3.4.74 \(\int (\frac {1}{1-x^4}-\frac {\sqrt {a x^6}}{x (1-x^4)}) \, dx\) [374]

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

[Out]

1/2*arctan(x)+1/2*arctanh(x)+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 = 49, normalized size of antiderivative = 1.00, 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 = {218, 212, 209, 15, 304} \begin {gather*} \frac {\sqrt {a x^6} \text {ArcTan}(x)}{2 x^3}-\frac {\sqrt {a x^6} \tanh ^{-1}(x)}{2 x^3}+\frac {\text {ArcTan}(x)}{2}+\frac {1}{2} \tanh ^{-1}(x) \end {gather*}

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 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 218

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] &&  !Gt
Q[a/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 \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*}

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Mathematica [A]
time = 0.05, size = 42, normalized size = 0.86 \begin {gather*} \frac {\left (x^3+\sqrt {a x^6}\right ) \tan ^{-1}(x)+\left (x^3-\sqrt {a x^6}\right ) \tanh ^{-1}(x)}{2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.23, size = 37, normalized size = 0.76

method	result	size
default	\(\frac {\arctan \left (x \right )}{2}+\frac {\arctanh \left (x \right )}{2}+\frac {\sqrt {a \,x^{6}}\, \left (\ln \left (-1+x \right )-\ln \left (1+x \right )+2 \arctan \left (x \right )\right )}{4 x^{3}}\)	\(37\)
meijerg	\(-\frac {x \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{4 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {a \,x^{6}}\, \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{4 \left (x^{4}\right )^{\frac {3}{4}}}\)	\(82\)
risch	\(\frac {i \ln \left (x +i\right ) x^{3}-i \ln \left (x -i\right ) x^{3}-\ln \left (-1+x \right ) x^{3}+\ln \left (1+x \right ) x^{3}+i \sqrt {a \,x^{6}}\, \ln \left (x +i\right )-i \sqrt {a \,x^{6}}\, \ln \left (x -i\right )+\sqrt {a \,x^{6}}\, \ln \left (-1+x \right )-\sqrt {a \,x^{6}}\, \ln \left (1+x \right )}{4 x^{3}}\)	\(101\)

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,method=_RETURNVERBOSE)

[Out]

1/2*arctan(x)+1/2*arctanh(x)+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 = 0.66, size = 42, normalized size = 0.86 \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 ) + \frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (37) = 74\).
time = 0.39, size = 256, normalized size = 5.22 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{x^{5} - x}\, dx - \int \left (- \frac {\sqrt {a x^{6}}}{x^{5} - x}\right )\, dx \end {gather*}

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 = 3.35, size = 48, normalized size = 0.98 \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} + \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 {gather*}

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {a\,x^6}}{x\,\left (x^4-1\right )}-\frac {1}{x^4-1} \,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)) - 1/(x^4 - 1),x)

[Out]

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

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