Optimal. Leaf size=85 \[ -\frac {1}{3 c x^3}-\frac {c}{x}-\frac {\sqrt {1-c x}}{3 c x^3 \sqrt {\frac {1}{1+c x}}}-\frac {2 c \sqrt {1-c x}}{3 x \sqrt {\frac {1}{1+c x}}}+c^2 \tanh ^{-1}(c x) \]
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Rubi [A]
time = 0.11, 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 = {6476, 1972,
105, 12, 97, 331, 212} \begin {gather*} 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 97
Rule 105
Rule 212
Rule 331
Rule 1972
Rule 6476
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 = 0.18, size = 90, normalized size = 1.06 \begin {gather*} -\frac {2+6 c^2 x^2+2 \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )+3 c^3 x^3 \log (1-c x)-3 c^3 x^3 \log (1+c x)}{6 c x^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.34, size = 90, normalized size = 1.06
method | result | size |
default | \(-\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \mathrm {csgn}\left (c \right )^{2} \left (2 c^{2} x^{2}+1\right )}{3 x^{2}}+\frac {\frac {c^{3} \ln \left (c x +1\right )}{2}-\frac {1}{3 x^{3}}-\frac {c^{2}}{x}-\frac {c^{3} \ln \left (c x -1\right )}{2}}{c}\) | \(90\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 89, normalized size = 1.05 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} - \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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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