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

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

[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)

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Rubi [A]  time = 0.100967, antiderivative size = 157, normalized size of antiderivative = 1., 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 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 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 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 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 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 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 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]

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]

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*}

Mathematica [A]  time = 0.105094, size = 134, normalized size = 0.85 \[ -\frac{\frac{32 a^5 x^3}{\left (a^4+x^4\right )^2}+\frac{104 a x^3}{a^4+x^4}+45 \sqrt{2} \log \left (a^2-\sqrt{2} a x+x^2\right )-45 \sqrt{2} \log \left (a^2+\sqrt{2} a x+x^2\right )+\frac{256 a}{x}-90 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} x}{a}\right )+90 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{a}+1\right )}{256 a^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a^4 + x^4)^3),x]

[Out]

-((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])/(256*a^13)

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Maple [A]  time = 0.012, size = 152, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{12}x}}-{\frac{13\,{x}^{7}}{32\,{a}^{12} \left ({a}^{4}+{x}^{4} \right ) ^{2}}}-{\frac{17\,{x}^{3}}{32\,{a}^{8} \left ({a}^{4}+{x}^{4} \right ) ^{2}}}-{\frac{45\,\sqrt{2}}{256\,{a}^{12}}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{a}^{4}}x\sqrt{2}+\sqrt{{a}^{4}} \right ) \left ({x}^{2}+\sqrt [4]{{a}^{4}}x\sqrt{2}+\sqrt{{a}^{4}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{a}^{4}}}}}-{\frac{45\,\sqrt{2}}{128\,{a}^{12}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{4}}}}}+1 \right ){\frac{1}{\sqrt [4]{{a}^{4}}}}}-{\frac{45\,\sqrt{2}}{128\,{a}^{12}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{4}}}}}-1 \right ){\frac{1}{\sqrt [4]{{a}^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^12/x-13/32/a^12/(a^4+x^4)^2*x^7-17/32/a^8/(a^4+x^4)^2*x^3-45/256/a^12/(a^4)^(1/4)*2^(1/2)*ln((x^2-(a^4)^(
1/4)*x*2^(1/2)+(a^4)^(1/2))/(x^2+(a^4)^(1/4)*x*2^(1/2)+(a^4)^(1/2)))-45/128/a^12/(a^4)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(a^4)^(1/4)*x+1)-45/128/a^12/(a^4)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a^4)^(1/4)*x-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.1156, size = 987, normalized size = 6.29 \begin{align*} -\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 )}} \end{align*}

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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Sympy [A]  time = 3.76579, size = 65, normalized size = 0.41 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 1.06128, size = 203, normalized size = 1.29 \begin{align*} -\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} \end{align*}

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)