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

Optimal. Leaf size=118 \[ -\frac{8}{15 a^3 \sqrt{a x^2+a}}-\frac{4}{45 a^2 \left (a x^2+a\right )^{3/2}}+\frac{8 x \cot ^{-1}(x)}{15 a^3 \sqrt{a x^2+a}}+\frac{4 x \cot ^{-1}(x)}{15 a^2 \left (a x^2+a\right )^{3/2}}-\frac{1}{25 a \left (a x^2+a\right )^{5/2}}+\frac{x \cot ^{-1}(x)}{5 a \left (a x^2+a\right )^{5/2}} \]

[Out]

-1/(25*a*(a + a*x^2)^(5/2)) - 4/(45*a^2*(a + a*x^2)^(3/2)) - 8/(15*a^3*Sqrt[a + a*x^2]) + (x*ArcCot[x])/(5*a*(
a + a*x^2)^(5/2)) + (4*x*ArcCot[x])/(15*a^2*(a + a*x^2)^(3/2)) + (8*x*ArcCot[x])/(15*a^3*Sqrt[a + a*x^2])

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Rubi [A]  time = 0.0670812, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4897, 4895} \[ -\frac{8}{15 a^3 \sqrt{a x^2+a}}-\frac{4}{45 a^2 \left (a x^2+a\right )^{3/2}}+\frac{8 x \cot ^{-1}(x)}{15 a^3 \sqrt{a x^2+a}}+\frac{4 x \cot ^{-1}(x)}{15 a^2 \left (a x^2+a\right )^{3/2}}-\frac{1}{25 a \left (a x^2+a\right )^{5/2}}+\frac{x \cot ^{-1}(x)}{5 a \left (a x^2+a\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]/(a + a*x^2)^(7/2),x]

[Out]

-1/(25*a*(a + a*x^2)^(5/2)) - 4/(45*a^2*(a + a*x^2)^(3/2)) - 8/(15*a^3*Sqrt[a + a*x^2]) + (x*ArcCot[x])/(5*a*(
a + a*x^2)^(5/2)) + (4*x*ArcCot[x])/(15*a^2*(a + a*x^2)^(3/2)) + (8*x*ArcCot[x])/(15*a^3*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 )^{7/2}} \, dx &=-\frac{1}{25 a \left (a+a x^2\right )^{5/2}}+\frac{x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac{4 \int \frac{\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx}{5 a}\\ &=-\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{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 \int \frac{\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx}{15 a^2}\\ &=-\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}}\\ \end{align*}

Mathematica [A]  time = 0.0378354, size = 47, normalized size = 0.4 \[ \frac{-120 x^4-260 x^2+15 \left (8 x^4+20 x^2+15\right ) x \cot ^{-1}(x)-149}{225 a \left (a \left (x^2+1\right )\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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))

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Maple [C]  time = 0.455, size = 289, normalized size = 2.5 \begin{align*}{\frac{ \left ( i+5\,{\rm arccot} \left (x\right ) \right ) \left ( 5\,i{x}^{4}+{x}^{5}-10\,i{x}^{2}-10\,{x}^{3}+i+5\,x \right ) }{800\, \left ({x}^{2}+1 \right ) ^{3}{a}^{4}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}-{\frac{ \left ( 5\,i+15\,{\rm arccot} \left (x\right ) \right ) \left ( 3\,i{x}^{2}+{x}^{3}-i-3\,x \right ) }{288\,{a}^{4} \left ({x}^{2}+1 \right ) ^{2}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}+{\frac{ \left ( 5\,{\rm arccot} \left (x\right )+5\,i \right ) \left ( x+i \right ) }{ \left ( 16\,{x}^{2}+16 \right ){a}^{4}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}+{\frac{ \left ( 5\,x-5\,i \right ) \left ({\rm arccot} \left (x\right )-i \right ) }{ \left ( 16\,{x}^{2}+16 \right ){a}^{4}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}-{\frac{ \left ( -5\,i+15\,{\rm arccot} \left (x\right ) \right ) \left ( -3\,x-3\,i{x}^{2}+{x}^{3}+i \right ) }{ \left ( 288\,{x}^{4}+576\,{x}^{2}+288 \right ){a}^{4}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}+{\frac{ \left ( -i+5\,{\rm arccot} \left (x\right ) \right ) \left ( -10\,{x}^{3}-5\,i{x}^{4}+{x}^{5}+5\,x+10\,i{x}^{2}-i \right ) }{ \left ( 800\,{x}^{6}+2400\,{x}^{4}+2400\,{x}^{2}+800 \right ){a}^{4}}\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)^(7/2),x)

[Out]

1/800*(I+5*arccot(x))*(5*I*x^4+x^5-10*I*x^2-10*x^3+I+5*x)*(a*(x+I)*(x-I))^(1/2)/(x^2+1)^3/a^4-5/288*(I+3*arcco
t(x))*(3*I*x^2+x^3-I-3*x)*(a*(x+I)*(x-I))^(1/2)/a^4/(x^2+1)^2+5/16*(arccot(x)+I)*(x+I)*(a*(x+I)*(x-I))^(1/2)/(
x^2+1)/a^4+5/16*(a*(x+I)*(x-I))^(1/2)*(x-I)*(arccot(x)-I)/(x^2+1)/a^4-5/288*(-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^4+1/800*(-I+5*arccot(x))*(a*(x+I)*(x-I))^(1/2)*(-10*x^3-5*I*x^4+x^5+
5*x+10*I*x^2-I)/(x^6+3*x^4+3*x^2+1)/a^4

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Maxima [A]  time = 1.53891, size = 126, normalized size = 1.07 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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/(s
qrt(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)

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Fricas [A]  time = 2.19193, size = 174, normalized size = 1.47 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.14537, size = 112, normalized size = 0.95 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(4*x^2*(2*x^2/a + 5/a) + 15/a)*x*arctan(1/x)/(a*x^2 + a)^(5/2) - 1/225*(120*(a*x^2 + a)^2 + 20*(a*x^2 + a
)*a + 9*a^2)/((a*x^2 + a)^(5/2)*a^3)