3.8 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^5} \, dx\)

Optimal. Leaf size=85 \[ -\frac{(-e)^{3/2} \sqrt{d+e x^2}}{6 d^2 x}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{12 d x^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{4 x^4} \]

[Out]

-(Sqrt[-e]*Sqrt[d + e*x^2])/(12*d*x^3) - ((-e)^(3/2)*Sqrt[d + e*x^2])/(6*d^2*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d +
e*x^2]]/(4*x^4)

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Rubi [A]  time = 0.0281823, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5151, 271, 264} \[ -\frac{(-e)^{3/2} \sqrt{d+e x^2}}{6 d^2 x}-\frac{\sqrt{-e} \sqrt{d+e x^2}}{12 d x^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^5,x]

[Out]

-(Sqrt[-e]*Sqrt[d + e*x^2])/(12*d*x^3) - ((-e)^(3/2)*Sqrt[d + e*x^2])/(6*d^2*x) - ArcTan[(Sqrt[-e]*x)/Sqrt[d +
e*x^2]]/(4*x^4)

Rule 5151

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*ArcTa
n[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; F
reeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{x^5} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{4 x^4}+\frac{1}{4} \sqrt{-e} \int \frac{1}{x^4 \sqrt{d+e x^2}} \, dx\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{12 d x^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{4 x^4}+\frac{(-e)^{3/2} \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{6 d}\\ &=-\frac{\sqrt{-e} \sqrt{d+e x^2}}{12 d x^3}-\frac{(-e)^{3/2} \sqrt{d+e x^2}}{6 d^2 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0441768, size = 67, normalized size = 0.79 \[ \frac{\sqrt{-e} x \sqrt{d+e x^2} \left (2 e x^2-d\right )-3 d^2 \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{12 d^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]]/x^5,x]

[Out]

(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-d + 2*e*x^2) - 3*d^2*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(12*d^2*x^4)

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Maple [A]  time = 0.039, size = 69, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{4}}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }+{\frac{e}{4\,{d}^{2}x}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{1}{12\,{d}^{2}{x}^{3}}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^5,x)

[Out]

-1/4*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^4+1/4*(-e)^(1/2)*e/d^2/x*(e*x^2+d)^(1/2)-1/12*(-e)^(1/2)/d^2/x^3*(
e*x^2+d)^(3/2)

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Maxima [A]  time = 1.01178, size = 92, normalized size = 1.08 \begin{align*} \frac{\sqrt{e x^{2} + d} \sqrt{-e} e}{4 \, d^{2} x} - \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} \sqrt{-e}}{12 \, d^{2} x^{3}} - \frac{\arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^5,x, algorithm="maxima")

[Out]

1/4*sqrt(e*x^2 + d)*sqrt(-e)*e/(d^2*x) - 1/12*(e*x^2 + d)^(3/2)*sqrt(-e)/(d^2*x^3) - 1/4*arctan(sqrt(-e)*x/sqr
t(e*x^2 + d))/x^4

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Fricas [A]  time = 2.5242, size = 139, normalized size = 1.64 \begin{align*} -\frac{3 \, d^{2} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) -{\left (2 \, e x^{3} - d x\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{12 \, d^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^5,x, algorithm="fricas")

[Out]

-1/12*(3*d^2*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (2*e*x^3 - d*x)*sqrt(e*x^2 + d)*sqrt(-e))/(d^2*x^4)

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Sympy [A]  time = 5.53503, size = 83, normalized size = 0.98 \begin{align*} - \frac{\operatorname{atan}{\left (\frac{x \sqrt{- e}}{\sqrt{d + e x^{2}}} \right )}}{4 x^{4}} - \frac{\sqrt{e} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{12 d x^{2}} + \frac{e^{\frac{3}{2}} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{6 d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(x*(-e)**(1/2)/(e*x**2+d)**(1/2))/x**5,x)

[Out]

-atan(x*sqrt(-e)/sqrt(d + e*x**2))/(4*x**4) - sqrt(e)*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(12*d*x**2) + e**(3/2)*sqr
t(-e)*sqrt(d/(e*x**2) + 1)/(6*d**2)

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Giac [B]  time = 1.20594, size = 267, normalized size = 3.14 \begin{align*} -\frac{x^{3}{\left (\frac{9 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{2} e^{\left (-2\right )}}{x^{2}} + e^{2}\right )} e^{6}}{96 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{3} d^{2}} - \frac{\arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right )}{4 \, x^{4}} + \frac{{\left (\frac{9 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )} d^{4} e^{6}}{x} + \frac{{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )}^{3} d^{4} e^{2}}{x^{3}}\right )} e^{\left (-6\right )}}{96 \, d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^5,x, algorithm="giac")

[Out]

-1/96*x^3*(9*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^2*e^(-2)/x^2 + e^2)*e^6/((sqrt(-x^2*e^2 - d*e)*e - sqrt(-
d*e)*e)^3*d^2) - 1/4*arctan(x*sqrt(-e)/sqrt(x^2*e + d))/x^4 + 1/96*(9*(sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)*
d^4*e^6/x + (sqrt(-x^2*e^2 - d*e)*e - sqrt(-d*e)*e)^3*d^4*e^2/x^3)*e^(-6)/d^6