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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {205} \[ \frac {\tan ^{-1}\left (\frac {x \sqrt {b-a c}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \]

Antiderivative was successfully verified.

[In]

Int[(a + (b - a*c)*x^2)^(-1),x]

[Out]

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

Rule 205

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

Rubi steps

\begin {align*} \int \frac {1}{a+(b-a c) x^2} \, dx &=\frac {\tan ^{-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 = 0.02, size = 36, normalized size = 1.06 \[ \frac {\tanh ^{-1}\left (\frac {x \sqrt {a c-b}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {a c-b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + (b - a*c)*x^2)^(-1),x]

[Out]

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

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fricas [A]  time = 0.87, size = 106, normalized size = 3.12 \[ \left [\frac {\log \left (\frac {{\left (a c - b\right )} x^{2} + 2 \, \sqrt {a^{2} c - a b} x + a}{{\left (a c - b\right )} x^{2} - a}\right )}{2 \, \sqrt {a^{2} c - a b}}, -\frac {\sqrt {-a^{2} c + a b} \arctan \left (\frac {\sqrt {-a^{2} c + a b} x}{a}\right )}{a^{2} c - a b}\right ] \]

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

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giac [A]  time = 0.63, size = 37, normalized size = 1.09 \[ -\frac {\arctan \left (\frac {a c x - b x}{\sqrt {-a^{2} c + a b}}\right )}{\sqrt {-a^{2} c + a b}} \]

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 = 0.01, size = 34, normalized size = 1.00 \[ \frac {\arctanh \left (\frac {\left (a c -b \right ) x}{\sqrt {\left (a c -b \right ) a}}\right )}{\sqrt {\left (a c -b \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 = 5.14, size = 38, normalized size = 1.12 \[ -\frac {\mathrm {atanh}\left (\frac {x\,\left (2\,b-2\,a\,c\right )}{2\,\sqrt {a}\,\sqrt {a\,c-b}}\right )}{\sqrt {a}\,\sqrt {a\,c-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + x^2*(b - a*c)),x)

[Out]

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

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sympy [B]  time = 0.25, size = 60, normalized size = 1.76 \[ - \frac {\sqrt {\frac {1}{a \left (a c - b\right )}} \log {\left (- a \sqrt {\frac {1}{a \left (a c - b\right )}} + x \right )}}{2} + \frac {\sqrt {\frac {1}{a \left (a c - b\right )}} \log {\left (a \sqrt {\frac {1}{a \left (a c - b\right )}} + x \right )}}{2} \]

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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