3.2.92 \(\int \frac {d+e x^2}{d^2-e^2 x^4} \, dx\) [192]

Optimal. Leaf size=24 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1164, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)/(d^2 - e^2*x^4),x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(Sqrt[d]*Sqrt[e])

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Maple [A]
time = 0.12, size = 16, normalized size = 0.67

method	result	size
default	\(\frac {\arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}\)	\(16\)
risch	\(\frac {\ln \left (e x +\sqrt {d e}\right )}{2 \sqrt {d e}}-\frac {\ln \left (-e x +\sqrt {d e}\right )}{2 \sqrt {d e}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).
time = 0.52, size = 34, normalized size = 1.42 \begin {gather*} -\frac {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 )}{2 \, \sqrt {d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 65, normalized size = 2.71 \begin {gather*} \left [\frac {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 )}{2 \, \sqrt {d}}, -\frac {\sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} x}{d}\right ) e^{\left (-1\right )}}{d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
time = 0.05, size = 46, normalized size = 1.92 \begin {gather*} - \frac {\sqrt {\frac {1}{d e}} \log {\left (- d \sqrt {\frac {1}{d e}} + x \right )}}{2} + \frac {\sqrt {\frac {1}{d e}} \log {\left (d \sqrt {\frac {1}{d e}} + x \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.04, size = 21, normalized size = 0.88 \begin {gather*} -\frac {\arctan \left (\frac {x e}{\sqrt {-d e}}\right )}{\sqrt {-d e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.06, size = 16, normalized size = 0.67 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )}{\sqrt {d}\,\sqrt {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2)/(d^2 - e^2*x^4),x)

[Out]

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

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