Optimal. Leaf size=34 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b-a c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \]
[Out]
________________________________________________________________________________________
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 = {214}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {x \sqrt {b-a c}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 36, normalized size = 1.06 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {-b+a c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {-b+a c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 34, normalized size = 1.00
method | result | size |
default | \(\frac {\arctan \left (\frac {\left (a c -b \right ) x}{\sqrt {a \left (a c -b \right )}}\right )}{\sqrt {a \left (a c -b \right )}}\) | \(34\) |
risch | \(-\frac {\ln \left (\left (a c -b \right ) x +\sqrt {-a \left (a c -b \right )}\right )}{2 \sqrt {-a \left (a c -b \right )}}+\frac {\ln \left (\left (-a c +b \right ) x +\sqrt {-a \left (a c -b \right )}\right )}{2 \sqrt {-a \left (a c -b \right )}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.46, size = 105, normalized size = 3.09 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (29) = 58\).
time = 0.10, size = 66, normalized size = 1.94 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.47, size = 36, normalized size = 1.06 \begin {gather*} \frac {\arctan \left (\frac {a c x - b x}{\sqrt {a^{2} c - a b}}\right )}{\sqrt {a^{2} c - a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.64, size = 38, normalized size = 1.12 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________