3.257 \(\int \frac{1}{-1+a+a x^2} \, dx\)

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

[Out]

-(ArcTanh[(Sqrt[a]*x)/Sqrt[1 - a]]/Sqrt[(1 - a)*a])

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Rubi [A]  time = 0.0267875, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{1-a}}\right )}{\sqrt{(1-a) a}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + a + a*x^2)^(-1),x]

[Out]

-(ArcTanh[(Sqrt[a]*x)/Sqrt[1 - a]]/Sqrt[(1 - a)*a])

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.0101971, size = 28, normalized size = 0.93 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-1}}\right )}{\sqrt{a-1} \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + a + a*x^2)^(-1),x]

[Out]

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

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Maple [A]  time = 0.005, size = 20, normalized size = 0.7 \begin{align*}{\arctan \left ({ax{\frac{1}{\sqrt{ \left ( a-1 \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a-1 \right ) a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.18032, size = 189, normalized size = 6.3 \begin{align*} \left [-\frac{\sqrt{-a^{2} + a} \log \left (\frac{a x^{2} - 2 \, \sqrt{-a^{2} + a} x - a + 1}{a x^{2} + a - 1}\right )}{2 \,{\left (a^{2} - a\right )}}, \frac{\arctan \left (\frac{\sqrt{a^{2} - a} x}{a - 1}\right )}{\sqrt{a^{2} - a}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.149289, size = 83, normalized size = 2.77 \begin{align*} - \frac{\sqrt{- \frac{1}{a \left (a - 1\right )}} \log{\left (- a \sqrt{- \frac{1}{a \left (a - 1\right )}} + x + \sqrt{- \frac{1}{a \left (a - 1\right )}} \right )}}{2} + \frac{\sqrt{- \frac{1}{a \left (a - 1\right )}} \log{\left (a \sqrt{- \frac{1}{a \left (a - 1\right )}} + x - \sqrt{- \frac{1}{a \left (a - 1\right )}} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(a*(a - 1)))*log(-a*sqrt(-1/(a*(a - 1))) + x + sqrt(-1/(a*(a - 1))))/2 + sqrt(-1/(a*(a - 1)))*log(a*s
qrt(-1/(a*(a - 1))) + x - sqrt(-1/(a*(a - 1))))/2

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Giac [A]  time = 2.22977, size = 31, normalized size = 1.03 \begin{align*} \frac{\arctan \left (\frac{a x}{\sqrt{a^{2} - a}}\right )}{\sqrt{a^{2} - a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(a*x/sqrt(a^2 - a))/sqrt(a^2 - a)