Optimal. Leaf size=86 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^{5/2}}-\frac {2 x^{3/2}}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2199, 2200,
2196} \begin {gather*} \frac {3 \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 )}{b^{5/2}}+\frac {3 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac {2 x^{3/2}}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2196
Rule 2199
Rule 2200
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx &=-\frac {2 x^{3/2}}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{b}\\ &=-\frac {2 x^{3/2}}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}-\frac {\left (3 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{2 b^2}\\ &=\frac {3 \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 )}{b^{5/2}}-\frac {2 x^{3/2}}{b \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 \sqrt {x} \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 81, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x} \left (-2 b x+3 \tanh ^{-1}(\tanh (a+b x))\right )}{b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \log \left (b \sqrt {x}+\sqrt {b} \sqrt {\tanh ^{-1}(\tanh (a+b x))}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 130, normalized size = 1.51
method | result | size |
derivativedivides | \(\frac {x^{\frac {3}{2}}}{b \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}+\frac {3 a \sqrt {x}}{b^{2} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {3 a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {5}{2}}}+\frac {3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}}{b^{2} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {5}{2}}}\) | \(130\) |
default | \(\frac {x^{\frac {3}{2}}}{b \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}+\frac {3 a \sqrt {x}}{b^{2} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {3 a \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {5}{2}}}+\frac {3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {x}}{b^{2} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \ln \left (\sqrt {b}\, \sqrt {x}+\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}\right )}{b^{\frac {5}{2}}}\) | \(130\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 145, normalized size = 1.69 \begin {gather*} \left [\frac {3 \, {\left (a b x + a^{2}\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (b^{2} x + 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{2 \, {\left (b^{4} x + a b^{3}\right )}}, \frac {3 \, {\left (a b x + a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (b^{2} x + 3 \, a b\right )} \sqrt {b x + a} \sqrt {x}}{b^{4} x + a b^{3}}\right ] \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 {x^{\frac {3}{2}}}{\operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 48, normalized size = 0.56 \begin {gather*} \frac {\sqrt {x} {\left (\frac {x}{b} + \frac {3 \, a}{b^{2}}\right )}}{\sqrt {b x + a}} + \frac {3 \, a \log \left ({\left | -\sqrt {b} \sqrt {x} + \sqrt {b x + a} \right |}\right )}{b^{\frac {5}{2}}} \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 = -0.01 \begin {gather*} \int \frac {x^{3/2}}{{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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