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

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Rubi [A]  time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {x \sqrt {c-b}}{\sqrt {c} \sqrt {a-d}}\right )}{\sqrt {c} \sqrt {a-d} \sqrt {c-b}} \]

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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fricas [B]  time = 0.83, size = 182, normalized size = 3.64 \[ \left [\frac {\log \left (\frac {{\left (b - c\right )} x^{2} + a c - c d + 2 \, \sqrt {a b c - a c^{2} - {\left (b c - c^{2}\right )} d} x}{{\left (b - c\right )} x^{2} - a c + c d}\right )}{2 \, \sqrt {a b c - a c^{2} - {\left (b c - c^{2}\right )} d}}, \frac {\sqrt {-a b c + a c^{2} + {\left (b c - c^{2}\right )} d} \arctan \left (-\frac {\sqrt {-a b c + a c^{2} + {\left (b c - c^{2}\right )} d} x}{a c - c d}\right )}{a b c - a c^{2} - {\left (b c - c^{2}\right )} d}\right ] \]

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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giac [A]  time = 0.64, size = 58, normalized size = 1.16 \[ -\frac {\arctan \left (\frac {b x - c x}{\sqrt {-a b c + a c^{2} + b c d - c^{2} d}}\right )}{\sqrt {-a b c + a c^{2} + b c d - c^{2} d}} \]

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)

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maple [A]  time = 0.01, size = 38, normalized size = 0.76 \[ \frac {\arctanh \left (\frac {\left (b -c \right ) x}{\sqrt {\left (a -d \right ) \left (b -c \right ) c}}\right )}{\sqrt {\left (a -d \right ) \left (b -c \right ) c}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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

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mupad [B]  time = 4.88, size = 46, normalized size = 0.92 \[ \frac {\mathrm {atanh}\left (\frac {x\,\left (2\,b-2\,c\right )}{2\,\sqrt {c}\,\sqrt {a-d}\,\sqrt {b-c}}\right )}{\sqrt {c}\,\sqrt {a-d}\,\sqrt {b-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*(a - d) - x^2*(b - c)),x)

[Out]

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

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sympy [B]  time = 0.32, size = 104, normalized size = 2.08 \[ - \frac {\sqrt {\frac {1}{c \left (a - d\right ) \left (b - c\right )}} \log {\left (- a c \sqrt {\frac {1}{c \left (a - d\right ) \left (b - c\right )}} + c d \sqrt {\frac {1}{c \left (a - d\right ) \left (b - c\right )}} + x \right )}}{2} + \frac {\sqrt {\frac {1}{c \left (a - d\right ) \left (b - c\right )}} \log {\left (a c \sqrt {\frac {1}{c \left (a - d\right ) \left (b - c\right )}} - c d \sqrt {\frac {1}{c \left (a - d\right ) \left (b - c\right )}} + x \right )}}{2} \]

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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