Optimal. Leaf size=81 \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt {\tanh ^{-1}(\tanh (a+b x))}-3 b \sqrt {b x-\tanh ^{-1}(\tanh (a+b x))} \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2168, 2159, 2161} \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt {\tanh ^{-1}(\tanh (a+b x))}-3 b \sqrt {b x-\tanh ^{-1}(\tanh (a+b x))} \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2161
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^2} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+\frac {1}{2} (3 b) \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{x} \, dx\\ &=3 b \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}-\frac {1}{2} \left (3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{x \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-3 b \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}+3 b \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 79, normalized size = 0.98 \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt {\tanh ^{-1}(\tanh (a+b x))}-3 b \tanh ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))-b x}}\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))-b x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 102, normalized size = 1.26 \[ \left [\frac {3 \, \sqrt {a} b x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, b x - a\right )} \sqrt {b x + a}}{2 \, x}, \frac {3 \, \sqrt {-a} b x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (2 \, b x - a\right )} \sqrt {b x + a}}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 69, normalized size = 0.85 \[ \frac {\sqrt {2} {\left (\frac {3 \, \sqrt {2} a b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {2} \sqrt {b x + a} b^{2} - \frac {\sqrt {2} \sqrt {b x + a} a b}{x}\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 85, normalized size = 1.05 \[ 2 b \left (\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {\left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{2}+\frac {b x}{2}\right ) \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{b x}-\frac {3 \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )-b x}\, \arctanh \left (\frac {\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )-b x}}\right )}{2}\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 459, normalized size = 5.67 \[ 3\,b\,\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}}+\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\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}}}{2\,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 {\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}}}{2\,x}+b\,\ln \left (-\frac {4\,\sqrt {2}\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-4\,\sqrt {2}\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+8\,\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}}\,\sqrt {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-2\,b\,x}+4\,\sqrt {2}\,b\,x}{x\,\sqrt {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-2\,b\,x}}\right )\,\sqrt {\frac {9\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{8}-\frac {9\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{8}-\frac {9\,b\,x}{4}} \]
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 \[ \int \frac {\operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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