3.2.96 \(\int \frac {\sqrt {d+e x^2}}{d^2-e^2 x^4} \, dx\) [196]

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]

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

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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {1164, 385, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {2} d \sqrt {e}} \end {gather*}

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 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 385

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 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 {\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\\ &=\text {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*}

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Mathematica [A]
time = 0.04, size = 50, normalized size = 1.32 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {d-e x^2+\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}\right )}{\sqrt {2} d \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(817\) vs. \(2(29)=58\).
time = 0.25, size = 818, normalized size = 21.53

method	result	size
default	\(-\frac {e \left (\sqrt {e \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}-2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}-\frac {\sqrt {-d e}\, \ln \left (\frac {-\sqrt {-d e}+e \left (x +\frac {\sqrt {-d e}}{e}\right )}{\sqrt {e}}+\sqrt {e \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}-2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}\right )}{\sqrt {e}}\right )}{2 \sqrt {-d e}\, \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right )}-\frac {e \left (\sqrt {e \left (x -\frac {\sqrt {d e}}{e}\right )^{2}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}+\frac {\sqrt {d e}\, \ln \left (\frac {\sqrt {d e}+e \left (x -\frac {\sqrt {d e}}{e}\right )}{\sqrt {e}}+\sqrt {e \left (x -\frac {\sqrt {d e}}{e}\right )^{2}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}\right )}{\sqrt {e}}-\sqrt {d}\, \sqrt {2}\, \ln \left (\frac {4 d +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {e \left (x -\frac {\sqrt {d e}}{e}\right )^{2}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}{x -\frac {\sqrt {d e}}{e}}\right )\right )}{2 \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}}+\frac {e \left (\sqrt {e \left (x +\frac {\sqrt {d e}}{e}\right )^{2}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}-\frac {\sqrt {d e}\, \ln \left (\frac {-\sqrt {d e}+e \left (x +\frac {\sqrt {d e}}{e}\right )}{\sqrt {e}}+\sqrt {e \left (x +\frac {\sqrt {d e}}{e}\right )^{2}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}\right )}{\sqrt {e}}-\sqrt {d}\, \sqrt {2}\, \ln \left (\frac {4 d -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {e \left (x +\frac {\sqrt {d e}}{e}\right )^{2}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}{x +\frac {\sqrt {d e}}{e}}\right )\right )}{2 \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}}+\frac {e \left (\sqrt {e \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}+2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}+\frac {\sqrt {-d e}\, \ln \left (\frac {\sqrt {-d e}+e \left (x -\frac {\sqrt {-d e}}{e}\right )}{\sqrt {e}}+\sqrt {e \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}+2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}\right )}{\sqrt {e}}\right )}{2 \sqrt {-d e}\, \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right )}\)	\(818\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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(x^2*e + d)/(x^4*e^2 - d^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (28) = 56\).
time = 0.33, size = 79, normalized size = 2.08 \begin {gather*} \frac {\sqrt {2} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {17 \, x^{4} e^{2} + 14 \, d x^{2} e + 4 \, \sqrt {2} {\left (3 \, x^{3} e + d x\right )} \sqrt {x^{2} e + d} e^{\frac {1}{2}} + d^{2}}{x^{4} e^{2} - 2 \, d x^{2} e + d^{2}}\right )}{8 \, d} \end {gather*}

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)*e^(-1/2)*log((17*x^4*e^2 + 14*d*x^2*e + 4*sqrt(2)*(3*x^3*e + d*x)*sqrt(x^2*e + d)*e^(1/2) + d^2)/(
x^4*e^2 - 2*d*x^2*e + d^2))/d

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- d \sqrt {d + e x^{2}} + e x^{2} \sqrt {d + e x^{2}}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (28) = 56\).
time = 3.52, size = 81, normalized size = 2.13 \begin {gather*} \frac {\sqrt {2} e^{\left (-\frac {1}{2}\right )} \log \left (\frac {{\left | 2 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{4 \, {\left | d \right |}} \end {gather*}

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)*e^(-1/2)*log(abs(2*(x*e^(1/2) - sqrt(x^2*e + d))^2 - 4*sqrt(2)*abs(d) - 6*d)/abs(2*(x*e^(1/2) - sq
rt(x^2*e + d))^2 + 4*sqrt(2)*abs(d) - 6*d))/abs(d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {e\,x^2+d}}{d^2-e^2\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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