3.174 \(\int \frac {1}{x^2 (a^4+x^4)^3} \, dx\)

Optimal. Leaf size=157 \[ \frac {45 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{64 \sqrt {2} a^{13}}-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {9}{32 a^8 x \left (a^4+x^4\right )} \]

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Rubi [A]  time = 0.10, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {290, 325, 297, 1162, 617, 204, 1165, 628} \[ \frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}-\frac {45}{32 a^{12} x}+\frac {45 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{64 \sqrt {2} a^{13}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a^4 + x^4)^3),x]

[Out]

-45/(32*a^12*x) + 1/(8*a^4*x*(a^4 + x^4)^2) + 9/(32*a^8*x*(a^4 + x^4)) + (45*ArcTan[1 - (Sqrt[2]*x)/a])/(64*Sq
rt[2]*a^13) - (45*ArcTan[1 + (Sqrt[2]*x)/a])/(64*Sqrt[2]*a^13) - (45*Log[a^2 - Sqrt[2]*a*x + x^2])/(128*Sqrt[2
]*a^13) + (45*Log[a^2 + Sqrt[2]*a*x + x^2])/(128*Sqrt[2]*a^13)

Rule 204

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

Rule 290

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

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 325

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]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a^4+x^4\right )^3} \, dx &=\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9 \int \frac {1}{x^2 \left (a^4+x^4\right )^2} \, dx}{8 a^4}\\ &=\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {45 \int \frac {1}{x^2 \left (a^4+x^4\right )} \, dx}{32 a^8}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}-\frac {45 \int \frac {x^2}{a^4+x^4} \, dx}{32 a^{12}}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {45 \int \frac {a^2-x^2}{a^4+x^4} \, dx}{64 a^{12}}-\frac {45 \int \frac {a^2+x^2}{a^4+x^4} \, dx}{64 a^{12}}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}-\frac {45 \int \frac {\sqrt {2} a+2 x}{-a^2-\sqrt {2} a x-x^2} \, dx}{128 \sqrt {2} a^{13}}-\frac {45 \int \frac {\sqrt {2} a-2 x}{-a^2+\sqrt {2} a x-x^2} \, dx}{128 \sqrt {2} a^{13}}-\frac {45 \int \frac {1}{a^2-\sqrt {2} a x+x^2} \, dx}{128 a^{12}}-\frac {45 \int \frac {1}{a^2+\sqrt {2} a x+x^2} \, dx}{128 a^{12}}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}-\frac {45 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}+\frac {45 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}\\ &=-\frac {45}{32 a^{12} x}+\frac {1}{8 a^4 x \left (a^4+x^4\right )^2}+\frac {9}{32 a^8 x \left (a^4+x^4\right )}+\frac {45 \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \tan ^{-1}\left (1+\frac {\sqrt {2} x}{a}\right )}{64 \sqrt {2} a^{13}}-\frac {45 \log \left (a^2-\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}+\frac {45 \log \left (a^2+\sqrt {2} a x+x^2\right )}{128 \sqrt {2} a^{13}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 134, normalized size = 0.85 \[ -\frac {\frac {104 a x^3}{a^4+x^4}+45 \sqrt {2} \log \left (a^2-\sqrt {2} a x+x^2\right )-45 \sqrt {2} \log \left (a^2+\sqrt {2} a x+x^2\right )+\frac {32 a^5 x^3}{\left (a^4+x^4\right )^2}+\frac {256 a}{x}-90 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} x}{a}\right )+90 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{a}+1\right )}{256 a^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a^4 + x^4)^3),x]

[Out]

-1/256*((256*a)/x + (32*a^5*x^3)/(a^4 + x^4)^2 + (104*a*x^3)/(a^4 + x^4) - 90*Sqrt[2]*ArcTan[1 - (Sqrt[2]*x)/a
] + 90*Sqrt[2]*ArcTan[1 + (Sqrt[2]*x)/a] + 45*Sqrt[2]*Log[a^2 - Sqrt[2]*a*x + x^2] - 45*Sqrt[2]*Log[a^2 + Sqrt
[2]*a*x + x^2])/a^13

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IntegrateAlgebraic [A]  time = 0.14, size = 107, normalized size = 0.68 \[ \frac {45 \tan ^{-1}\left (\frac {\frac {a}{\sqrt {2}}-\frac {x^2}{\sqrt {2} a}}{x}\right )}{64 \sqrt {2} a^{13}}+\frac {45 \tanh ^{-1}\left (\frac {\sqrt {2} a x}{a^2+x^2}\right )}{64 \sqrt {2} a^{13}}+\frac {-32 a^8-81 a^4 x^4-45 x^8}{32 a^{12} x \left (a^4+x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^2*(a^4 + x^4)^3),x]

[Out]

