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

Optimal. Leaf size=38 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.0256809, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1150, 377, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/(Sqrt[2]*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 377

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

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

Mathematica [A]  time = 0.156767, size = 38, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.027, size = 986, normalized size = 26. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x)

[Out]

1/2*e/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/(-d*e)^(1/2)*((x+(-d*e)^(1/2)/e)^2*e-2*(-d*e)^(1/2
)*(x+(-d*e)^(1/2)/e))^(1/2)-1/2*e^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))*ln(((x+(-d*e)^(1
/2)/e)*e-(-d*e)^(1/2))/e^(1/2)+((x+(-d*e)^(1/2)/e)^2*e-2*(-d*e)^(1/2)*(x+(-d*e)^(1/2)/e))^(1/2))-1/2*e/((-d*e)
^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))/(-d*e)^(1/2)*((x-(-d*e)^(1/2)/e)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(
1/2)/e))^(1/2)-1/2*e^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))*ln(((x-(-d*e)^(1/2)/e)*e+(-d*
e)^(1/2))/e^(1/2)+((x-(-d*e)^(1/2)/e)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2))-1/2*e/(d*e)^(1/2)/((-d*e)^
(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))*((x+(d*e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*d)^(1/2
)+1/2*e^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))*ln(((x+(d*e)^(1/2)/e)*e-(d*e)^(1/2))/e^(1/
2)+((x+(d*e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*d)^(1/2))+1/2*e/(d*e)^(1/2)/((-d*e)^(1/2)+(d*e)^(1
/2))/((-d*e)^(1/2)-(d*e)^(1/2))*d^(1/2)*2^(1/2)*ln((4*d-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*2^(1/2)*d^(1/2)*((x+
(d*e)^(1/2)/e)^2*e-2*(d*e)^(1/2)*(x+(d*e)^(1/2)/e)+2*d)^(1/2))/(x+(d*e)^(1/2)/e))+1/2*e/(d*e)^(1/2)/((-d*e)^(1
/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))*((x-(d*e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*d)^(1/2)+
1/2*e^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2))/((-d*e)^(1/2)-(d*e)^(1/2))*ln(((x-(d*e)^(1/2)/e)*e+(d*e)^(1/2))/e^(1/2)
+((x-(d*e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*d)^(1/2))-1/2*e/(d*e)^(1/2)/((-d*e)^(1/2)+(d*e)^(1/2
))/((-d*e)^(1/2)-(d*e)^(1/2))*d^(1/2)*2^(1/2)*ln((4*d+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*2^(1/2)*d^(1/2)*((x-(d
*e)^(1/2)/e)^2*e+2*(d*e)^(1/2)*(x-(d*e)^(1/2)/e)+2*d)^(1/2))/(x-(d*e)^(1/2)/e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\sqrt{e x^{2} + d}}{e^{2} x^{4} - d^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.15559, size = 340, normalized size = 8.95 \begin{align*} \left [\frac{\sqrt{2} \log \left (\frac{17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt{2}{\left (3 \, e x^{3} + d x\right )} \sqrt{e x^{2} + d} \sqrt{e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )}{8 \, d \sqrt{e}}, -\frac{\sqrt{2} \sqrt{-e} \arctan \left (\frac{\sqrt{2}{\left (3 \, e x^{2} + d\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{4 \,{\left (e^{2} x^{3} + d e x\right )}}\right )}{4 \, d e}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="fricas")

[Out]

[1/8*sqrt(2)*log((17*e^2*x^4 + 14*d*e*x^2 + 4*sqrt(2)*(3*e*x^3 + d*x)*sqrt(e*x^2 + d)*sqrt(e) + d^2)/(e^2*x^4
- 2*d*e*x^2 + d^2))/(d*sqrt(e)), -1/4*sqrt(2)*sqrt(-e)*arctan(1/4*sqrt(2)*(3*e*x^2 + d)*sqrt(e*x^2 + d)*sqrt(-
e)/(e^2*x^3 + d*e*x))/(d*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- d \sqrt{d + e x^{2}} + e x^{2} \sqrt{d + e x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(1/2)/(-e**2*x**4+d**2),x)

[Out]

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

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Giac [B]  time = 1.45393, size = 177, normalized size = 4.66 \begin{align*} -\frac{{\left (\sqrt{2} i \arctan \left (\frac{e^{\frac{1}{2}}}{\sqrt{-\frac{d e + \sqrt{d^{2}} e}{d}}}\right ) e^{\frac{1}{2}} - \sqrt{2} i \arctan \left (\frac{e^{\frac{1}{2}}}{\sqrt{-\frac{d e - \sqrt{d^{2}} e}{d}}}\right ) e^{\frac{1}{2}}\right )} e^{\left (-1\right )} \mathrm{sgn}\left (x\right )}{4 \,{\left | d \right |}} + \frac{\sqrt{2} i \arctan \left (\frac{\sqrt{\frac{d}{x^{2}} + e}}{\sqrt{-\frac{d e \mathrm{sgn}\left (x\right ) + \sqrt{d^{2}} e}{d \mathrm{sgn}\left (x\right )}}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \,{\left | d \right |}{\left | \mathrm{sgn}\left (x\right ) \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)/(-e^2*x^4+d^2),x, algorithm="giac")

[Out]

-1/4*(sqrt(2)*i*arctan(e^(1/2)/sqrt(-(d*e + sqrt(d^2)*e)/d))*e^(1/2) - sqrt(2)*i*arctan(e^(1/2)/sqrt(-(d*e - s
qrt(d^2)*e)/d))*e^(1/2))*e^(-1)*sgn(x)/abs(d) + 1/2*sqrt(2)*i*arctan(sqrt(d/x^2 + e)/sqrt(-(d*e*sgn(x) + sqrt(
d^2)*e)/(d*sgn(x))))*e^(-1/2)/(abs(d)*abs(sgn(x)))