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

Optimal. Leaf size=193 \[ -\frac{\sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{5/4}}-\frac{1}{a x} \]

[Out]

-(1/(a*x)) + (c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4)) - (c^(1/4)*ArcTan[1 + (Sqrt
[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4)) - (c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]
)/(4*Sqrt[2]*a^(5/4)) + (c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(5/4))

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Rubi [A]  time = 0.12092, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {325, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{5/4}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + c*x^4)),x]

[Out]

-(1/(a*x)) + (c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4)) - (c^(1/4)*ArcTan[1 + (Sqrt
[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4)) - (c^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]
)/(4*Sqrt[2]*a^(5/4)) + (c^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(5/4))

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+c x^4\right )} \, dx &=-\frac{1}{a x}-\frac{c \int \frac{x^2}{a+c x^4} \, dx}{a}\\ &=-\frac{1}{a x}+\frac{\sqrt{c} \int \frac{\sqrt{a}-\sqrt{c} x^2}{a+c x^4} \, dx}{2 a}-\frac{\sqrt{c} \int \frac{\sqrt{a}+\sqrt{c} x^2}{a+c x^4} \, dx}{2 a}\\ &=-\frac{1}{a x}-\frac{\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 a}-\frac{\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 a}-\frac{\sqrt [4]{c} \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{c} \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt{2} a^{5/4}}\\ &=-\frac{1}{a x}-\frac{\sqrt [4]{c} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{c} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4}}\\ &=-\frac{1}{a x}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{c} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{5/4}}\\ \end{align*}

Mathematica [A]  time = 0.0260654, size = 179, normalized size = 0.93 \[ \frac{-\sqrt{2} \sqrt [4]{c} x \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+\sqrt{2} \sqrt [4]{c} x \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+2 \sqrt{2} \sqrt [4]{c} x \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-2 \sqrt{2} \sqrt [4]{c} x \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-8 \sqrt [4]{a}}{8 a^{5/4} x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + c*x^4)),x]

[Out]

(-8*a^(1/4) + 2*Sqrt[2]*c^(1/4)*x*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 2*Sqrt[2]*c^(1/4)*x*ArcTan[1 + (Sq
rt[2]*c^(1/4)*x)/a^(1/4)] - Sqrt[2]*c^(1/4)*x*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sqrt[2]
*c^(1/4)*x*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(8*a^(5/4)*x)

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Maple [A]  time = 0.004, size = 136, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{2}}{8\,a}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{\sqrt{2}}{4\,a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{\sqrt{2}}{4\,a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{1}{ax}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/a/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))
-1/4/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)-1/4/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4
)*x-1)-1/a/x

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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/(c*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88008, size = 302, normalized size = 1.56 \begin{align*} \frac{4 \, a x \left (-\frac{c}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (-a x \left (-\frac{c}{a^{5}}\right )^{\frac{1}{4}} + a \sqrt{-\frac{a^{3} \sqrt{-\frac{c}{a^{5}}} - c x^{2}}{c}} \left (-\frac{c}{a^{5}}\right )^{\frac{1}{4}}\right ) - a x \left (-\frac{c}{a^{5}}\right )^{\frac{1}{4}} \log \left (a^{4} \left (-\frac{c}{a^{5}}\right )^{\frac{3}{4}} + c x\right ) + a x \left (-\frac{c}{a^{5}}\right )^{\frac{1}{4}} \log \left (-a^{4} \left (-\frac{c}{a^{5}}\right )^{\frac{3}{4}} + c x\right ) - 4}{4 \, a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*a*x*(-c/a^5)^(1/4)*arctan(-a*x*(-c/a^5)^(1/4) + a*sqrt(-(a^3*sqrt(-c/a^5) - c*x^2)/c)*(-c/a^5)^(1/4)) -
a*x*(-c/a^5)^(1/4)*log(a^4*(-c/a^5)^(3/4) + c*x) + a*x*(-c/a^5)^(1/4)*log(-a^4*(-c/a^5)^(3/4) + c*x) - 4)/(a*
x)

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Sympy [A]  time = 0.474391, size = 29, normalized size = 0.15 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{5} + c, \left ( t \mapsto t \log{\left (- \frac{64 t^{3} a^{4}}{c} + x \right )} \right )\right )} - \frac{1}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4*a**5 + c, Lambda(_t, _t*log(-64*_t**3*a**4/c + x))) - 1/(a*x)

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Giac [A]  time = 1.12785, size = 252, normalized size = 1.31 \begin{align*} -\frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2} c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2} c^{2}} + \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a^{2} c^{2}} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a^{2} c^{2}} - \frac{1}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^2) - 1/4*sqrt(2)
*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^2) + 1/8*sqrt(2)*(a*c^3)^(3/
4)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^2) - 1/8*sqrt(2)*(a*c^3)^(3/4)*log(x^2 - sqrt(2)*x*(a/c
)^(1/4) + sqrt(a/c))/(a^2*c^2) - 1/(a*x)