3.2.91 \(\int \frac {(d+e x^2)^2}{d^2-e^2 x^4} \, dx\) [191]

Optimal. Leaf size=29 \[ -x+\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1164, 396, 214} \begin {gather*} \frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^2/(d^2 - e^2*x^4),x]

[Out]

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 396

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

Rule 1164

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p + q)*(a/d + (c/e)
*x^2)^p, x] /; FreeQ[{a, c, d, e, q}, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^2}{d^2-e^2 x^4} \, dx &=\int \frac {d+e x^2}{d-e x^2} \, dx\\ &=-x+(2 d) \int \frac {1}{d-e x^2} \, dx\\ &=-x+\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} -x+\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^2/(d^2 - e^2*x^4),x]

[Out]

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

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Maple [A]
time = 0.13, size = 22, normalized size = 0.76

method	result	size
default	\(-x +\frac {2 d \arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}\)	\(22\)
risch	\(-x -\frac {\sqrt {d e}\, \ln \left (\sqrt {d e}\, x -d \right )}{e}+\frac {\sqrt {d e}\, \ln \left (-\sqrt {d e}\, x -d \right )}{e}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2/(-e^2*x^4+d^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.52, size = 38, normalized size = 1.31 \begin {gather*} -\sqrt {d} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {x e - \sqrt {d} e^{\frac {1}{2}}}{x e + \sqrt {d} e^{\frac {1}{2}}}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2/(-e^2*x^4+d^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 69, normalized size = 2.38 \begin {gather*} \left [\sqrt {d} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {x^{2} e + 2 \, \sqrt {d} x e^{\frac {1}{2}} + d}{x^{2} e - d}\right ) - x, -2 \, \sqrt {-d e^{\left (-1\right )}} \arctan \left (\frac {\sqrt {-d e^{\left (-1\right )}} x e}{d}\right ) - x\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2/(-e^2*x^4+d^2),x, algorithm="fricas")

[Out]

[sqrt(d)*e^(-1/2)*log((x^2*e + 2*sqrt(d)*x*e^(1/2) + d)/(x^2*e - d)) - x, -2*sqrt(-d*e^(-1))*arctan(sqrt(-d*e^
(-1))*x*e/d) - x]

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Sympy [A]
time = 0.06, size = 34, normalized size = 1.17 \begin {gather*} - x - \sqrt {\frac {d}{e}} \log {\left (x - \sqrt {\frac {d}{e}} \right )} + \sqrt {\frac {d}{e}} \log {\left (x + \sqrt {\frac {d}{e}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2/(-e**2*x**4+d**2),x)

[Out]

-x - sqrt(d/e)*log(x - sqrt(d/e)) + sqrt(d/e)*log(x + sqrt(d/e))

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Giac [A]
time = 4.68, size = 26, normalized size = 0.90 \begin {gather*} -\frac {2 \, d \arctan \left (\frac {x e}{\sqrt {-d e}}\right )}{\sqrt {-d e}} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2/(-e^2*x^4+d^2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.43, size = 21, normalized size = 0.72 \begin {gather*} \frac {2\,\sqrt {d}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{\sqrt {e}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)^2/(d^2 - e^2*x^4),x)

[Out]

(2*d^(1/2)*atanh((e^(1/2)*x)/d^(1/2)))/e^(1/2) - x

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