∫ esech−1(ax2)x2 dx [50]

Optimal. Leaf size=67 \[ \frac {2 x}{3 a}+\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{3 a^{3/2}} \]

2/3*x/a+1/3*(1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^3+2/3*EllipticF(x*a^(1/2),I)*(1/(a*x^2+1))^(1/2)*(

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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6470, 8, 254, 227} \begin {gather*} \frac {2 \sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} F\left (\left .\text {ArcSin}\left (\sqrt {a} x\right )\right |-1\right )}{3 a^{3/2}}+\frac {1}{3} x^3 e^{\text {sech}^{-1}\left (a x^2\right )}+\frac {2 x}{3 a} \end {gather*}

(2*x)/(3*a) + (E^ArcSech[a*x^2]*x^3)/3 + (2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*EllipticF[ArcSin[Sqrt[a]*x]

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_.)*((a2_.) + (b2_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[(a1*a2 + b1*b2*x^(2*

n))^p, x] /; FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ArcSech[a*x^p]/(m + 1)), x] + (Dist

[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[p*(Sqrt[1 + a*x^p]/(a*(m + 1)))*Sqrt[1/(1 + a*x^p)], Int[x^(m - p

)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]

\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} x^2 \, dx &=\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {2 \int 1 \, dx}{3 a}+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{3 a}\\ &=\frac {2 x}{3 a}+\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {\left (2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {1}{\sqrt {1-a^2 x^4}} \, dx}{3 a}\\ &=\frac {2 x}{3 a}+\frac {1}{3} e^{\text {sech}^{-1}\left (a x^2\right )} x^3+\frac {2 \sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt {a} x\right )\right |-1\right )}{3 a^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 116, normalized size = 1.73 \begin {gather*} \frac {x}{a}+\frac {\sqrt {\frac {1-a x^2}{1+a x^2}} \left (x+a x^3\right )}{3 a}-\frac {2 i \sqrt {\frac {1-a x^2}{1+a x^2}} \sqrt {1-a^2 x^4} F\left (\left .i \sinh ^{-1}\left (\sqrt {-a} x\right )\right |-1\right )}{3 (-a)^{3/2} \left (-1+a x^2\right )} \end {gather*}

x/a + (Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(x + a*x^3))/(3*a) - (((2*I)/3)*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*Sqrt[1 - a^

2*x^4]*EllipticF[I*ArcSinh[Sqrt[-a]*x], -1])/((-a)^(3/2)*(-1 + a*x^2))

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method	result	size

default	\(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (a^{\frac {5}{2}} x^{5}-2 \EllipticF \left (x \sqrt {a}, i\right ) \sqrt {-a \,x^{2}+1}\, \sqrt {a \,x^{2}+1}-x \sqrt {a}\right )}{3 \left (a^{2} x^{4}-1\right ) \sqrt {a}}+\frac {x}{a}\)	\(102\)

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

1/3*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(a^(5/2)*x^5-2*EllipticF(x*a^(1/2),I)*(-a*x^2+1)^(1/2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.13, size = 47, normalized size = 0.70 \begin {gather*} \frac {a x^{3} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} + 3 \, x}{3 \, a} \end {gather*}

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

1/3*(a*x^3*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)) + 3*x)/a

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int 1\, dx + \int a x^{2} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \end {gather*}

integrate((1/a/x**2+(1/a/x**2-1)**(1/2)*(1/a/x**2+1)**(1/2))*x**2,x)

(Integral(1, x) + Integral(a*x**2*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x**2)), x))/a

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,\left (\sqrt {\frac {1}{a\,x^2}-1}\,\sqrt {\frac {1}{a\,x^2}+1}+\frac {1}{a\,x^2}\right ) \,d x \end {gather*}

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

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

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3.1.50 \(\int e^{\text {sech}^{-1}(a x^2)} x^2 \, dx\) [50]