Optimal. Leaf size=16 \[ -\frac {2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2188, 30}
\begin {gather*} -\frac {2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2188
Rubi steps
\begin {align*} \int \frac {1}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b}\\ &=-\frac {2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 15, normalized size = 0.94
method | result | size |
derivativedivides | \(-\frac {2}{b \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}\) | \(15\) |
default | \(-\frac {2}{b \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{\sqrt {b x + a} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 20, normalized size = 1.25 \begin {gather*} -\frac {2 \, \sqrt {b x + a}}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 45.83, size = 26, normalized size = 1.62 \begin {gather*} \begin {cases} - \frac {2}{b \sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}} & \text {for}\: b \neq 0 \\\frac {x}{\operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a \right )} \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{\sqrt {b x + a} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.31, size = 97, normalized size = 6.06 \begin {gather*} \frac {4\,\sqrt {\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}}{b\,\left (\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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