Optimal. Leaf size=121 \[ 2 \sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {2}{3} \tanh ^{-1}(\tanh (a+b x))^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {2}{5} \tanh ^{-1}(\tanh (a+b x))^{5/2}-2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2159, 2161} \[ 2 \sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {2}{3} \tanh ^{-1}(\tanh (a+b x))^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac {2}{5} \tanh ^{-1}(\tanh (a+b x))^{5/2}-2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2} \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2161
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x} \, dx &=\frac {2}{5} \tanh ^{-1}(\tanh (a+b x))^{5/2}-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x} \, dx\\ &=-\frac {2}{3} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}+\frac {2}{5} \tanh ^{-1}(\tanh (a+b x))^{5/2}-\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{x} \, dx\\ &=2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {2}{3} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}+\frac {2}{5} \tanh ^{-1}(\tanh (a+b x))^{5/2}+\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{x \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-2 \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}+2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}-\frac {2}{3} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}+\frac {2}{5} \tanh ^{-1}(\tanh (a+b x))^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 99, normalized size = 0.82 \[ \frac {2}{15} \left (15 b^2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}+23 \tanh ^{-1}(\tanh (a+b x))^{5/2}-35 b x \tanh ^{-1}(\tanh (a+b x))^{3/2}-15 \tanh ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))-b x}}\right ) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{5/2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 114, normalized size = 0.94 \[ \left [a^{\frac {5}{2}} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \frac {2}{15} \, {\left (3 \, b^{2} x^{2} + 11 \, a b x + 23 \, a^{2}\right )} \sqrt {b x + a}, 2 \, \sqrt {-a} a^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + \frac {2}{15} \, {\left (3 \, b^{2} x^{2} + 11 \, a b x + 23 \, a^{2}\right )} \sqrt {b x + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 73, normalized size = 0.60 \[ \frac {1}{15} \, \sqrt {2} {\left (\frac {15 \, \sqrt {2} a^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 3 \, \sqrt {2} {\left (b x + a\right )}^{\frac {5}{2}} + 5 \, \sqrt {2} {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {2} \sqrt {b x + a} a^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 222, normalized size = 1.83 \[ \frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{5}+\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}} a}{3}+\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{3}+2 a^{2} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}+4 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}+2 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}-\frac {2 \left (a^{3}+3 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+3 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}\right ) \arctanh \left (\frac {\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )-b x}}\right )}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )-b x}} \]
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 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.82, size = 789, normalized size = 6.52 \[ \text {result too large to display} \]
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 \[ \int \frac {\operatorname {atanh}^{\frac {5}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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