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

Optimal. Leaf size=49 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {c}}+\frac {x^2}{4 a \left (a+c x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 49, 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 = {275, 199, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {c}}+\frac {x^2}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + c*x^4)^2,x]

[Out]

x^2/(4*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*a^(3/2)*Sqrt[c])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 49, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {c}}+\frac {x^2}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + c*x^4)^2,x]

[Out]

x^2/(4*a*(a + c*x^4)) + ArcTan[(Sqrt[c]*x^2)/Sqrt[a]]/(4*a^(3/2)*Sqrt[c])

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fricas [A]  time = 0.63, size = 129, normalized size = 2.63 \[ \left [\frac {2 \, a c x^{2} - {\left (c x^{4} + a\right )} \sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{8 \, {\left (a^{2} c^{2} x^{4} + a^{3} c\right )}}, \frac {a c x^{2} - {\left (c x^{4} + a\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{4 \, {\left (a^{2} c^{2} x^{4} + a^{3} c\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 39, normalized size = 0.80 \[ \frac {x^{2}}{4 \, {\left (c x^{4} + a\right )} a} + \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2/((c*x^4 + a)*a) + 1/4*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a)

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maple [A]  time = 0.01, size = 40, normalized size = 0.82 \[ \frac {x^{2}}{4 \left (c \,x^{4}+a \right ) a}+\frac {\arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{4 \sqrt {a c}\, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.89, size = 39, normalized size = 0.80 \[ \frac {x^{2}}{4 \, {\left (a c x^{4} + a^{2}\right )}} + \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2/(a*c*x^4 + a^2) + 1/4*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a)

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mupad [B]  time = 0.04, size = 37, normalized size = 0.76 \[ \frac {x^2}{4\,a\,\left (c\,x^4+a\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{4\,a^{3/2}\,\sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + c*x^4)^2,x)

[Out]

x^2/(4*a*(a + c*x^4)) + atan((c^(1/2)*x^2)/a^(1/2))/(4*a^(3/2)*c^(1/2))

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sympy [B]  time = 0.53, size = 83, normalized size = 1.69 \[ \frac {x^{2}}{4 a^{2} + 4 a c x^{4}} - \frac {\sqrt {- \frac {1}{a^{3} c}} \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} c}} + x^{2} \right )}}{8} + \frac {\sqrt {- \frac {1}{a^{3} c}} \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} c}} + x^{2} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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