Optimal. Leaf size=65 \[ -\frac {\left (1-a^2 x^2\right )^{3/2}}{a c (1-a x)^2}-\frac {2 \sqrt {1-a^2 x^2}}{a c}+\frac {2 \sin ^{-1}(a x)}{a c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6131, 6128, 793, 665, 216} \[ -\frac {\left (1-a^2 x^2\right )^{3/2}}{a c (1-a x)^2}-\frac {2 \sqrt {1-a^2 x^2}}{a c}+\frac {2 \sin ^{-1}(a x)}{a c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 665
Rule 793
Rule 6128
Rule 6131
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}}{(1-a x)^2} \, dx}{c}\\ &=-\frac {\left (1-a^2 x^2\right )^{3/2}}{a c (1-a x)^2}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{1-a x} \, dx}{c}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a c}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a c (1-a x)^2}+\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a c}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a c (1-a x)^2}+\frac {2 \sin ^{-1}(a x)}{a c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 52, normalized size = 0.80 \[ \frac {\frac {(a x-3) \sqrt {a x+1}}{\sqrt {1-a x}}-4 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 68, normalized size = 1.05 \[ -\frac {3 \, a x + 4 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 3\right )} - 3}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 73, normalized size = 1.12 \[ \frac {2 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} - \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 96, normalized size = 1.48 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{a c}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{2} c \left (x -\frac {1}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 91, normalized size = 1.40 \[ \frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a \left (\int \frac {x}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________