Optimal. Leaf size=108 \[ -\frac{1}{2} c \log \left (1-c^2 x^2\right )-\frac{\sqrt{1-c x}}{2 c x^2 \sqrt{\frac{1}{c x+1}}}-\frac{1}{2 c x^2}+c \log (x)-\frac{1}{2} c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.176461, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6341, 1956, 103, 12, 92, 208, 266, 44} \[ -\frac{1}{2} c \log \left (1-c^2 x^2\right )-\frac{\sqrt{1-c x}}{2 c x^2 \sqrt{\frac{1}{c x+1}}}-\frac{1}{2 c x^2}+c \log (x)-\frac{1}{2} c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{c x+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6341
Rule 1956
Rule 103
Rule 12
Rule 92
Rule 208
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}(c x)}}{x^2 \left (1-c^2 x^2\right )} \, dx &=\frac{\int \frac{\sqrt{\frac{1}{1+c x}}}{x^3 \sqrt{1-c x}} \, dx}{c}+\frac{\int \frac{1}{x^3 \left (1-c^2 x^2\right )} \, dx}{c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 c}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^3 \sqrt{1-c x} \sqrt{1+c x}} \, dx}{c}\\ &=-\frac{\sqrt{1-c x}}{2 c x^2 \sqrt{\frac{1}{1+c x}}}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c^2}{x}-\frac{c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{2 c}+\frac{\left (\sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{c^2}{x \sqrt{1-c x} \sqrt{1+c x}} \, dx}{2 c}\\ &=-\frac{1}{2 c x^2}-\frac{\sqrt{1-c x}}{2 c x^2 \sqrt{\frac{1}{1+c x}}}+c \log (x)-\frac{1}{2} c \log \left (1-c^2 x^2\right )+\frac{1}{2} \left (c \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=-\frac{1}{2 c x^2}-\frac{\sqrt{1-c x}}{2 c x^2 \sqrt{\frac{1}{1+c x}}}+c \log (x)-\frac{1}{2} c \log \left (1-c^2 x^2\right )-\frac{1}{2} \left (c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{c-c x^2} \, dx,x,\sqrt{1-c x} \sqrt{1+c x}\right )\\ &=-\frac{1}{2 c x^2}-\frac{\sqrt{1-c x}}{2 c x^2 \sqrt{\frac{1}{1+c x}}}-\frac{1}{2} c \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\sqrt{1-c x} \sqrt{1+c x}\right )+c \log (x)-\frac{1}{2} c \log \left (1-c^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.151469, size = 108, normalized size = 1. \[ \frac{1}{2} \left (-c \log \left (1-c^2 x^2\right )-\frac{\sqrt{\frac{1-c x}{c x+1}} (c x+1)}{c x^2}-\frac{1}{c x^2}+3 c \log (x)-c \log \left (c x \sqrt{\frac{1-c x}{c x+1}}+\sqrt{\frac{1-c x}{c x+1}}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.227, size = 111, normalized size = 1. \begin{align*} -{\frac{1}{2\,x}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ({c}^{2}{x}^{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) +\sqrt{-{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}-{\frac{c\ln \left ( cx-1 \right ) }{2}}-{\frac{1}{2\,c{x}^{2}}}+c\ln \left ( x \right ) -{\frac{c\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*} c \int \frac{1}{x}\,{d x} - \frac{1}{2} \, c \log \left (c x + 1\right ) - \frac{1}{2} \, c \log \left (c x - 1\right ) + \frac{-\frac{1}{2 \, x^{2}}}{c} - \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1}}{c^{3} x^{5} - c x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.12026, size = 344, normalized size = 3.19 \begin{align*} -\frac{2 \, c^{2} x^{2} \log \left (c^{2} x^{2} - 1\right ) + c^{2} x^{2} \log \left (c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + 1\right ) - c^{2} x^{2} \log \left (c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} - 1\right ) - 4 \, c^{2} x^{2} \log \left (x\right ) + 2 \, c x \sqrt{\frac{c x + 1}{c x}} \sqrt{-\frac{c x - 1}{c x}} + 2}{4 \, c x^{2}} \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^{5} - x^{3}}\, dx + \int \frac{1}{c^{2} x^{5} - x^{3}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]