Optimal. Leaf size=85 \[ c^2 \tanh ^{-1}(c x)-\frac {\sqrt {1-c x}}{3 c x^3 \sqrt {\frac {1}{c x+1}}}-\frac {1}{3 c x^3}-\frac {2 c \sqrt {1-c x}}{3 x \sqrt {\frac {1}{c x+1}}}-\frac {c}{x} \]
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Rubi [A] time = 0.17, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6341, 1956, 103, 12, 95, 325, 206} \[ c^2 \tanh ^{-1}(c x)-\frac {\sqrt {1-c x}}{3 c x^3 \sqrt {\frac {1}{c x+1}}}-\frac {1}{3 c x^3}-\frac {2 c \sqrt {1-c x}}{3 x \sqrt {\frac {1}{c x+1}}}-\frac {c}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 103
Rule 206
Rule 325
Rule 1956
Rule 6341
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(c x)}}{x^3 \left (1-c^2 x^2\right )} \, dx &=\frac {\int \frac {\sqrt {\frac {1}{1+c x}}}{x^4 \sqrt {1-c x}} \, dx}{c}+\frac {\int \frac {1}{x^4 \left (1-c^2 x^2\right )} \, dx}{c}\\ &=-\frac {1}{3 c x^3}+c \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^4 \sqrt {1-c x} \sqrt {1+c x}} \, dx}{c}\\ &=-\frac {1}{3 c x^3}-\frac {c}{x}-\frac {\sqrt {1-c x}}{3 c x^3 \sqrt {\frac {1}{1+c x}}}+c^3 \int \frac {1}{1-c^2 x^2} \, dx-\frac {\left (\sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {2 c^2}{x^2 \sqrt {1-c x} \sqrt {1+c x}} \, dx}{3 c}\\ &=-\frac {1}{3 c x^3}-\frac {c}{x}-\frac {\sqrt {1-c x}}{3 c x^3 \sqrt {\frac {1}{1+c x}}}+c^2 \tanh ^{-1}(c x)+\frac {1}{3} \left (2 c \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\\ &=-\frac {1}{3 c x^3}-\frac {c}{x}-\frac {\sqrt {1-c x}}{3 c x^3 \sqrt {\frac {1}{1+c x}}}-\frac {2 c \sqrt {1-c x}}{3 x \sqrt {\frac {1}{1+c x}}}+c^2 \tanh ^{-1}(c x)\\ \end {align*}
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Mathematica [A] time = 0.26, size = 90, normalized size = 1.06 \[ -\frac {3 c^3 x^3 \log (1-c x)-3 c^3 x^3 \log (c x+1)+6 c^2 x^2+2 \sqrt {\frac {1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )+2}{6 c x^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.86, size = 89, normalized size = 1.05 \[ \frac {3 \, c^{3} x^{3} \log \left (c x + 1\right ) - 3 \, c^{3} x^{3} \log \left (c x - 1\right ) - 6 \, c^{2} x^{2} - 2 \, {\left (2 \, c^{3} x^{3} + c x\right )} \sqrt {\frac {c x + 1}{c x}} \sqrt {-\frac {c x - 1}{c x}} - 2}{6 \, c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}}{{\left (c^{2} x^{2} - 1\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 86, normalized size = 1.01 \[ -\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \mathrm {csgn}\relax (c )^{2} \left (2 c^{2} x^{2}+1\right )}{3 x^{2}}-\frac {1}{3 c \,x^{3}}-\frac {c}{x}+\frac {c^{2} \ln \left (c x +1\right )}{2}-\frac {c^{2} \ln \left (c x -1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, c^{2} \log \left (c x + 1\right ) - \frac {1}{2} \, c^{2} \log \left (c x - 1\right ) + c \int \frac {1}{x^{2}}\,{d x} + \frac {-\frac {1}{3 \, x^{3}}}{c} - \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c^{3} x^{6} - c x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.51, size = 75, normalized size = 0.88 \[ c^2\,\mathrm {atanh}\left (c\,x\right )-\frac {\left (\frac {\sqrt {\frac {1}{c\,x}+1}}{3}+\frac {2\,c^2\,x^2\,\sqrt {\frac {1}{c\,x}+1}}{3}\right )\,\sqrt {\frac {1}{c\,x}-1}}{x^2}-\frac {c^2\,x^2+\frac {1}{3}}{c\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {c x \sqrt {-1 + \frac {1}{c x}} \sqrt {1 + \frac {1}{c x}}}{c^{2} x^{6} - x^{4}}\, dx + \int \frac {1}{c^{2} x^{6} - x^{4}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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