\(\int \frac {\cot ^{-1}(x)}{(a+a x^2)^{5/2}} \, dx\) [68]

Optimal result

Integrand size = 14, antiderivative size = 79 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=-\frac {1}{9 a \left (a+a x^2\right )^{3/2}}-\frac {2}{3 a^2 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a+a x^2}} \]

[Out]

-1/9/a/(a*x^2+a)^(3/2)+1/3*x*arccot(x)/a/(a*x^2+a)^(3/2)-2/3/a^2/(a*x^2+a)^(1/2)+2/3*x*arccot(x)/a^2/(a*x^2+a)
^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5017, 5015} \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=-\frac {2}{3 a^2 \sqrt {a x^2+a}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a x^2+a}}-\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}} \]

[In]

Int[ArcCot[x]/(a + a*x^2)^(5/2),x]

[Out]

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

Rule 5015

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

Rule 5017

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] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcCot[c*x]), x], x] -
Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcCot[c*x])/(2*d*(q + 1))), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^
2*d] && LtQ[q, -1] && NeQ[q, -3/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{9 a \left (a+a x^2\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx}{3 a} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.47 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {-7-6 x^2+\left (9 x+6 x^3\right ) \cot ^{-1}(x)}{9 a \left (a \left (1+x^2\right )\right )^{3/2}} \]

[In]

Integrate[ArcCot[x]/(a + a*x^2)^(5/2),x]

[Out]

(-7 - 6*x^2 + (9*x + 6*x^3)*ArcCot[x])/(9*a*(a*(1 + x^2))^(3/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.28

method	result	size
risch	\(\frac {i x \left (2 x^{2}+3\right ) \ln \left (i x +1\right )}{6 a^{2} \left (x^{2}+1\right ) \sqrt {a \left (x^{2}+1\right )}}+\frac {-6 i x^{3} \ln \left (-i x +1\right )+6 \pi \,x^{3}-9 i x \ln \left (-i x +1\right )+9 \pi x -12 x^{2}-14}{18 a^{2} \left (x^{2}+1\right ) \sqrt {a \left (x^{2}+1\right )}}\)	\(101\)
default	\(-\frac {\left (i+3 \,\operatorname {arccot}\left (x \right )\right ) \left (x^{3}+3 i x^{2}-3 x -i\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{72 \left (x^{2}+1\right )^{2} a^{3}}+\frac {3 \left (\operatorname {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{8 a^{3} \left (x^{2}+1\right )}+\frac {3 \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\operatorname {arccot}\left (x \right )-i\right )}{8 a^{3} \left (x^{2}+1\right )}-\frac {\left (-i+3 \,\operatorname {arccot}\left (x \right )\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x^{3}-3 i x^{2}-3 x +i\right )}{72 \left (x^{4}+2 x^{2}+1\right ) a^{3}}\)	\(165\)

[In]

int(arccot(x)/(a*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6*I/a^2*x*(2*x^2+3)/(x^2+1)/(a*(x^2+1))^(1/2)*ln(1+I*x)+1/18/a^2*(-6*I*x^3*ln(1-I*x)+6*Pi*x^3-9*I*x*ln(1-I*x
)+9*Pi*x-12*x^2-14)/(x^2+1)/(a*(x^2+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a x^{2} + a} {\left (6 \, x^{2} - 3 \, {\left (2 \, x^{3} + 3 \, x\right )} \operatorname {arccot}\left (x\right ) + 7\right )}}{9 \, {\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \]

[In]

integrate(arccot(x)/(a*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

-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)

Sympy [F]

\[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(acot(x)/(a*x**2+a)**(5/2),x)

[Out]

Integral(acot(x)/(a*(x**2 + 1))**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {a x^{2} + a} a^{2}} + \frac {x}{{\left (a x^{2} + a\right )}^{\frac {3}{2}} a}\right )} \operatorname {arccot}\left (x\right ) - \frac {2}{3 \, \sqrt {a x^{2} + a} a^{2}} - \frac {1}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a} \]

[In]

integrate(arccot(x)/(a*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\frac {x {\left (\frac {2 \, x^{2}}{a} + \frac {3}{a}\right )} \arctan \left (\frac {1}{x}\right )}{3 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}}} - \frac {6 \, a x^{2} + 7 \, a}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} \]

[In]

integrate(arccot(x)/(a*x^2+a)^(5/2),x, algorithm="giac")

[Out]

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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{5/2}} \,d x \]

[In]

int(acot(x)/(a + a*x^2)^(5/2),x)

[Out]

int(acot(x)/(a + a*x^2)^(5/2), x)