Optimal. Leaf size=23 \[ 4 b \sqrt {x}-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 30} \[ 4 b \sqrt {x}-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^{3/2}} \, dx &=-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt {x}}+(2 b) \int \frac {1}{\sqrt {x}} \, dx\\ &=4 b \sqrt {x}-\frac {2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 20, normalized size = 0.87 \[ \frac {4 b x-2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 12, normalized size = 0.52 \[ \frac {2 \, {\left (b x - a\right )}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 13, normalized size = 0.57 \[ 2 \, b \sqrt {x} - \frac {2 \, a}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 20, normalized size = 0.87 \[ -\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )}{\sqrt {x}}+4 b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 19, normalized size = 0.83 \[ 4 \, b \sqrt {x} - \frac {2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 56, normalized size = 2.43 \[ \frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{\sqrt {x}}+4\,b\,\sqrt {x}-\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.05, size = 22, normalized size = 0.96 \[ 4 b \sqrt {x} - \frac {2 \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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