^{2}+1\right )}}\)	\(130\)
default	\(\frac {\left (i+5 \,\operatorname {arccot}\left (x \right )\right ) \left (x^{5}+5 i x^{4}-10 x^{3}-10 i x^{2}+5 x +i\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{800 \left (x^{2}+1\right )^{3} a^{4}}+\frac {5 \left (\operatorname {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{16 \left (x^{2}+1\right ) a^{4}}+\frac {5 \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\operatorname {arccot}\left (x \right )-i\right )}{16 \left (x^{2}+1\right ) a^{4}}-\frac {5 \left (-i+3 \,\operatorname {arccot}\left (x \right )\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x^{3}-3 i x^{2}-3 x +i\right )}{288 \left (x^{4}+2 x^{2}+1\right ) a^{4}}-\frac {\sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (67 i+165 \,\operatorname {arccot}\left (x \right )\right ) \cos \left (4 \,\operatorname {arccot}\left (x \right )\right )}{3600 \left (x^{2}+1\right ) a^{4}}-\frac {\sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (i x +1\right ) \left (29 i+105 \,\operatorname {arccot}\left (x \right )\right ) \sin \left (4 \,\operatorname {arccot}\left (x \right )\right )}{1800 \left (x^{2}+1\right ) a^{4}}\)	\(258\)

Input:

int(arccot(x)/(a*x^2+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

(64/15*x^7+616/45*x^5+3388/225*x^3+1268/225*x)*arccot(x)/(a*x^2+a)^(7/2)+1 
/225*(120*x^4+260*x^2+149)*(x^2+1)^2*(-1/(x^2+1)/(a*x^2+a)^(7/2)-7*arccot( 
x)/(a*x^2+a)^(9/2)*a*x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.59 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=-\frac {{\left (120 \, x^{4} + 260 \, x^{2} - 15 \, {\left (8 \, x^{5} + 20 \, x^{3} + 15 \, x\right )} \operatorname {arccot}\left (x\right ) + 149\right )} \sqrt {a x^{2} + a}}{225 \, {\left (a^{4} x^{6} + 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} + a^{4}\right )}} \] Input:

integrate(arccot(x)/(a*x^2+a)^(7/2),x, algorithm="fricas")
 

Output:

-1/225*(120*x^4 + 260*x^2 - 15*(8*x^5 + 20*x^3 + 15*x)*arccot(x) + 149)*sq 
rt(a*x^2 + a)/(a^4*x^6 + 3*a^4*x^4 + 3*a^4*x^2 + a^4)
 

Sympy [F]

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

integrate(acot(x)/(a*x**2+a)**(7/2),x)
 

Output:

Integral(acot(x)/(a*(x**2 + 1))**(7/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {a x^{2} + a} a^{3}} + \frac {4 \, x}{{\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {3 \, x}{{\left (a x^{2} + a\right )}^{\frac {5}{2}} a}\right )} \operatorname {arccot}\left (x\right ) - \frac {8}{15 \, \sqrt {a x^{2} + a} a^{3}} - \frac {4}{45 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {1}{25 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}} a} \] Input:

integrate(arccot(x)/(a*x^2+a)^(7/2),x, algorithm="maxima")
 

Output:

1/15*(8*x/(sqrt(a*x^2 + a)*a^3) + 4*x/((a*x^2 + a)^(3/2)*a^2) + 3*x/((a*x^ 
2 + a)^(5/2)*a))*arccot(x) - 8/15/(sqrt(a*x^2 + a)*a^3) - 4/45/((a*x^2 + a 
)^(3/2)*a^2) - 1/25/((a*x^2 + a)^(5/2)*a)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {{\left (4 \, x^{2} {\left (\frac {2 \, x^{2}}{a} + \frac {5}{a}\right )} + \frac {15}{a}\right )} x \arctan \left (\frac {1}{x}\right )}{15 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}}} - \frac {120 \, {\left (a x^{2} + a\right )}^{2} + 20 \, {\left (a x^{2} + a\right )} a + 9 \, a^{2}}{225 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}} a^{3}} \] Input:

integrate(arccot(x)/(a*x^2+a)^(7/2),x, algorithm="giac")
 

Output:

1/15*(4*x^2*(2*x^2/a + 5/a) + 15/a)*x*arctan(1/x)/(a*x^2 + a)^(5/2) - 1/22 
5*(120*(a*x^2 + a)^2 + 20*(a*x^2 + a)*a + 9*a^2)/((a*x^2 + a)^(5/2)*a^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{7/2}} \,d x \] Input:

int(acot(x)/(a + a*x^2)^(7/2),x)
 

Output:

int(acot(x)/(a + a*x^2)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.56 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{7/2}} \, dx=\frac {\sqrt {a}\, \left (240 \mathit {atan} \left (\sqrt {x^{2}+1}+x \right ) x^{6}+720 \mathit {atan} \left (\sqrt {x^{2}+1}+x \right ) x^{4}+720 \mathit {atan} \left (\sqrt {x^{2}+1}+x \right ) x^{2}+240 \mathit {atan} \left (\sqrt {x^{2}+1}+x \right )+120 \sqrt {x^{2}+1}\, \mathit {atan} \left (\frac {1}{x}\right ) x^{5}+300 \sqrt {x^{2}+1}\, \mathit {atan} \left (\frac {1}{x}\right ) x^{3}+225 \sqrt {x^{2}+1}\, \mathit {atan} \left (\frac {1}{x}\right ) x +120 \mathit {atan} \left (\frac {1}{x}\right ) x^{6}+360 \mathit {atan} \left (\frac {1}{x}\right ) x^{4}+360 \mathit {atan} \left (\frac {1}{x}\right ) x^{2}+120 \mathit {atan} \left (\frac {1}{x}\right )-120 \sqrt {x^{2}+1}\, x^{4}-260 \sqrt {x^{2}+1}\, x^{2}-149 \sqrt {x^{2}+1}\right )}{225 a^{4} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )} \] Input:

int(acot(x)/(a*x^2+a)^(7/2),x)
 

Output:

(sqrt(a)*(240*atan(sqrt(x**2 + 1) + x)*x**6 + 720*atan(sqrt(x**2 + 1) + x) 
*x**4 + 720*atan(sqrt(x**2 + 1) + x)*x**2 + 240*atan(sqrt(x**2 + 1) + x) + 
 120*sqrt(x**2 + 1)*atan(1/x)*x**5 + 300*sqrt(x**2 + 1)*atan(1/x)*x**3 + 2 
25*sqrt(x**2 + 1)*atan(1/x)*x + 120*atan(1/x)*x**6 + 360*atan(1/x)*x**4 + 
360*atan(1/x)*x**2 + 120*atan(1/x) - 120*sqrt(x**2 + 1)*x**4 - 260*sqrt(x* 
*2 + 1)*x**2 - 149*sqrt(x**2 + 1)))/(225*a**4*(x**6 + 3*x**4 + 3*x**2 + 1) 
)