Optimal. Leaf size=104 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{4 b^{3/2}}+\frac {1}{2} x^{3/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{4 b} \]
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Rubi [A] time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2169, 2165} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{4 b^{3/2}}+\frac {1}{2} x^{3/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 2165
Rule 2169
Rubi steps
\begin {align*} \int \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac {1}{2} x^{3/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {1}{4} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac {\sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=\frac {1}{2} x^{3/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{4 b}-\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{8 b}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}{4 b^{3/2}}+\frac {1}{2} x^{3/2} \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {\sqrt {x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 0.81 \[ \frac {\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (\tanh ^{-1}(\tanh (a+b x))+b x\right )}{4 b}-\frac {\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2 \log \left (\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}+b \sqrt {x}\right )}{4 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 114, normalized size = 1.10 \[ \left [\frac {a^{2} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x + a b\right )} \sqrt {b x + a} \sqrt {x}}{8 \, b^{2}}, \frac {a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (2 \, b^{2} x + a b\right )} \sqrt {b x + a} \sqrt {x}}{4 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 48, normalized size = 0.46 \[ \frac {1}{4} \, \sqrt {b x + a} {\left (2 \, x + \frac {a}{b}\right )} \sqrt {x} + \frac {a^{2} \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{4 \, b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 174, normalized size = 1.67 \[ \frac {\sqrt {x}\, \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}{2 b}-\frac {a \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{4 b}-\frac {\ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) a^{2}}{4 b^{\frac {3}{2}}}-\frac {a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{2 b^{\frac {3}{2}}}-\frac {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}\, \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{4 b}-\frac {\ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{4 b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x} \sqrt {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {x}\,\sqrt {\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )} \,d x \]
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 \[ \int \sqrt {x} \sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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