3.1.97 \(\int \frac {x (-1+a e^{\text {sech}^{-1}(a x)} x)}{1-a^2 x^2} \, dx\) [97]

Optimal. Leaf size=12 \[ -\frac {e^{\text {sech}^{-1}(a x)} x}{a} \]

[Out]

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

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(26\) vs. \(2(12)=24\).
time = 0.74, antiderivative size = 26, normalized size of antiderivative = 2.17, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6857, 266, 6476, 1972, 75} \begin {gather*} -\frac {\sqrt {1-a x}}{a^2 \sqrt {\frac {1}{a x+1}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(x*(-1 + a*E^ArcSech[a*x]*x))/(1 - a^2*x^2),x]

[Out]

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

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1972

Int[(u_.)*((c_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(a + b*x^n)
^(p*q)], Int[u*(a + b*x^n)^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] && GeQ[a, 0]

Rule 6476

Int[(E^ArcSech[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[d/(a*c), Int[(d*x)^(m
- 1)*(Sqrt[1/(1 + c*x)]/Sqrt[1 - c*x]), x], x] + Dist[d/c, Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a,
b, c, d, m}, x] && EqQ[b + a*c^2, 0]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x \left (-1+a e^{\text {sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx &=\int \left (\frac {x}{-1+a^2 x^2}-\frac {a e^{\text {sech}^{-1}(a x)} x^2}{-1+a^2 x^2}\right ) \, dx\\ &=-\left (a \int \frac {e^{\text {sech}^{-1}(a x)} x^2}{-1+a^2 x^2} \, dx\right )+\int \frac {x}{-1+a^2 x^2} \, dx\\ &=\frac {\log \left (1-a^2 x^2\right )}{2 a^2}+\int \frac {x \sqrt {\frac {1}{1+a x}}}{\sqrt {1-a x}} \, dx-\int \frac {x}{-1+a^2 x^2} \, dx\\ &=\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {x}{\sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=-\frac {\sqrt {1-a x}}{a^2 \sqrt {\frac {1}{1+a x}}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(28\) vs. \(2(12)=24\).
time = 0.17, size = 28, normalized size = 2.33 \begin {gather*} -\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{a^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*(-1 + a*E^ArcSech[a*x]*x))/(1 - a^2*x^2),x]

[Out]

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

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Maple [A]
time = 0.30, size = 36, normalized size = 3.00

method	result	size
gosper	\(-\frac {x \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{a}\)	\(36\)
default	\(-\frac {x \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{a}\)	\(36\)
risch	\(-\frac {x \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}}{a}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*x*(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.51, size = 35, normalized size = 2.92 \begin {gather*} -\frac {x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.96, size = 76, normalized size = 6.33 \begin {gather*} \frac {\ln \left (\frac {1}{x}\right )}{a^2}-\frac {\ln \left (a+\frac {1}{x}\right )}{2\,a^2}-\frac {\ln \left (\frac {1}{x}-a\right )}{2\,a^2}+\frac {\ln \left (a^2\,x^2-1\right )}{2\,a^2}-\frac {x\,\sqrt {\frac {1}{a\,x}-1}\,\sqrt {\frac {1}{a\,x}+1}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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