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

Optimal. Leaf size=37 \[ \frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac {\sqrt {a x^6} \tan ^{-1}(x)}{2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 298, 203, 206} \[ \frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac {\sqrt {a x^6} \tan ^{-1}(x)}{2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^6]/(x*(1 - x^4)),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 33, normalized size = 0.89 \[ -\frac {\sqrt {a x^6} \left (\log (1-x)-\log (x+1)+2 \tan ^{-1}(x)\right )}{4 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Sqrt[a*x^6]*(2*ArcTan[x] + Log[1 - x] - Log[1 + x]))/x^3

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fricas [A]  time = 0.46, size = 29, normalized size = 0.78 \[ -\frac {\sqrt {a x^{6}} {\left (2 \, \arctan \relax (x) - \log \left (\frac {x + 1}{x - 1}\right )\right )}}{4 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 29, normalized size = 0.78 \[ -\frac {1}{4} \, {\left (2 \, \arctan \relax (x) \mathrm {sgn}\relax (x) - \log \left ({\left | x + 1 \right |}\right ) \mathrm {sgn}\relax (x) + \log \left ({\left | x - 1 \right |}\right ) \mathrm {sgn}\relax (x)\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [A]  time = 0.01, size = 28, normalized size = 0.76 \[ -\frac {\sqrt {a \,x^{6}}\, \left (2 \arctan \relax (x )+\ln \left (x -1\right )-\ln \left (x +1\right )\right )}{4 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.53, size = 26, normalized size = 0.70 \[ -\frac {1}{2} \, \sqrt {a} \arctan \relax (x) + \frac {1}{4} \, \sqrt {a} \log \left (x + 1\right ) - \frac {1}{4} \, \sqrt {a} \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {\sqrt {a\,x^6}}{x\,\left (x^4-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*x^6)^(1/2)/(x*(x^4 - 1)),x)

[Out]

-int((a*x^6)^(1/2)/(x*(x^4 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {a x^{6}}}{x^{5} - x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(sqrt(a*x**6)/(x**5 - x), x)

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