3.3.11 \(\int \frac {r}{\sqrt {-\alpha ^2+2 e r^2-2 k r^4}} \, dr\) [211]

Optimal. Leaf size=56 \[ -\frac {\tan ^{-1}\left (\frac {e-2 k r^2}{\sqrt {2} \sqrt {k} \sqrt {-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt {2} \sqrt {k}} \]

[Out]

-1/4*arctan(1/2*(-2*k*r^2+e)*2^(1/2)/k^(1/2)/(-2*k*r^4+2*e*r^2-alpha^2)^(1/2))*2^(1/2)/k^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1121, 635, 210} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {e-2 k r^2}{\sqrt {2} \sqrt {k} \sqrt {-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt {2} \sqrt {k}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[r/Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4],r]

[Out]

-1/2*ArcTan[(e - 2*k*r^2)/(Sqrt[2]*Sqrt[k]*Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4])]/(Sqrt[2]*Sqrt[k])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rubi steps

\begin {align*} \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2-2 k r^4}} \, dr &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-\alpha ^2+2 e r-2 k r^2}} \, dr,r,r^2\right )\\ &=\text {Subst}\left (\int \frac {1}{-8 k-r^2} \, dr,r,\frac {2 \left (e-2 k r^2\right )}{\sqrt {-\alpha ^2+2 e r^2-2 k r^4}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {e-2 k r^2}{\sqrt {2} \sqrt {k} \sqrt {-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt {2} \sqrt {k}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(56)=112\).
time = 0.18, size = 150, normalized size = 2.68 \begin {gather*} -\frac {2 \sqrt {-k} \tan ^{-1}\left (\frac {\sqrt {k} \left (2 \sqrt {-k} r^2-\sqrt {2} \sqrt {-\alpha ^2+2 e r^2-2 k r^4}\right )}{e}\right )+\sqrt {k} \log \left (e^2+4 e k r^2-2 k \left (\alpha ^2+4 k r^4+2 \sqrt {2} \sqrt {-k} r^2 \sqrt {-\alpha ^2+2 e r^2-2 k r^4}\right )\right )}{4 \sqrt {2} \sqrt {-k^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[r/Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4],r]

[Out]

-1/4*(2*Sqrt[-k]*ArcTan[(Sqrt[k]*(2*Sqrt[-k]*r^2 - Sqrt[2]*Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4]))/e] + Sqrt[k]*L
og[e^2 + 4*e*k*r^2 - 2*k*(alpha^2 + 4*k*r^4 + 2*Sqrt[2]*Sqrt[-k]*r^2*Sqrt[-alpha^2 + 2*e*r^2 - 2*k*r^4])])/(Sq
rt[2]*Sqrt[-k^2])

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[r/Sqrt[2*e*r^2-alpha^2-2*k*r^4],r]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.03, size = 47, normalized size = 0.84

method	result	size
default	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {k}\, \left (r^{2}-\frac {e}{2 k}\right )}{\sqrt {-2 k \,r^{4}+2 e \,r^{2}-\alpha ^{2}}}\right )}{4 \sqrt {k}}\)	\(47\)
elliptic	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {k}\, \left (r^{2}-\frac {e}{2 k}\right )}{\sqrt {-2 k \,r^{4}+2 e \,r^{2}-\alpha ^{2}}}\right )}{4 \sqrt {k}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(r/(-2*k*r^4+2*e*r^2-alpha^2)^(1/2),r,method=_RETURNVERBOSE)

[Out]

1/4*2^(1/2)/k^(1/2)*arctan(2^(1/2)*k^(1/2)*(r^2-1/2*e/k)/(-2*k*r^4+2*e*r^2-alpha^2)^(1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(-2*k*r^4+2*e*r^2-alpha^2)^(1/2),r, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(2*alpha^2*k-%e^2>0)', see `ass
ume?` for mo

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Fricas [A]
time = 0.33, size = 155, normalized size = 2.77 \begin {gather*} \left [-\frac {\sqrt {2} \sqrt {-k} \log \left (8 \, k^{2} r^{4} - 8 \, k r^{2} e + 2 \, \alpha ^{2} k - 2 \, \sqrt {2} \sqrt {-2 \, k r^{4} + 2 \, r^{2} e - \alpha ^{2}} {\left (2 \, k r^{2} - e\right )} \sqrt {-k} + e^{2}\right )}{8 \, k}, -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, k r^{4} + 2 \, r^{2} e - \alpha ^{2}} {\left (2 \, k r^{2} - e\right )} \sqrt {k}}{2 \, {\left (2 \, k^{2} r^{4} - 2 \, k r^{2} e + \alpha ^{2} k\right )}}\right )}{4 \, \sqrt {k}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(-2*k*r^4+2*e*r^2-alpha^2)^(1/2),r, algorithm="fricas")

[Out]

[-1/8*sqrt(2)*sqrt(-k)*log(8*k^2*r^4 - 8*k*r^2*e + 2*alpha^2*k - 2*sqrt(2)*sqrt(-2*k*r^4 + 2*r^2*e - alpha^2)*
(2*k*r^2 - e)*sqrt(-k) + e^2)/k, -1/4*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-2*k*r^4 + 2*r^2*e - alpha^2)*(2*k*r^2 -
e)*sqrt(k)/(2*k^2*r^4 - 2*k*r^2*e + alpha^2*k))/sqrt(k)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {r}{\sqrt {- \alpha ^{2} + 2 e r^{2} - 2 k r^{4}}}\, dr \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(-2*k*r**4+2*e*r**2-alpha**2)**(1/2),r)

[Out]

Integral(r/sqrt(-alpha**2 + 2*e*r**2 - 2*k*r**4), r)

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Giac [A]
time = 0.01, size = 72, normalized size = 1.29 \begin {gather*} -\frac {\ln \left |-\sqrt {2} \sqrt {-k} \left (\sqrt {-\alpha ^{2}-2 k r^{4}+2 r^{2} \mathrm {e}}-\sqrt {-2 k} r^{2}\right )+\mathrm {e}\right |}{2 \sqrt {2} \sqrt {-k}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(r/(-2*k*r^4+2*e*r^2-alpha^2)^(1/2),r)

[Out]

-1/4*sqrt(2)*log(abs(sqrt(2)*(sqrt(2)*sqrt(-k)*r^2 - sqrt(-2*k*r^4 + 2*e*r^2 - alpha^2))*sqrt(-k) + e))/sqrt(-
k)

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Mupad [B]
time = 0.98, size = 50, normalized size = 0.89 \begin {gather*} \frac {\sqrt {2}\,\ln \left (\sqrt {-\alpha ^2-2\,k\,r^4+2\,e\,r^2}+\frac {\sqrt {2}\,\left (e-2\,k\,r^2\right )}{2\,\sqrt {-k}}\right )}{4\,\sqrt {-k}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(r/(2*e*r^2 - 2*k*r^4 - alpha^2)^(1/2),r)

[Out]

(2^(1/2)*log((2*e*r^2 - 2*k*r^4 - alpha^2)^(1/2) + (2^(1/2)*(e - 2*k*r^2))/(2*(-k)^(1/2))))/(4*(-k)^(1/2))

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