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

Optimal. Leaf size=50 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-c}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{b-c}} \]

[Out]

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

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Rubi [A]  time = 0.0533083, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {208} \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-c}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{b-c}} \]

Antiderivative was successfully verified.

[In]

Int[(c*(a - d) - (b - c)*x^2)^(-1),x]

[Out]

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

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

Mathematica [A]  time = 0.0202594, size = 50, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{c-b}}{\sqrt{c} \sqrt{a-d}}\right )}{\sqrt{c} \sqrt{a-d} \sqrt{c-b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*(a - d) - (b - c)*x^2)^(-1),x]

[Out]

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

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Maple [A]  time = 0.007, size = 38, normalized size = 0.8 \begin{align*}{{\it Artanh} \left ({ \left ( b-c \right ) x{\frac{1}{\sqrt{c \left ( a-d \right ) \left ( b-c \right ) }}}} \right ){\frac{1}{\sqrt{c \left ( a-d \right ) \left ( b-c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.52494, size = 370, normalized size = 7.4 \begin{align*} \left [\frac{\log \left (\frac{{\left (b - c\right )} x^{2} + a c - c d + 2 \, \sqrt{a b c - a c^{2} -{\left (b c - c^{2}\right )} d} x}{{\left (b - c\right )} x^{2} - a c + c d}\right )}{2 \, \sqrt{a b c - a c^{2} -{\left (b c - c^{2}\right )} d}}, \frac{\sqrt{-a b c + a c^{2} +{\left (b c - c^{2}\right )} d} \arctan \left (-\frac{\sqrt{-a b c + a c^{2} +{\left (b c - c^{2}\right )} d} x}{a c - c d}\right )}{a b c - a c^{2} -{\left (b c - c^{2}\right )} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.352276, size = 104, normalized size = 2.08 \begin{align*} - \frac{\sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} \log{\left (- a c \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + c d \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} \log{\left (a c \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} - c d \sqrt{\frac{1}{c \left (a - d\right ) \left (b - c\right )}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.47143, size = 78, normalized size = 1.56 \begin{align*} -\frac{\arctan \left (\frac{b x - c x}{\sqrt{-a b c + a c^{2} + b c d - c^{2} d}}\right )}{\sqrt{-a b c + a c^{2} + b c d - c^{2} d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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