Integrand size = 14, antiderivative size = 79 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=-\frac {1}{9 a \left (a+a x^2\right )^{3/2}}-\frac {2}{3 a^2 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a+a x^2}} \] Output:
-1/9/a/(a*x^2+a)^(3/2)-2/3/a^2/(a*x^2+a)^(1/2)+1/3*x*arccot(x)/a/(a*x^2+a) ^(3/2)+2/3*x*arccot(x)/a^2/(a*x^2+a)^(1/2)
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.47 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {-7-6 x^2+\left (9 x+6 x^3\right ) \cot ^{-1}(x)}{9 a \left (a \left (1+x^2\right )\right )^{3/2}} \] Input:
Integrate[ArcCot[x]/(a + a*x^2)^(5/2),x]
Output:
(-7 - 6*x^2 + (9*x + 6*x^3)*ArcCot[x])/(9*a*(a*(1 + x^2))^(3/2))
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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.
\(\displaystyle \int \frac {\cot ^{-1}(x)}{\left (a x^2+a\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5432 |
\(\displaystyle \frac {2 \int \frac {\cot ^{-1}(x)}{\left (a x^2+a\right )^{3/2}}dx}{3 a}-\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}}\) |
\(\Big \downarrow \) 5430 |
\(\displaystyle -\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}}+\frac {2 \left (\frac {x \cot ^{-1}(x)}{a \sqrt {a x^2+a}}-\frac {1}{a \sqrt {a x^2+a}}\right )}{3 a}\) |
Input:
Int[ArcCot[x]/(a + a*x^2)^(5/2),x]
Output:
-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)
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]
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]
Time = 1.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99
method | result | size |
orering | \(\frac {\left (4 x^{5}+\frac {80}{9} x^{3}+\frac {44}{9} x \right ) \operatorname {arccot}\left (x \right )}{\left (a \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\left (6 x^{2}+7\right ) \left (x^{2}+1\right )^{2} \left (-\frac {1}{\left (x^{2}+1\right ) \left (a \,x^{2}+a \right )^{\frac {5}{2}}}-\frac {5 \,\operatorname {arccot}\left (x \right ) a x}{\left (a \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{9}\) | \(78\) |
risch | \(\frac {i x \left (2 x^{2}+3\right ) \ln \left (i x +1\right )}{6 a^{2} \left (x^{2}+1\right ) \sqrt {a \left (x^{2}+1\right )}}+\frac {-6 i x^{3} \ln \left (-i x +1\right )+6 \pi \,x^{3}-9 i \ln \left (-i x +1\right ) x +9 \pi x -12 x^{2}-14}{18 a^{2} \left (x^{2}+1\right ) \sqrt {a \left (x^{2}+1\right )}}\) | \(101\) |
default | \(-\frac {\left (i+3 \,\operatorname {arccot}\left (x \right )\right ) \left (x^{3}+3 i x^{2}-3 x -i\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{72 \left (x^{2}+1\right )^{2} a^{3}}+\frac {3 \left (\operatorname {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{8 a^{3} \left (x^{2}+1\right )}+\frac {3 \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\operatorname {arccot}\left (x \right )-i\right )}{8 a^{3} \left (x^{2}+1\right )}-\frac {\left (-i+3 \,\operatorname {arccot}\left (x \right )\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x^{3}-3 i x^{2}-3 x +i\right )}{72 \left (x^{4}+2 x^{2}+1\right ) a^{3}}\) | \(165\) |
Input:
int(arccot(x)/(a*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(4*x^5+80/9*x^3+44/9*x)*arccot(x)/(a*x^2+a)^(5/2)+1/9*(6*x^2+7)*(x^2+1)^2* (-1/(x^2+1)/(a*x^2+a)^(5/2)-5*arccot(x)/(a*x^2+a)^(7/2)*a*x)
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a x^{2} + a} {\left (6 \, x^{2} - 3 \, {\left (2 \, x^{3} + 3 \, x\right )} \operatorname {arccot}\left (x\right ) + 7\right )}}{9 \, {\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \] Input:
integrate(arccot(x)/(a*x^2+a)^(5/2),x, algorithm="fricas")
Output:
-1/9*sqrt(a*x^2 + a)*(6*x^2 - 3*(2*x^3 + 3*x)*arccot(x) + 7)/(a^3*x^4 + 2* a^3*x^2 + a^3)
\[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(acot(x)/(a*x**2+a)**(5/2),x)
Output:
Integral(acot(x)/(a*(x**2 + 1))**(5/2), x)
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {a x^{2} + a} a^{2}} + \frac {x}{{\left (a x^{2} + a\right )}^{\frac {3}{2}} a}\right )} \operatorname {arccot}\left (x\right ) - \frac {2}{3 \, \sqrt {a x^{2} + a} a^{2}} - \frac {1}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a} \] Input:
integrate(arccot(x)/(a*x^2+a)^(5/2),x, algorithm="maxima")
Output:
1/3*(2*x/(sqrt(a*x^2 + a)*a^2) + x/((a*x^2 + a)^(3/2)*a))*arccot(x) - 2/3/ (sqrt(a*x^2 + a)*a^2) - 1/9/((a*x^2 + a)^(3/2)*a)
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {2 \, x^{2}}{a} + \frac {3}{a}\right )} \arctan \left (\frac {1}{x}\right )}{3 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}}} - \frac {6 \, a x^{2} + 7 \, a}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} \] Input:
integrate(arccot(x)/(a*x^2+a)^(5/2),x, algorithm="giac")
Output:
1/3*x*(2*x^2/a + 3/a)*arctan(1/x)/(a*x^2 + a)^(3/2) - 1/9*(6*a*x^2 + 7*a)/ ((a*x^2 + a)^(3/2)*a^2)
Timed out. \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(acot(x)/(a + a*x^2)^(5/2),x)
Output:
int(acot(x)/(a + a*x^2)^(5/2), x)
Time = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.65 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {\sqrt {a}\, \left (12 \mathit {atan} \left (\sqrt {x^{2}+1}+x \right ) x^{4}+24 \mathit {atan} \left (\sqrt {x^{2}+1}+x \right ) x^{2}+12 \mathit {atan} \left (\sqrt {x^{2}+1}+x \right )+6 \sqrt {x^{2}+1}\, \mathit {atan} \left (\frac {1}{x}\right ) x^{3}+9 \sqrt {x^{2}+1}\, \mathit {atan} \left (\frac {1}{x}\right ) x +6 \mathit {atan} \left (\frac {1}{x}\right ) x^{4}+12 \mathit {atan} \left (\frac {1}{x}\right ) x^{2}+6 \mathit {atan} \left (\frac {1}{x}\right )-6 \sqrt {x^{2}+1}\, x^{2}-7 \sqrt {x^{2}+1}\right )}{9 a^{3} \left (x^{4}+2 x^{2}+1\right )} \] Input:
int(acot(x)/(a*x^2+a)^(5/2),x)
Output:
(sqrt(a)*(12*atan(sqrt(x**2 + 1) + x)*x**4 + 24*atan(sqrt(x**2 + 1) + x)*x **2 + 12*atan(sqrt(x**2 + 1) + x) + 6*sqrt(x**2 + 1)*atan(1/x)*x**3 + 9*sq rt(x**2 + 1)*atan(1/x)*x + 6*atan(1/x)*x**4 + 12*atan(1/x)*x**2 + 6*atan(1 /x) - 6*sqrt(x**2 + 1)*x**2 - 7*sqrt(x**2 + 1)))/(9*a**3*(x**4 + 2*x**2 + 1))