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

Optimal. Leaf size=29 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {275, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + c*x^4),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + c*x^4),x]

[Out]

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

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fricas [A]  time = 0.54, size = 72, normalized size = 2.48 \[ \left [-\frac {\sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{4 \, a c}, -\frac {\sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{2 \, a c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 18, normalized size = 0.62 \[ \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 19, normalized size = 0.66 \[ \frac {\arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{2 \sqrt {a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.94, size = 18, normalized size = 0.62 \[ \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, \sqrt {a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 19, normalized size = 0.66 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{2\,\sqrt {a}\,\sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + c*x^4),x)

[Out]

atan((c^(1/2)*x^2)/a^(1/2))/(2*a^(1/2)*c^(1/2))

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sympy [B]  time = 0.22, size = 56, normalized size = 1.93 \[ - \frac {\sqrt {- \frac {1}{a c}} \log {\left (- a \sqrt {- \frac {1}{a c}} + x^{2} \right )}}{4} + \frac {\sqrt {- \frac {1}{a c}} \log {\left (a \sqrt {- \frac {1}{a c}} + x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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