3.2.97 \(\int \frac {1}{\sqrt {d+e x^2} (d^2-e^2 x^4)} \, dx\) [197]

Optimal. Leaf size=61 \[ \frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1164, 390, 385, 214} \begin {gather*} \frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x^2]*(d^2 - e^2*x^4)),x]

[Out]

x/(2*d^2*Sqrt[d + e*x^2]) + ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]/(2*Sqrt[2]*d^2*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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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 {1}{\sqrt {d+e x^2} \left (d^2-e^2 x^4\right )} \, dx &=\int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\int \frac {1}{\left (d-e x^2\right ) \sqrt {d+e x^2}} \, dx}{2 d}\\ &=\frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\text {Subst}\left (\int \frac {1}{d-2 d e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{2 d}\\ &=\frac {x}{2 d^2 \sqrt {d+e x^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d^2 \sqrt {e}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 69, normalized size = 1.13 \begin {gather*} \frac {\frac {2 x}{\sqrt {d+e x^2}}+\frac {\sqrt {2} \tanh ^{-1}\left (\frac {d-e x^2+\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}\right )}{\sqrt {e}}}{4 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x^2]*(d^2 - e^2*x^4)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs. \(2(45)=90\).
time = 0.24, size = 441, normalized size = 7.23

method	result	size
default	\(\frac {\sqrt {e \left (x +\frac {\sqrt {-d e}}{e}\right )^{2}-2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}{2 d \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \left (x +\frac {\sqrt {-d e}}{e}\right )}+\frac {e \sqrt {2}\, \ln \left (\frac {4 d +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {e \left (x -\frac {\sqrt {d e}}{e}\right )^{2}+2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}{x -\frac {\sqrt {d e}}{e}}\right )}{4 \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}\, \sqrt {d}}-\frac {e \sqrt {2}\, \ln \left (\frac {4 d -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {e \left (x +\frac {\sqrt {d e}}{e}\right )^{2}-2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}{x +\frac {\sqrt {d e}}{e}}\right )}{4 \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}\, \sqrt {d}}+\frac {\sqrt {e \left (x -\frac {\sqrt {-d e}}{e}\right )^{2}+2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}{2 d \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \left (x -\frac {\sqrt {-d e}}{e}\right )}\)	\(441\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (45) = 90\).
time = 0.34, size = 119, normalized size = 1.95 \begin {gather*} \frac {\sqrt {2} {\left (x^{2} e + d\right )} e^{\frac {1}{2}} \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 \, \sqrt {x^{2} e + d} x e}{16 \, {\left (d^{2} x^{2} e^{2} + d^{3} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(sqrt(2)*(x^2*e + d)*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)) + 8*sqrt(x^2*e + d)*x*e)/(d^2*x^2*e^2 + d^3*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- d^{2} \sqrt {d + e x^{2}} + e^{2} x^{4} \sqrt {d + e x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(-d**2*sqrt(d + e*x**2) + e**2*x**4*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. 101 vs. \(2 (45) = 90\).
time = 3.21, size = 101, normalized size = 1.66 \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 )}{8 \, d {\left | d \right |}} + \frac {x}{2 \, \sqrt {x^{2} e + d} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*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))/(d*abs(d)) + 1/2*x/(sqrt(x^2*e + d)*d^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\left (d^2-e^2\,x^4\right )\,\sqrt {e\,x^2+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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