(-32*a^8 - 81*a^4*x^4 - 45*x^8)/(32*a^12*x*(a^4 + x^4)^2) + (45*ArcTan[(a/Sqrt[2] - x^2/(Sqrt[2]*a))/x])/(64*S
qrt[2]*a^13) + (45*ArcTanh[(Sqrt[2]*a*x)/(a^2 + x^2)])/(64*Sqrt[2]*a^13)

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fricas [B]  time = 1.46, size = 338, normalized size = 2.15 \[ -\frac {256 \, a^{8} + 648 \, a^{4} x^{4} + 360 \, x^{8} - 180 \, \sqrt {2} {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \frac {1}{a^{52}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a^{12} \frac {1}{a^{52}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {\sqrt {2} a^{40} \frac {1}{a^{52}}^{\frac {3}{4}} x + a^{28} \sqrt {\frac {1}{a^{52}}} + x^{2}} a^{12} \frac {1}{a^{52}}^{\frac {1}{4}} - 1\right ) - 180 \, \sqrt {2} {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \frac {1}{a^{52}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a^{12} \frac {1}{a^{52}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {-\sqrt {2} a^{40} \frac {1}{a^{52}}^{\frac {3}{4}} x + a^{28} \sqrt {\frac {1}{a^{52}}} + x^{2}} a^{12} \frac {1}{a^{52}}^{\frac {1}{4}} + 1\right ) - 45 \, \sqrt {2} {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \frac {1}{a^{52}}^{\frac {1}{4}} \log \left (\sqrt {2} a^{40} \frac {1}{a^{52}}^{\frac {3}{4}} x + a^{28} \sqrt {\frac {1}{a^{52}}} + x^{2}\right ) + 45 \, \sqrt {2} {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )} \frac {1}{a^{52}}^{\frac {1}{4}} \log \left (-\sqrt {2} a^{40} \frac {1}{a^{52}}^{\frac {3}{4}} x + a^{28} \sqrt {\frac {1}{a^{52}}} + x^{2}\right )}{256 \, {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a^4+x^4)^3,x, algorithm="fricas")

[Out]

-1/256*(256*a^8 + 648*a^4*x^4 + 360*x^8 - 180*sqrt(2)*(a^20*x + 2*a^16*x^5 + a^12*x^9)*(a^(-52))^(1/4)*arctan(
-sqrt(2)*a^12*(a^(-52))^(1/4)*x + sqrt(2)*sqrt(sqrt(2)*a^40*(a^(-52))^(3/4)*x + a^28*sqrt(a^(-52)) + x^2)*a^12
*(a^(-52))^(1/4) - 1) - 180*sqrt(2)*(a^20*x + 2*a^16*x^5 + a^12*x^9)*(a^(-52))^(1/4)*arctan(-sqrt(2)*a^12*(a^(
-52))^(1/4)*x + sqrt(2)*sqrt(-sqrt(2)*a^40*(a^(-52))^(3/4)*x + a^28*sqrt(a^(-52)) + x^2)*a^12*(a^(-52))^(1/4)
+ 1) - 45*sqrt(2)*(a^20*x + 2*a^16*x^5 + a^12*x^9)*(a^(-52))^(1/4)*log(sqrt(2)*a^40*(a^(-52))^(3/4)*x + a^28*s
qrt(a^(-52)) + x^2) + 45*sqrt(2)*(a^20*x + 2*a^16*x^5 + a^12*x^9)*(a^(-52))^(1/4)*log(-sqrt(2)*a^40*(a^(-52))^
(3/4)*x + a^28*sqrt(a^(-52)) + x^2))/(a^20*x + 2*a^16*x^5 + a^12*x^9)

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giac [A]  time = 0.93, size = 150, normalized size = 0.96 \[ -\frac {45 \, \sqrt {2} {\left | a \right |} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} + 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{128 \, a^{14}} - \frac {45 \, \sqrt {2} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left | a \right |} - 2 \, x\right )}}{2 \, {\left | a \right |}}\right )}{128 \, a^{14}} + \frac {45 \, \sqrt {2} {\left | a \right |} \log \left (\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{256 \, a^{14}} - \frac {45 \, \sqrt {2} {\left | a \right |} \log \left (-\sqrt {2} x {\left | a \right |} + x^{2} + {\left | a \right |}^{2}\right )}{256 \, a^{14}} - \frac {17 \, a^{4} x^{3} + 13 \, x^{7}}{32 \, {\left (a^{4} + x^{4}\right )}^{2} a^{12}} - \frac {1}{a^{12} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a^4+x^4)^3,x, algorithm="giac")

[Out]

