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

Optimal. Leaf size=79 \[ -\frac{2}{3 a^2 \sqrt{a x^2+a}}+\frac{2 x \cot ^{-1}(x)}{3 a^2 \sqrt{a x^2+a}}-\frac{1}{9 a \left (a x^2+a\right )^{3/2}}+\frac{x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}} \]

[Out]

-1/(9*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])

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Rubi [A]  time = 0.0440098, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4897, 4895} \[ -\frac{2}{3 a^2 \sqrt{a x^2+a}}+\frac{2 x \cot ^{-1}(x)}{3 a^2 \sqrt{a x^2+a}}-\frac{1}{9 a \left (a x^2+a\right )^{3/2}}+\frac{x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]/(a + a*x^2)^(5/2),x]

[Out]

-1/(9*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 4897

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] - S
imp[(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]

Rule 4895

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*} \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{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]  time = 0.0305638, size = 37, normalized size = 0.47 \[ \frac{-6 x^2+\left (6 x^3+9 x\right ) \cot ^{-1}(x)-7}{9 a \left (a \left (x^2+1\right )\right )^{3/2}} \]

Antiderivative was successfully verified.

[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))

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Maple [C]  time = 0.386, size = 165, normalized size = 2.1 \begin{align*} -{\frac{ \left ( i+3\,{\rm arccot} \left (x\right ) \right ) \left ( 3\,i{x}^{2}+{x}^{3}-i-3\,x \right ) }{72\, \left ({x}^{2}+1 \right ) ^{2}{a}^{3}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}+{\frac{ \left ( 3\,{\rm arccot} \left (x\right )+3\,i \right ) \left ( x+i \right ) }{8\,{a}^{3} \left ({x}^{2}+1 \right ) }\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}+{\frac{ \left ( 3\,x-3\,i \right ) \left ({\rm arccot} \left (x\right )-i \right ) }{8\,{a}^{3} \left ({x}^{2}+1 \right ) }\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}-{\frac{ \left ( -i+3\,{\rm arccot} \left (x\right ) \right ) \left ( -3\,x-3\,i{x}^{2}+{x}^{3}+i \right ) }{ \left ( 72\,{x}^{4}+144\,{x}^{2}+72 \right ){a}^{3}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(I+3*arccot(x))*(3*I*x^2+x^3-I-3*x)*(a*(x+I)*(x-I))^(1/2)/(x^2+1)^2/a^3+3/8*(arccot(x)+I)*(x+I)*(a*(x+I)
*(x-I))^(1/2)/a^3/(x^2+1)+3/8*(a*(x+I)*(x-I))^(1/2)*(x-I)*(arccot(x)-I)/a^3/(x^2+1)-1/72*(-I+3*arccot(x))*(a*(
x+I)*(x-I))^(1/2)*(-3*x-3*I*x^2+x^3+I)/(x^4+2*x^2+1)/a^3

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Maxima [A]  time = 1.45611, size = 85, normalized size = 1.08 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.20266, size = 122, normalized size = 1.54 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14799, size = 74, normalized size = 0.94 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)