3.127 \(\int \frac {1}{a^4-x^4} \, dx\)

Optimal. Leaf size=27 \[ \frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^3}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {212, 206, 203} \[ \frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^3}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Int[(a^4 - x^4)^(-1),x]

[Out]

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

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 38, normalized size = 1.41 \[ -\frac {\log (a-x)}{4 a^3}+\frac {\log (a+x)}{4 a^3}+\frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^4 - x^4)^(-1),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.01, size = 27, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {x}{a}\right )}{2 a^3}+\frac {\tanh ^{-1}\left (\frac {x}{a}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^4 - x^4)^(-1),x]

[Out]

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

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fricas [A]  time = 1.04, size = 26, normalized size = 0.96 \[ \frac {2 \, \arctan \left (\frac {x}{a}\right ) + \log \left (a + x\right ) - \log \left (-a + x\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^4-x^4),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.86, size = 34, normalized size = 1.26 \[ \frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{3}} + \frac {\log \left ({\left | a + x \right |}\right )}{4 \, a^{3}} - \frac {\log \left ({\left | -a + x \right |}\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^4-x^4),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.28, size = 33, normalized size = 1.22

method	result	size
default	\(\frac {\arctan \left (\frac {x}{a}\right )}{2 a^{3}}+\frac {\ln \left (a +x \right )}{4 a^{3}}-\frac {\ln \left (a -x \right )}{4 a^{3}}\)	\(33\)
risch	\(\frac {\arctan \left (\frac {x}{a}\right )}{2 a^{3}}-\frac {\ln \left (-a +x \right )}{4 a^{3}}+\frac {\ln \left (a +x \right )}{4 a^{3}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^4-x^4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.97, size = 32, normalized size = 1.19 \[ \frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{3}} + \frac {\log \left (a + x\right )}{4 \, a^{3}} - \frac {\log \left (-a + x\right )}{4 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^4-x^4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.07, size = 18, normalized size = 0.67 \[ \frac {\mathrm {atan}\left (\frac {x}{a}\right )+\mathrm {atanh}\left (\frac {x}{a}\right )}{2\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^4 - x^4),x)

[Out]

(atan(x/a) + atanh(x/a))/(2*a^3)

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sympy [C]  time = 0.15, size = 37, normalized size = 1.37 \[ - \frac {\frac {\log {\left (- a + x \right )}}{4} - \frac {\log {\left (a + x \right )}}{4} + \frac {i \log {\left (- i a + x \right )}}{4} - \frac {i \log {\left (i a + x \right )}}{4}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**4-x**4),x)

[Out]

-(log(-a + x)/4 - log(a + x)/4 + I*log(-I*a + x)/4 - I*log(I*a + x)/4)/a**3

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