Optimal. Leaf size=31 \[ -\frac {\tanh (x)}{\sqrt {a \tanh ^4(x)}}+\frac {x \tanh ^2(x)}{\sqrt {a \tanh ^4(x)}} \]
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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 = {3739, 3554, 8}
\begin {gather*} \frac {x \tanh ^2(x)}{\sqrt {a \tanh ^4(x)}}-\frac {\tanh (x)}{\sqrt {a \tanh ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3739
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 \begin {gather*} \frac {\tanh (x) (-1+x \tanh (x))}{\sqrt {a \tanh ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 32, normalized size = 1.03
method | result | size |
derivativedivides | \(\frac {\tanh \left (x \right ) \left (\ln \left (1+\tanh \left (x \right )\right ) \tanh \left (x \right )-\ln \left (\tanh \left (x \right )-1\right ) \tanh \left (x \right )-2\right )}{2 \sqrt {a \left (\tanh ^{4}\left (x \right )\right )}}\) | \(32\) |
default | \(\frac {\tanh \left (x \right ) \left (\ln \left (1+\tanh \left (x \right )\right ) \tanh \left (x \right )-\ln \left (\tanh \left (x \right )-1\right ) \tanh \left (x \right )-2\right )}{2 \sqrt {a \left (\tanh ^{4}\left (x \right )\right )}}\) | \(32\) |
risch | \(\frac {\left ({\mathrm e}^{2 x}-1\right )^{2} x}{\sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{4}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {2 \left ({\mathrm e}^{2 x}-1\right )}{\sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{4}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 23, normalized size = 0.74 \begin {gather*} \frac {x}{\sqrt {a}} + \frac {2 \, \sqrt {a}}{a e^{\left (-2 \, x\right )} - a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 238 vs.
\(2 (27) = 54\).
time = 0.36, size = 238, normalized size = 7.68 \begin {gather*} \frac {{\left (x \cosh \left (x\right )^{2} + {\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )^{2} + {\left (x \cosh \left (x\right )^{2} - x - 2\right )} e^{\left (4 \, x\right )} + 2 \, {\left (x \cosh \left (x\right )^{2} - x - 2\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, x \cosh \left (x\right ) e^{\left (2 \, x\right )} + x \cosh \left (x\right )\right )} \sinh \left (x\right ) - 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 \left (x\right )^{2} + {\left (a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (a \cosh \left (x\right )^{2} - a\right )} e^{\left (4 \, x\right )} - 2 \, {\left (a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \left (x\right ) e^{\left (4 \, x\right )} - 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a} \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*} \int \frac {1}{\sqrt {a \tanh ^{4}{\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 19, normalized size = 0.61 \begin {gather*} \frac {x}{\sqrt {a}} - \frac {2}{\sqrt {a} {\left (e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a\,{\mathrm {tanh}\left (x\right )}^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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