Optimal. Leaf size=31 \[ \frac {x \tanh ^2(x)}{\sqrt {a \tanh ^4(x)}}-\frac {\tanh (x)}{\sqrt {a \tanh ^4(x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3658, 3473, 8} \[ \frac {x \tanh ^2(x)}{\sqrt {a \tanh ^4(x)}}-\frac {\tanh (x)}{\sqrt {a \tanh ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \tanh ^4(x)}} \, dx &=\frac {\tanh ^2(x) \int \coth ^2(x) \, dx}{\sqrt {a \tanh ^4(x)}}\\ &=-\frac {\tanh (x)}{\sqrt {a \tanh ^4(x)}}+\frac {\tanh ^2(x) \int 1 \, dx}{\sqrt {a \tanh ^4(x)}}\\ &=-\frac {\tanh (x)}{\sqrt {a \tanh ^4(x)}}+\frac {x \tanh ^2(x)}{\sqrt {a \tanh ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 19, normalized size = 0.61 \[ \frac {\tanh (x) (x \tanh (x)-1)}{\sqrt {a \tanh ^4(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 238, normalized size = 7.68 \[ \frac {{\left (x \cosh \relax (x)^{2} + {\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sinh \relax (x)^{2} + {\left (x \cosh \relax (x)^{2} - x - 2\right )} e^{\left (4 \, x\right )} + 2 \, {\left (x \cosh \relax (x)^{2} - x - 2\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \relax (x) e^{\left (4 \, x\right )} + 2 \, x \cosh \relax (x) e^{\left (2 \, x\right )} + x \cosh \relax (x)\right )} \sinh \relax (x) - x - 2\right )} \sqrt {\frac {a e^{\left (8 \, x\right )} - 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} - 4 \, a e^{\left (2 \, x\right )} + a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}}}{a \cosh \relax (x)^{2} + {\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{2} + {\left (a \cosh \relax (x)^{2} - a\right )} e^{\left (4 \, x\right )} - 2 \, {\left (a \cosh \relax (x)^{2} - a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \relax (x) e^{\left (4 \, x\right )} - 2 \, a \cosh \relax (x) e^{\left (2 \, x\right )} + a \cosh \relax (x)\right )} \sinh \relax (x) - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 19, normalized size = 0.61 \[ \frac {x}{\sqrt {a}} - \frac {2}{\sqrt {a} {\left (e^{\left (2 \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 32, normalized size = 1.03 \[ -\frac {\tanh \relax (x ) \left (\ln \left (\tanh \relax (x )-1\right ) \tanh \relax (x )-\ln \left (1+\tanh \relax (x )\right ) \tanh \relax (x )+2\right )}{2 \sqrt {a \left (\tanh ^{4}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 23, normalized size = 0.74 \[ \frac {x}{\sqrt {a}} + \frac {2 \, \sqrt {a}}{a e^{\left (-2 \, x\right )} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {a\,{\mathrm {tanh}\relax (x)}^4}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \tanh ^{4}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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