3.262 \(\int \frac {1}{a-(b-a c) x^2} \, dx\)

Optimal. Leaf size=34 \[ \frac {\tanh ^{-1}\left (\frac {x \sqrt {b-a c}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {208} \[ \frac {\tanh ^{-1}\left (\frac {x \sqrt {b-a c}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \]

Antiderivative was successfully verified.

[In]

Int[(a - (b - a*c)*x^2)^(-1),x]

[Out]

ArcTanh[(Sqrt[b - a*c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[b - a*c])

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

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.06 \[ \frac {\tan ^{-1}\left (\frac {x \sqrt {a c-b}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {a c-b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - (b - a*c)*x^2)^(-1),x]

[Out]

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

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fricas [A]  time = 0.83, size = 105, normalized size = 3.09 \[ \left [-\frac {\sqrt {-a^{2} c + a b} \log \left (\frac {{\left (a c - b\right )} x^{2} - 2 \, \sqrt {-a^{2} c + a b} x - a}{{\left (a c - b\right )} x^{2} + a}\right )}{2 \, {\left (a^{2} c - a b\right )}}, \frac {\arctan \left (\frac {\sqrt {a^{2} c - a b} x}{a}\right )}{\sqrt {a^{2} c - a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.62, size = 36, normalized size = 1.06 \[ \frac {\arctan \left (\frac {a c x - b x}{\sqrt {a^{2} c - a b}}\right )}{\sqrt {a^{2} c - a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 34, normalized size = 1.00 \[ \frac {\arctan \left (\frac {\left (a c -b \right ) x}{\sqrt {\left (a c -b \right ) a}}\right )}{\sqrt {\left (a c -b \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*c-b>0)', see `assume?` for m
ore details)Is a*c-b positive or negative?

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mupad [B]  time = 4.64, size = 38, normalized size = 1.12 \[ -\frac {\mathrm {atan}\left (\frac {x\,\left (2\,b-2\,a\,c\right )}{2\,\sqrt {a^2\,c-a\,b}}\right )}{\sqrt {a^2\,c-a\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - x^2*(b - a*c)),x)

[Out]

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

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sympy [B]  time = 0.23, size = 66, normalized size = 1.94 \[ - \frac {\sqrt {- \frac {1}{a \left (a c - b\right )}} \log {\left (- a \sqrt {- \frac {1}{a \left (a c - b\right )}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a \left (a c - b\right )}} \log {\left (a \sqrt {- \frac {1}{a \left (a c - b\right )}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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