\(\int \frac {\cot ^{-1}(x)}{(a+a x^2)^{3/2}} \, dx\) [67]

Optimal result

Integrand size = 14, antiderivative size = 35 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=-\frac {1}{a \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{a \sqrt {a+a x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5015} \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {x \cot ^{-1}(x)}{a \sqrt {a x^2+a}}-\frac {1}{a \sqrt {a x^2+a}} \]

[In]

Int[ArcCot[x]/(a + a*x^2)^(3/2),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{a \sqrt {a+a x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {-1+x \cot ^{-1}(x)}{a \sqrt {a \left (1+x^2\right )}} \]

[In]

Integrate[ArcCot[x]/(a + a*x^2)^(3/2),x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57

method	result	size
risch	\(\frac {i x \ln \left (i x +1\right )}{2 a \sqrt {a \left (x^{2}+1\right )}}+\frac {-i x \ln \left (-i x +1\right )+\pi x -2}{2 a \sqrt {a \left (x^{2}+1\right )}}\)	\(55\)
default	\(\frac {\left (\operatorname {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{2 \left (x^{2}+1\right ) a^{2}}+\frac {\sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\operatorname {arccot}\left (x \right )-i\right )}{2 \left (x^{2}+1\right ) a^{2}}\)	\(68\)

[In]

int(arccot(x)/(a*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*I/a*x/(a*(x^2+1))^(1/2)*ln(1+I*x)+1/2/a*(-I*x*ln(1-I*x)+Pi*x-2)/(a*(x^2+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {\sqrt {a x^{2} + a} {\left (x \operatorname {arccot}\left (x\right ) - 1\right )}}{a^{2} x^{2} + a^{2}} \]

[In]

integrate(arccot(x)/(a*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(acot(x)/(a*x**2+a)**(3/2),x)

[Out]

Integral(acot(x)/(a*(x**2 + 1))**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {x \operatorname {arccot}\left (x\right )}{\sqrt {a x^{2} + a} a} - \frac {1}{\sqrt {a x^{2} + a} a} \]

[In]

integrate(arccot(x)/(a*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\frac {x \arctan \left (\frac {1}{x}\right )}{\sqrt {a x^{2} + a} a} - \frac {1}{\sqrt {a x^{2} + a} a} \]

[In]

integrate(arccot(x)/(a*x^2+a)^(3/2),x, algorithm="giac")

[Out]

x*arctan(1/x)/(sqrt(a*x^2 + a)*a) - 1/(sqrt(a*x^2 + a)*a)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{3/2}} \,d x \]

[In]

int(acot(x)/(a + a*x^2)^(3/2),x)

[Out]

int(acot(x)/(a + a*x^2)^(3/2), x)