3.11.6 \(\int \frac {x}{\sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx\) [1006]

Optimal. Leaf size=30 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4, 281, 223, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]

[Out]

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

Rule 4

Int[(u_.)*((a_.) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(a + c*x^(2*n))^p, x] /; Fre
eQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[b, 0]

Rule 212

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 281

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

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Mathematica [A]
time = 0.10, size = 30, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {c} x^2}\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + (2 + 2*b - 2*(1 + b))*x^2 + c*x^4],x]

[Out]

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

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Maple [A]
time = 0.13, size = 24, normalized size = 0.80

method	result	size
default	\(\frac {\ln \left (x^{2} \sqrt {c}+\sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}\)	\(24\)
elliptic	\(\frac {\ln \left (x^{2} \sqrt {c}+\sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
time = 0.49, size = 45, normalized size = 1.50 \begin {gather*} -\frac {\log \left (-\frac {\sqrt {c} - \frac {\sqrt {c x^{4} + a}}{x^{2}}}{\sqrt {c} + \frac {\sqrt {c x^{4} + a}}{x^{2}}}\right )}{4 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 63, normalized size = 2.10 \begin {gather*} \left [\frac {\log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right )}{4 \, \sqrt {c}}, -\frac {\sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2}}{\sqrt {c x^{4} + a}}\right )}{2 \, c}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.44, size = 20, normalized size = 0.67 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(sqrt(c)*x**2/sqrt(a))/(2*sqrt(c))

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Giac [A]
time = 3.95, size = 25, normalized size = 0.83 \begin {gather*} -\frac {\log \left ({\left | -\sqrt {c} x^{2} + \sqrt {c x^{4} + a} \right |}\right )}{2 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x}{\sqrt {c\,x^4+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + c*x^4)^(1/2),x)

[Out]

int(x/(a + c*x^4)^(1/2), x)

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