Optimal. Leaf size=42 \[ -\frac{\sqrt{1-c x}}{c x \sqrt{\frac{1}{c x+1}}}-\frac{1}{c x}+\tanh ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.142613, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6341, 1956, 95, 325, 206} \[ -\frac{\sqrt{1-c x}}{c x \sqrt{\frac{1}{c x+1}}}-\frac{1}{c x}+\tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6341
Rule 1956
Rule 95
Rule 325
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)}}{x \left (1-c^2 x^2\right )} \, dx &=\frac{\int \frac{\sqrt{\frac{1}{1+c x}}}{x^2 \sqrt{1-c x}} \, dx}{c}+\frac{\int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx}{c}\\ &=-\frac{1}{c x}+c \int \frac{1}{1-c^2 x^2} \, dx+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=-\frac{1}{c x}-\frac{\sqrt{1-c x}}{c x \sqrt{\frac{1}{1+c x}}}+\tanh ^{-1}(c x)\\ \end{align*}
Mathematica [A] time = 0.163771, size = 59, normalized size = 1.4 \[ -\sqrt{\frac{1-c x}{c x+1}} \left (\frac{1}{c x}+1\right )-\frac{1}{c x}-\frac{1}{2} \log (1-c x)+\frac{1}{2} \log (c x+1) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.21, size = 61, normalized size = 1.5 \begin{align*} -\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ({\it csgn} \left ( c \right ) \right ) ^{2}-{\frac{\ln \left ( cx-1 \right ) }{2}}-{\frac{1}{cx}}+{\frac{\ln \left ( cx+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{1}{x}}{c} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1}}{c^{3} x^{4} - c x^{2}}\,{d x} + \frac{1}{2} \, \log \left (c x + 1\right ) - \frac{1}{2} \, \log \left (c x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.16174, size = 144, normalized size = 3.43 \begin{align*} -\frac{2 \, c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} - c x \log \left (c x + 1\right ) + c x \log \left (c x - 1\right ) + 2}{2 \, c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{c x \sqrt{-1 + \frac{1}{c x}} \sqrt{1 + \frac{1}{c x}}}{c^{2} x^{4} - x^{2}}\, dx + \int \frac{1}{c^{2} x^{4} - x^{2}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}}{{\left (c^{2} x^{2} - 1\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]