3.2.34 \(\int \frac {d+e x^2}{d^2-e^2 x^4} \, dx\)

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

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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 = {1150, 208} \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 208

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

Rule 1150

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p + q)*(a/d + (c*x^
2)/e)^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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^2}{d^2-e^2 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.09, size = 68, normalized size = 2.83 \begin {gather*} \left [\frac {\sqrt {d e} \log \left (\frac {e x^{2} + 2 \, \sqrt {d e} x + d}{e x^{2} - d}\right )}{2 \, d e}, -\frac {\sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} x}{d}\right )}{d e}\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*sqrt(d*e)*log((e*x^2 + 2*sqrt(d*e)*x + d)/(e*x^2 - d))/(d*e), -sqrt(-d*e)*arctan(sqrt(-d*e)*x/d)/(d*e)]

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giac [B]  time = 0.29, size = 116, normalized size = 4.83 \begin {gather*} \frac {{\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {7}{2}} - {\left (d^{2}\right )}^{\frac {1}{4}} {\left | d \right |} e^{\frac {7}{2}}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{{\left (d^{2}\right )}^{\frac {1}{4}}}\right ) e^{\left (-4\right )}}{2 \, d^{2}} + \frac {{\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {11}{2}} + {\left (d^{2}\right )}^{\frac {3}{4}} e^{\frac {11}{2}}\right )} e^{\left (-6\right )} \log \left ({\left | {\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right )}{4 \, d^{2}} - \frac {{\left ({\left (d^{2}\right )}^{\frac {1}{4}} d e^{\frac {7}{2}} + {\left (d^{2}\right )}^{\frac {1}{4}} {\left | d \right |} e^{\frac {7}{2}}\right )} e^{\left (-4\right )} \log \left ({\left | -{\left (d^{2}\right )}^{\frac {1}{4}} e^{\left (-\frac {1}{2}\right )} + x \right |}\right )}{4 \, d^{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, algorithm="giac")

[Out]

1/2*((d^2)^(1/4)*d*e^(7/2) - (d^2)^(1/4)*abs(d)*e^(7/2))*arctan(x*e^(1/2)/(d^2)^(1/4))*e^(-4)/d^2 + 1/4*((d^2)
^(1/4)*d*e^(11/2) + (d^2)^(3/4)*e^(11/2))*e^(-6)*log(abs((d^2)^(1/4)*e^(-1/2) + x))/d^2 - 1/4*((d^2)^(1/4)*d*e
^(7/2) + (d^2)^(1/4)*abs(d)*e^(7/2))*e^(-4)*log(abs(-(d^2)^(1/4)*e^(-1/2) + x))/d^2

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maple [A]  time = 0.00, size = 16, normalized size = 0.67 \begin {gather*} \frac {\arctanh \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.35, size = 31, normalized size = 1.29 \begin {gather*} -\frac {\log \left (\frac {e x - \sqrt {d e}}{e x + \sqrt {d e}}\right )}{2 \, \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="maxima")

[Out]

-1/2*log((e*x - sqrt(d*e))/(e*x + 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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sympy [B]  time = 0.15, 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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