3.4.71 \(\int \frac {\sqrt {a x^6}}{x (1-x^4)} \, dx\) [371]

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

[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, 304, 209, 212} \begin {gather*} \frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac {\sqrt {a x^6} \text {ArcTan}(x)}{2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^6]/(x*(1 - x^4)),x]

[Out]

-1/2*(Sqrt[a*x^6]*ArcTan[x])/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 209

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

Rule 212

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

Rule 304

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 \begin {gather*} -\frac {\sqrt {a x^6} \left (2 \tan ^{-1}(x)+\log (1-x)-\log (1+x)\right )}{4 x^3} \end {gather*}

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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Maple [A]
time = 0.20, size = 28, normalized size = 0.76

method	result	size
default	\(-\frac {\sqrt {a \,x^{6}}\, \left (\ln \left (-1+x \right )-\ln \left (1+x \right )+2 \arctan \left (x \right )\right )}{4 x^{3}}\)	\(28\)
meijerg	\(-\frac {\sqrt {a \,x^{6}}\, \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{4 \left (x^{4}\right )^{\frac {3}{4}}}\)	\(44\)
risch	\(-\frac {i \sqrt {a \,x^{6}}\, \ln \left (x +i\right )}{4 x^{3}}+\frac {i \sqrt {a \,x^{6}}\, \ln \left (x -i\right )}{4 x^{3}}+\frac {\sqrt {a \,x^{6}}\, \ln \left (1+x \right )}{4 x^{3}}-\frac {\sqrt {a \,x^{6}}\, \ln \left (-1+x \right )}{4 x^{3}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 26, normalized size = 0.70 \begin {gather*} -\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 ) \end {gather*}

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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Fricas [A]
time = 0.36, size = 29, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {a x^{6}} {\left (2 \, \arctan \left (x\right ) - \log \left (\frac {x + 1}{x - 1}\right )\right )}}{4 \, x^{3}} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {a x^{6}}}{x^{5} - x}\, dx \end {gather*}

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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Giac [A]
time = 6.82, size = 29, normalized size = 0.78 \begin {gather*} -\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} \end {gather*}

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

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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