3.374 \(\int (\frac{1}{1-x^4}-\frac{\sqrt{a x^6}}{x (1-x^4)}) \, dx\)

Optimal. Leaf size=49 \[ \frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

ArcTan[x]/2 + (Sqrt[a*x^6]*ArcTan[x])/(2*x^3) + ArcTanh[x]/2 - (Sqrt[a*x^6]*ArcTanh[x])/(2*x^3)

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Rubi [A]  time = 0.0133674, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {212, 206, 203, 15, 298} \[ \frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)^(-1) - Sqrt[a*x^6]/(x*(1 - x^4)),x]

[Out]

ArcTan[x]/2 + (Sqrt[a*x^6]*ArcTan[x])/(2*x^3) + ArcTanh[x]/2 - (Sqrt[a*x^6]*ArcTanh[x])/(2*x^3)

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

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

Rule 203

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rubi steps

\begin{align*} \int \left (\frac{1}{1-x^4}-\frac{\sqrt{a x^6}}{x \left (1-x^4\right )}\right ) \, dx &=\int \frac{1}{1-x^4} \, dx-\int \frac{\sqrt{a x^6}}{x \left (1-x^4\right )} \, dx\\ &=\frac{1}{2} \int \frac{1}{1-x^2} \, dx+\frac{1}{2} \int \frac{1}{1+x^2} \, dx-\frac{\sqrt{a x^6} \int \frac{x^2}{1-x^4} \, dx}{x^3}\\ &=\frac{1}{2} \tan ^{-1}(x)+\frac{1}{2} \tanh ^{-1}(x)-\frac{\sqrt{a x^6} \int \frac{1}{1-x^2} \, dx}{2 x^3}+\frac{\sqrt{a x^6} \int \frac{1}{1+x^2} \, dx}{2 x^3}\\ &=\frac{1}{2} \tan ^{-1}(x)+\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3}+\frac{1}{2} \tanh ^{-1}(x)-\frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0636909, size = 29, normalized size = 0.59 \[ \frac{1}{2} \left (\frac{\sqrt{a x^6} \left (\tan ^{-1}(x)-\tanh ^{-1}(x)\right )}{x^3}+\tan ^{-1}(x)+\tanh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)^(-1) - Sqrt[a*x^6]/(x*(1 - x^4)),x]

[Out]

(ArcTan[x] + (Sqrt[a*x^6]*(ArcTan[x] - ArcTanh[x]))/x^3 + ArcTanh[x])/2

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Maple [A]  time = 0.003, size = 37, normalized size = 0.8 \begin{align*}{\frac{{\it Artanh} \left ( x \right ) }{2}}+{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( x-1 \right ) -\ln \left ( 1+x \right ) +2\,\arctan \left ( x \right ) }{4\,{x}^{3}}\sqrt{a{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^4+1)-(a*x^6)^(1/2)/x/(-x^4+1),x)

[Out]

1/2*arctanh(x)+1/2*arctan(x)+1/4*(a*x^6)^(1/2)*(ln(x-1)-ln(1+x)+2*arctan(x))/x^3

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Maxima [A]  time = 1.66504, size = 57, normalized size = 1.16 \begin{align*} \frac{1}{2} \, \sqrt{a} \arctan \left (x\right ) - \frac{1}{4} \, \sqrt{a} \log \left (x + 1\right ) + \frac{1}{4} \, \sqrt{a} \log \left (x - 1\right ) + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+1)-(a*x^6)^(1/2)/x/(-x^4+1),x, algorithm="maxima")

[Out]

1/2*sqrt(a)*arctan(x) - 1/4*sqrt(a)*log(x + 1) + 1/4*sqrt(a)*log(x - 1) + 1/2*arctan(x) + 1/4*log(x + 1) - 1/4
*log(x - 1)

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Fricas [B]  time = 1.50862, size = 618, normalized size = 12.61 \begin{align*} \left [\frac{x^{3} \sqrt{-\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}} \log \left (\frac{{\left (a - 1\right )} x^{4} -{\left (a - 1\right )} x^{2} - 2 \,{\left (x^{3} - \sqrt{a x^{6}}\right )} \sqrt{-\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}}}{x^{4} + x^{2}}\right ) + x^{3} \log \left (x + 1\right ) - x^{3} \log \left (x - 1\right ) - \sqrt{a x^{6}}{\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )}}{4 \, x^{3}}, \frac{2 \, x^{3} \sqrt{\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}} \arctan \left (-\frac{{\left (x^{3} - \sqrt{a x^{6}}\right )} \sqrt{\frac{{\left (a + 1\right )} x^{3} + 2 \, \sqrt{a x^{6}}}{x^{3}}}}{{\left (a - 1\right )} x^{2}}\right ) + x^{3} \log \left (x + 1\right ) - x^{3} \log \left (x - 1\right ) - \sqrt{a x^{6}}{\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )}}{4 \, x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+1)-(a*x^6)^(1/2)/x/(-x^4+1),x, algorithm="fricas")

[Out]

[1/4*(x^3*sqrt(-((a + 1)*x^3 + 2*sqrt(a*x^6))/x^3)*log(((a - 1)*x^4 - (a - 1)*x^2 - 2*(x^3 - sqrt(a*x^6))*sqrt
(-((a + 1)*x^3 + 2*sqrt(a*x^6))/x^3))/(x^4 + x^2)) + x^3*log(x + 1) - x^3*log(x - 1) - sqrt(a*x^6)*(log(x + 1)
- log(x - 1)))/x^3, 1/4*(2*x^3*sqrt(((a + 1)*x^3 + 2*sqrt(a*x^6))/x^3)*arctan(-(x^3 - sqrt(a*x^6))*sqrt(((a +
1)*x^3 + 2*sqrt(a*x^6))/x^3)/((a - 1)*x^2)) + x^3*log(x + 1) - x^3*log(x - 1) - sqrt(a*x^6)*(log(x + 1) - log
(x - 1)))/x^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{x^{5} - x}\, dx - \int - \frac{\sqrt{a x^{6}}}{x^{5} - x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**4+1)-(a*x**6)**(1/2)/x/(-x**4+1),x)

[Out]

-Integral(x/(x**5 - x), x) - Integral(-sqrt(a*x**6)/(x**5 - x), x)

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Giac [A]  time = 1.13407, size = 65, normalized size = 1.33 \begin{align*} \frac{1}{4} \,{\left (2 \, \arctan \left (x\right ) \mathrm{sgn}\left (x\right ) - \log \left ({\left | x + 1 \right |}\right ) \mathrm{sgn}\left (x\right ) + \log \left ({\left | x - 1 \right |}\right ) \mathrm{sgn}\left (x\right )\right )} \sqrt{a} + \frac{1}{2} \, \arctan \left (x\right ) + \frac{1}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^4+1)-(a*x^6)^(1/2)/x/(-x^4+1),x, algorithm="giac")

[Out]

1/4*(2*arctan(x)*sgn(x) - log(abs(x + 1))*sgn(x) + log(abs(x - 1))*sgn(x))*sqrt(a) + 1/2*arctan(x) + 1/4*log(a
bs(x + 1)) - 1/4*log(abs(x - 1))