3.2.9 \(\int \frac {1}{-2014-15 x+x^2} \, dx\) [109]

Optimal. Leaf size=19 \[ \frac {1}{91} \log (53-x)-\frac {1}{91} \log (38+x) \]

[Out]

1/91*ln(53-x)-1/91*ln(38+x)

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {630, 31} \begin {gather*} \frac {1}{91} \log (53-x)-\frac {1}{91} \log (x+38) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2014 - 15*x + x^2)^(-1),x]

[Out]

Log[53 - x]/91 - Log[38 + x]/91

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 630

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{91} \int \frac {1}{-53+x} \, dx-\frac {1}{91} \int \frac {1}{38+x} \, dx\\ &=\frac {1}{91} \log (53-x)-\frac {1}{91} \log (38+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{91} \log (53-x)-\frac {1}{91} \log (38+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2014 - 15*x + x^2)^(-1),x]

[Out]

Log[53 - x]/91 - Log[38 + x]/91

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Maple [A]
time = 0.08, size = 14, normalized size = 0.74

method	result	size
default	\(\frac {\ln \left (x -53\right )}{91}-\frac {\ln \left (38+x \right )}{91}\)	\(14\)
norman	\(\frac {\ln \left (x -53\right )}{91}-\frac {\ln \left (38+x \right )}{91}\)	\(14\)
risch	\(\frac {\ln \left (x -53\right )}{91}-\frac {\ln \left (38+x \right )}{91}\)	\(14\)
parallelrisch	\(\frac {\ln \left (x -53\right )}{91}-\frac {\ln \left (38+x \right )}{91}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2-15*x-2014),x,method=_RETURNVERBOSE)

[Out]

1/91*ln(x-53)-1/91*ln(38+x)

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Maxima [A]
time = 0.35, size = 13, normalized size = 0.68 \begin {gather*} -\frac {1}{91} \, \log \left (x + 38\right ) + \frac {1}{91} \, \log \left (x - 53\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-15*x-2014),x, algorithm="maxima")

[Out]

-1/91*log(x + 38) + 1/91*log(x - 53)

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Fricas [A]
time = 0.58, size = 13, normalized size = 0.68 \begin {gather*} -\frac {1}{91} \, \log \left (x + 38\right ) + \frac {1}{91} \, \log \left (x - 53\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-15*x-2014),x, algorithm="fricas")

[Out]

-1/91*log(x + 38) + 1/91*log(x - 53)

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Sympy [A]
time = 0.04, size = 12, normalized size = 0.63 \begin {gather*} \frac {\log {\left (x - 53 \right )}}{91} - \frac {\log {\left (x + 38 \right )}}{91} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2-15*x-2014),x)

[Out]

log(x - 53)/91 - log(x + 38)/91

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Giac [A]
time = 0.45, size = 15, normalized size = 0.79 \begin {gather*} -\frac {1}{91} \, \log \left ({\left | x + 38 \right |}\right ) + \frac {1}{91} \, \log \left ({\left | x - 53 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-15*x-2014),x, algorithm="giac")

[Out]

-1/91*log(abs(x + 38)) + 1/91*log(abs(x - 53))

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Mupad [B]
time = 0.09, size = 8, normalized size = 0.42 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,x}{91}-\frac {15}{91}\right )}{91} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(15*x - x^2 + 2014),x)

[Out]

-(2*atanh((2*x)/91 - 15/91))/91

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Chatgpt [F] Failed to verify
time = 1.00, size = 14, normalized size = 0.74 \begin {gather*} \frac {2 \sqrt {4013}\, \arctan \left (\frac {2 \left (x -\frac {15}{2}\right ) \sqrt {4013}}{4013}\right )}{4013} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(1/(x^2-15*x-2014),x)

[Out]

2/4013*4013^(1/2)*arctan(2/4013*(x-15/2)*4013^(1/2))

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