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]

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

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 1.00 \[ \frac {\sqrt {d+e x^2}}{\sqrt {-e}}+x \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]

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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fricas [A]  time = 0.44, size = 41, normalized size = 0.95 \[ \frac {e x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \sqrt {e x^{2} + d} \sqrt {-e}}{e} \]

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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giac [A]  time = 0.18, size = 41, normalized size = 0.95 \[ x \arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right ) - \sqrt {-x^{2} e^{2} - d e} e^{\left (-1\right )} \]

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)

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maple [B]  time = 0.04, size = 84, normalized size = 1.95 \[ x \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )+\frac {\sqrt {-e}\, x^{2} \sqrt {e \,x^{2}+d}}{3 d}-\frac {2 \sqrt {-e}\, \sqrt {e \,x^{2}+d}}{3 e}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 d e} \]

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 = 0.34, size = 77, normalized size = 1.79 \[ x \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} \sqrt {-e}}{3 \, d e} + \frac {{\left ({\left (e x^{2} + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x^{2} + d} d\right )} \sqrt {-e}}{3 \, d e} \]

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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mupad [B]  time = 0.66, size = 35, normalized size = 0.81 \[ \frac {\sqrt {e\,x^2+d}}{\sqrt {-e}}+x\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.06, size = 39, normalized size = 0.91 \[ \begin {cases} i x \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )} - \frac {i \sqrt {d + e x^{2}}}{\sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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