Optimal. Leaf size=221 \[ \frac {3 b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {3 b^5 \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}+\frac {b^4}{128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^3}{64 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{16 x^3}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4} \]
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Rubi [A] time = 0.17, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2168, 2163, 2161} \[ -\frac {b^3}{64 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{16 x^3}+\frac {3 b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {b^4}{128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {3 b^5 \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 2161
Rule 2163
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^6} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}+\frac {1}{2} b \int \frac {\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^5} \, dx\\ &=-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}+\frac {1}{16} \left (3 b^2\right ) \int \frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{x^4} \, dx\\ &=-\frac {b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{16 x^3}-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}+\frac {1}{32} b^3 \int \frac {1}{x^3 \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-\frac {b^3}{64 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{16 x^3}-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}-\frac {1}{128} b^4 \int \frac {1}{x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx\\ &=\frac {b^4}{128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^3}{64 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{16 x^3}-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}+\frac {1}{256} \left (3 b^5\right ) \int \frac {1}{x \tanh ^{-1}(\tanh (a+b x))^{5/2}} \, dx\\ &=\frac {b^4}{128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^3}{64 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{16 x^3}-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}-\frac {\left (3 b^5\right ) \int \frac {1}{x \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{256 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b^4}{128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^3}{64 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{16 x^3}-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}-\frac {\left (3 b^5\right ) \int \frac {1}{x \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{256 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {3 b^5 \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}+\frac {b^4}{128 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^3}{64 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 b^5}{128 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{16 x^3}-\frac {b \tanh ^{-1}(\tanh (a+b x))^{3/2}}{8 x^4}-\frac {\tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^5}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 150, normalized size = 0.68 \[ \frac {1}{640} \left (-\frac {15 b^5 \tanh ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{5/2}}-\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))} \left (10 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+8 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-176 b x \tanh ^{-1}(\tanh (a+b x))^3+128 \tanh ^{-1}(\tanh (a+b x))^4+15 b^4 x^4\right )}{x^5 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 189, normalized size = 0.86 \[ \left [\frac {15 \, \sqrt {a} b^{5} x^{5} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (15 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} - 248 \, a^{3} b^{2} x^{2} - 336 \, a^{4} b x - 128 \, a^{5}\right )} \sqrt {b x + a}}{1280 \, a^{3} x^{5}}, \frac {15 \, \sqrt {-a} b^{5} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (15 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} - 248 \, a^{3} b^{2} x^{2} - 336 \, a^{4} b x - 128 \, a^{5}\right )} \sqrt {b x + a}}{640 \, a^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 123, normalized size = 0.56 \[ \frac {\sqrt {2} {\left (\frac {15 \, \sqrt {2} b^{6} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {\sqrt {2} {\left (15 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{6} - 70 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{6} - 128 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{6} + 70 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{6} - 15 \, \sqrt {b x + a} a^{4} b^{6}\right )}}{a^{2} b^{5} x^{5}}\right )}}{1280 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 262, normalized size = 1.19 \[ 2 b^{5} \left (\frac {\frac {3 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {9}{2}}}{256 \left (a^{2}+2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right )}-\frac {7 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{128 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right )}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{10}+\left (\frac {7 \arctanh \left (\tanh \left (b x +a \right )\right )}{128}-\frac {7 b x}{128}\right ) \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}+\left (-\frac {3 a^{2}}{256}-\frac {3 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{128}-\frac {3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{256}\right ) \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{b^{5} x^{5}}-\frac {3 \arctanh \left (\frac {\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )-b x}}\right )}{256 \left (a^{2}+2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )-b x}}\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{\frac {5}{2}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.51, size = 1292, normalized size = 5.85 \[ \text {result too large to display} \]
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 \[ \int \frac {\operatorname {atanh}^{\frac {5}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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