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

Optimal. Leaf size=43 \[ \frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0101861, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5147, 261} \[ \frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5147

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]], x_Symbol] :> Simp[x*ArcTan[(c*x)/Sqrt[a + b*x^2]], x] - D
ist[c, Int[x/Sqrt[a + b*x^2], x], x] /; FreeQ[{a, b, c}, x] && EqQ[b + c^2, 0]

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0195625, size = 43, normalized size = 1. \[ \frac{\sqrt{d+e x^2}}{\sqrt{-e}}+x \tan ^{-1}\left (\frac{\sqrt{-e} x}{\sqrt{d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.038, size = 84, normalized size = 2. \begin{align*} x\arctan \left ({x\sqrt{-e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) +{\frac{{x}^{2}}{3\,d}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{2}{3\,e}\sqrt{-e}\sqrt{e{x}^{2}+d}}-{\frac{1}{3\,de}\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)

[Out]

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

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Maxima [B]  time = 1.00229, size = 104, normalized size = 2.42 \begin{align*} x \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} \sqrt{-e}}{3 \, d e} + \frac{{\left ({\left (e x^{2} + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x^{2} + d} d\right )} \sqrt{-e}}{3 \, d e} \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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.36114, size = 96, normalized size = 2.23 \begin{align*} \frac{e x \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - \sqrt{e x^{2} + d} \sqrt{-e}}{e} \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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.768269, size = 39, normalized size = 0.91 \begin{align*} \begin{cases} i x \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )} - \frac{i \sqrt{d + e x^{2}}}{\sqrt{e}} & \text{for}\: e \neq 0 \\0 & \text{otherwise} \end{cases} \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)

[Out]

Piecewise((I*x*atanh(sqrt(e)*x/sqrt(d + e*x**2)) - I*sqrt(d + e*x**2)/sqrt(e), Ne(e, 0)), (0, True))

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Giac [A]  time = 1.13894, size = 55, normalized size = 1.28 \begin{align*} x \arctan \left (\frac{x \sqrt{-e}}{\sqrt{x^{2} e + d}}\right ) - \sqrt{-x^{2} e^{2} - d e} e^{\left (-1\right )} \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, algorithm="giac")

[Out]

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