Optimal. Leaf size=25 \[ 2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {4}{3} b x^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 25, 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} \[ 2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-\frac {4}{3} b x^{3/2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))}{\sqrt {x}} \, dx &=2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))-(2 b) \int \sqrt {x} \, dx\\ &=-\frac {4}{3} b x^{3/2}+2 \sqrt {x} \tanh ^{-1}(\tanh (a+b x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.92 \[ \frac {2}{3} \sqrt {x} \left (3 \tanh ^{-1}(\tanh (a+b x))-2 b x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 12, normalized size = 0.48 \[ \frac {2}{3} \, {\left (b x + 3 \, a\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 13, normalized size = 0.52 \[ \frac {2}{3} \, b x^{\frac {3}{2}} + 2 \, a \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 20, normalized size = 0.80 \[ -\frac {4 b \,x^{\frac {3}{2}}}{3}+2 \arctanh \left (\tanh \left (b x +a \right )\right ) \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 19, normalized size = 0.76 \[ -\frac {4}{3} \, b x^{\frac {3}{2}} + 2 \, \sqrt {x} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 56, normalized size = 2.24 \[ \sqrt {x}\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\frac {4\,b\,x^{3/2}}{3}-\sqrt {x}\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right ) \]
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 \[ \int \frac {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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