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]

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

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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]

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]

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*}

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Mathematica [A]  time = 0.04, 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]

-1/2*(Sqrt[-e]*x*Sqrt[d + e*x^2] + d*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/(d*x^2)

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fricas [A]  time = 0.62, size = 44, normalized size = 0.77 \[ -\frac {\sqrt {e x^{2} + d} \sqrt {-e} x + d \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{2 \, d x^{2}} \]

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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giac [B]  time = 0.26, size = 105, normalized size = 1.84 \[ \frac {x e^{3}}{4 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )} d} - \frac {{\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )} e^{\left (-1\right )}}{4 \, d x} - \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{2 \, x^{2}} \]

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

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maple [A]  time = 0.04, size = 67, normalized size = 1.18 \[ -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{2 x^{2}}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d^{2} x}+\frac {\sqrt {-e}\, e x \sqrt {e \,x^{2}+d}}{2 d^{2}} \]

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 = 0.37, size = 58, normalized size = 1.02 \[ -\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{2 \, x^{2}} - \frac {\sqrt {-e} e x^{2} + d \sqrt {-e}}{2 \, \sqrt {e x^{2} + d} d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x^3,x)

[Out]

int(atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2))/x^3, x)

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sympy [A]  time = 4.60, size = 53, normalized size = 0.93 \[ - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{2 x^{2}} - \frac {\sqrt {e} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{2 d} \]

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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