\(\int \frac {1}{x^4 (a^4-x^4)} \, dx\) [132]

Optimal result

Integrand size = 15, antiderivative size = 37 \[ \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx=-\frac {1}{3 a^4 x^3}+\frac {\arctan \left (\frac {x}{a}\right )}{2 a^7}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^7} \]

[Out]

-1/3/a^4/x^3+1/2*arctan(x/a)/a^7+1/2*arctanh(x/a)/a^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {331, 218, 212, 209} \[ \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{2 a^7}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^7}-\frac {1}{3 a^4 x^3} \]

[In]

Int[1/(x^4*(a^4 - x^4)),x]

[Out]

-1/3*1/(a^4*x^3) + ArcTan[x/a]/(2*a^7) + ArcTanh[x/a]/(2*a^7)

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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3 a^4 x^3}+\frac {\int \frac {1}{a^4-x^4} \, dx}{a^4} \\ & = -\frac {1}{3 a^4 x^3}+\frac {\int \frac {1}{a^2-x^2} \, dx}{2 a^6}+\frac {\int \frac {1}{a^2+x^2} \, dx}{2 a^6} \\ & = -\frac {1}{3 a^4 x^3}+\frac {\arctan \left (\frac {x}{a}\right )}{2 a^7}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx=-\frac {1}{3 a^4 x^3}+\frac {\arctan \left (\frac {x}{a}\right )}{2 a^7}-\frac {\log (a-x)}{4 a^7}+\frac {\log (a+x)}{4 a^7} \]

[In]

Integrate[1/(x^4*(a^4 - x^4)),x]

[Out]

-1/3*1/(a^4*x^3) + ArcTan[x/a]/(2*a^7) - Log[a - x]/(4*a^7) + Log[a + x]/(4*a^7)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {\ln \left (a -x \right )}{4 a^{7}}+\frac {\arctan \left (\frac {x}{a}\right )}{2 a^{7}}-\frac {1}{3 a^{4} x^{3}}+\frac {\ln \left (a +x \right )}{4 a^{7}}\)	\(41\)
parallelrisch	\(-\frac {3 i \ln \left (-i a +x \right ) x^{3}-3 i \ln \left (i a +x \right ) x^{3}+3 \ln \left (-a +x \right ) x^{3}-3 \ln \left (a +x \right ) x^{3}+4 a^{3}}{12 a^{7} x^{3}}\)	\(61\)
risch	\(-\frac {1}{3 a^{4} x^{3}}-\frac {\ln \left (a -x \right )}{4 a^{7}}+\frac {\ln \left (-a -x \right )}{4 a^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{14} \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4} a^{28}+4\right ) x -a^{8} \textit {\_R} \right )\right )}{4}\)	\(71\)

[In]

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

[Out]

-1/4/a^7*ln(a-x)+1/2*arctan(x/a)/a^7-1/3/a^4/x^3+1/4*ln(a+x)/a^7

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx=\frac {6 \, x^{3} \arctan \left (\frac {x}{a}\right ) + 3 \, x^{3} \log \left (a + x\right ) - 3 \, x^{3} \log \left (-a + x\right ) - 4 \, a^{3}}{12 \, a^{7} x^{3}} \]

[In]

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

[Out]

1/12*(6*x^3*arctan(x/a) + 3*x^3*log(a + x) - 3*x^3*log(-a + x) - 4*a^3)/(a^7*x^3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx=- \frac {1}{3 a^{4} x^{3}} - \frac {\frac {\log {\left (- a + x \right )}}{4} - \frac {\log {\left (a + x \right )}}{4} + \frac {i \log {\left (- i a + x \right )}}{4} - \frac {i \log {\left (i a + x \right )}}{4}}{a^{7}} \]

[In]

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

[Out]

-1/(3*a**4*x**3) - (log(-a + x)/4 - log(a + x)/4 + I*log(-I*a + x)/4 - I*log(I*a + x)/4)/a**7

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{7}} + \frac {\log \left (a + x\right )}{4 \, a^{7}} - \frac {\log \left (-a + x\right )}{4 \, a^{7}} - \frac {1}{3 \, a^{4} x^{3}} \]

[In]

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

[Out]

1/2*arctan(x/a)/a^7 + 1/4*log(a + x)/a^7 - 1/4*log(-a + x)/a^7 - 1/3/(a^4*x^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx=\frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{7}} + \frac {\log \left ({\left | a + x \right |}\right )}{4 \, a^{7}} - \frac {\log \left ({\left | -a + x \right |}\right )}{4 \, a^{7}} - \frac {1}{3 \, a^{4} x^{3}} \]

[In]

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

[Out]

1/2*arctan(x/a)/a^7 + 1/4*log(abs(a + x))/a^7 - 1/4*log(abs(-a + x))/a^7 - 1/3/(a^4*x^3)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^4 \left (a^4-x^4\right )} \, dx=\frac {\mathrm {atan}\left (\frac {x}{a}\right )}{2\,a^7}+\frac {\mathrm {atanh}\left (\frac {x}{a}\right )}{2\,a^7}-\frac {1}{3\,a^4\,x^3} \]

[In]

int(1/(x^4*(a^4 - x^4)),x)

[Out]

atan(x/a)/(2*a^7) + atanh(x/a)/(2*a^7) - 1/(3*a^4*x^3)