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

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

[Out]

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

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Rubi [A]  time = 0.0122413, antiderivative size = 24, normalized size of antiderivative = 1., 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} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}} \]

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 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]

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]

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*}

Mathematica [A]  time = 0.0043265, size = 24, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}} \]

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 = 0.003, size = 16, normalized size = 0.7 \begin{align*}{{\it Artanh} \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.90441, size = 151, normalized size = 6.29 \begin{align*} \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{align*}

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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Sympy [B]  time = 0.186596, size = 46, normalized size = 1.92 \begin{align*} - \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{align*}

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 [B]  time = 1.16564, size = 157, normalized size = 6.54 \begin{align*} \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{align*}

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