3.7 \(\int \frac{\tan ^{-1}(\frac{\sqrt{-e} x}{\sqrt{d+e x^2}})}{x^3} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\sqrt{-e} \sqrt{d+e x^2}}{2 d x}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0187291, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {5151, 264} \[ -\frac{\sqrt{-e} \sqrt{d+e x^2}}{2 d x}-\frac{\tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0363215, size = 54, normalized size = 0.95 \[ -\frac{\sqrt{-e} x \sqrt{d+e x^2}+d \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right )}{2 d x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 67, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,{x}^{2}}\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }-{\frac{1}{2\,{d}^{2}x}\sqrt{-e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{ex}{2\,{d}^{2}}\sqrt{-e}\sqrt{e{x}^{2}+d}} \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^3,x)

[Out]

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

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Maxima [A]  time = 1.14299, size = 78, normalized size = 1.37 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{2 \, x^{2}} - \frac{\sqrt{-e} e x^{2} + d \sqrt{-e}}{2 \, \sqrt{e x^{2} + d} d x} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.35543, size = 111, normalized size = 1.95 \begin{align*} -\frac{\sqrt{e x^{2} + d} \sqrt{-e} x + d \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right )}{2 \, d x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.24586, size = 53, normalized size = 0.93 \begin{align*} - \frac{\operatorname{atan}{\left (\frac{x \sqrt{- e}}{\sqrt{d + e x^{2}}} \right )}}{2 x^{2}} - \frac{\sqrt{e} \sqrt{- e} \sqrt{\frac{d}{e x^{2}} + 1}}{2 d} \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**3,x)

[Out]

-atan(x*sqrt(-e)/sqrt(d + e*x**2))/(2*x**2) - sqrt(e)*sqrt(-e)*sqrt(d/(e*x**2) + 1)/(2*d)

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Giac [B]  time = 1.2364, size = 142, normalized size = 2.49 \begin{align*} \frac{x e^{3}}{4 \,{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )} d} - \frac{{\left (\sqrt{-x^{2} e^{2} - d e} e - \sqrt{-d e} e\right )} e^{\left (-1\right )}}{4 \, d x} - \frac{\arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right )}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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