-45/128*sqrt(2)*abs(a)*arctan(1/2*sqrt(2)*(sqrt(2)*abs(a) + 2*x)/abs(a))/a^14 - 45/128*sqrt(2)*abs(a)*arctan(-
1/2*sqrt(2)*(sqrt(2)*abs(a) - 2*x)/abs(a))/a^14 + 45/256*sqrt(2)*abs(a)*log(sqrt(2)*x*abs(a) + x^2 + abs(a)^2)
/a^14 - 45/256*sqrt(2)*abs(a)*log(-sqrt(2)*x*abs(a) + x^2 + abs(a)^2)/a^14 - 1/32*(17*a^4*x^3 + 13*x^7)/((a^4
+ x^4)^2*a^12) - 1/(a^12*x)

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maple [C]  time = 0.28, size = 75, normalized size = 0.48

method	result	size
risch	\(\frac {-\frac {45 x^{8}}{32 a^{12}}-\frac {81 x^{4}}{32 a^{8}}-\frac {1}{a^{4}}}{x \left (a^{4}+x^{4}\right )^{2}}+\frac {45 \left (\munderset {\textit {\_R} =\RootOf \left (a^{52} \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4} a^{52}+4\right ) x +\textit {\_R}^{3} a^{40}\right )\right )}{128}\)	\(75\)
default	\(-\frac {\frac {\frac {17}{32} a^{4} x^{3}+\frac {13}{32} x^{7}}{\left (a^{4}+x^{4}\right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}{x^{2}+\left (a^{4}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {a^{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (a^{4}\right )^{\frac {1}{4}}}-1\right )\right )}{256 \left (a^{4}\right )^{\frac {1}{4}}}}{a^{12}}-\frac {1}{a^{12} x}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a^4+x^4)^3,x,method=_RETURNVERBOSE)

[Out]

(-45/32/a^12*x^8-81/32/a^8*x^4-1/a^4)/x/(a^4+x^4)^2+45/128*sum(_R*ln((5*_R^4*a^52+4)*x+_R^3*a^40),_R=RootOf(_Z
^4*a^52+1))

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maxima [A]  time = 1.30, size = 147, normalized size = 0.94 \[ -\frac {32 \, a^{8} + 81 \, a^{4} x^{4} + 45 \, x^{8}}{32 \, {\left (a^{20} x + 2 \, a^{16} x^{5} + a^{12} x^{9}\right )}} - \frac {45 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a + 2 \, x\right )}}{2 \, a}\right )}{a} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a - 2 \, x\right )}}{2 \, a}\right )}{a} - \frac {\sqrt {2} \log \left (\sqrt {2} a x + a^{2} + x^{2}\right )}{a} + \frac {\sqrt {2} \log \left (-\sqrt {2} a x + a^{2} + x^{2}\right )}{a}\right )}}{256 \, a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a^4+x^4)^3,x, algorithm="maxima")

[Out]

-1/32*(32*a^8 + 81*a^4*x^4 + 45*x^8)/(a^20*x + 2*a^16*x^5 + a^12*x^9) - 45/256*(2*sqrt(2)*arctan(1/2*sqrt(2)*(
sqrt(2)*a + 2*x)/a)/a + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a - 2*x)/a)/a - sqrt(2)*log(sqrt(2)*a*x + a^2 +
x^2)/a + sqrt(2)*log(-sqrt(2)*a*x + a^2 + x^2)/a)/a^12

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mupad [B]  time = 0.11, size = 76, normalized size = 0.48 \[ \frac {45\,{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{64\,a^{13}}-\frac {45\,{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,x}{a}\right )}{64\,a^{13}}-\frac {\frac {1}{a^4}+\frac {81\,x^4}{32\,a^8}+\frac {45\,x^8}{32\,a^{12}}}{a^8\,x+2\,a^4\,x^5+x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a^4 + x^4)^3),x)

[Out]

(45*(-1)^(1/4)*atanh(((-1)^(1/4)*x)/a))/(64*a^13) - (45*(-1)^(1/4)*atan(((-1)^(1/4)*x)/a))/(64*a^13) - (1/a^4
+ (81*x^4)/(32*a^8) + (45*x^8)/(32*a^12))/(a^8*x + x^9 + 2*a^4*x^5)

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sympy [A]  time = 0.44, size = 66, normalized size = 0.42 \[ \frac {- 32 a^{8} - 81 a^{4} x^{4} - 45 x^{8}}{32 a^{20} x + 64 a^{16} x^{5} + 32 a^{12} x^{9}} + \frac {\operatorname {RootSum} {\left (268435456 t^{4} + 4100625, \left (t \mapsto t \log {\left (- \frac {2097152 t^{3} a}{91125} + x \right )} \right )\right )}}{a^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a**4+x**4)**3,x)

[Out]

(-32*a**8 - 81*a**4*x**4 - 45*x**8)/(32*a**20*x + 64*a**16*x**5 + 32*a**12*x**9) + RootSum(268435456*_t**4 + 4
100625, Lambda(_t, _t*log(-2097152*_t**3*a/91125 + x)))/a**13

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