Optimal. Leaf size=21 \[ \frac {\sqrt {1-a^2 x^2}}{a c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6266, 6263,
267} \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 267
Rule 6263
Rule 6266
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\frac {a \int \frac {e^{-\tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac {a \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {\sqrt {1-a^2 x^2}}{a c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.73, size = 150, normalized size = 7.14
method | result | size |
gosper | \(\frac {\sqrt {-a^{2} x^{2}+1}}{a c}\) | \(20\) |
trager | \(\frac {\sqrt {-a^{2} x^{2}+1}}{a c}\) | \(20\) |
risch | \(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c}\) | \(30\) |
default | \(\frac {a \left (\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}}{2 a^{2}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}}{2 a^{2}}\right )}{c}\) | \(150\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 22, normalized size = 1.05 \begin {gather*} \frac {\sqrt {a x + 1} \sqrt {-a x + 1}}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 19, normalized size = 0.90 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 5.76, size = 53, normalized size = 2.52 \begin {gather*} a \left (\begin {cases} \tilde {\infty } \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c = 0 \\- \frac {x^{2}}{2 c} & \text {for}\: a^{2} = 0 \\\frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2} c} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 19, normalized size = 0.90 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 19, normalized size = 0.90 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________