Optimal. Leaf size=158 \[ \frac {3 b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}-\frac {1}{2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {b}{4 x \tanh ^{-1}(\tanh (a+b x))^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2168, 2163, 2161} \[ \frac {3 b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}-\frac {1}{2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {b}{4 x \tanh ^{-1}(\tanh (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2161
Rule 2163
Rule 2168
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx &=-\frac {1}{2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {1}{4} b \int \frac {1}{x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx\\ &=\frac {b}{4 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {1}{2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {1}{8} \left (3 b^2\right ) \int \frac {1}{x \tanh ^{-1}(\tanh (a+b x))^{5/2}} \, dx\\ &=\frac {b}{4 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {1}{2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {\left (3 b^2\right ) \int \frac {1}{x \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{8 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b}{4 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {1}{2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {\left (3 b^2\right ) \int \frac {1}{x \sqrt {\tanh ^{-1}(\tanh (a+b x))}} \, dx}{8 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{5/2}}+\frac {b}{4 x \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {1}{2 x^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}+\frac {3 b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 98, normalized size = 0.62 \[ \frac {1}{4} \left (\frac {\left (5 b x-2 \tanh ^{-1}(\tanh (a+b x))\right ) \sqrt {\tanh ^{-1}(\tanh (a+b x))}}{x^2 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {\tanh ^{-1}(\tanh (a+b x))}}{\sqrt {\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{5/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 123, normalized size = 0.78 \[ \left [\frac {3 \, \sqrt {a} b^{2} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, a b x - 2 \, a^{2}\right )} \sqrt {b x + a}}{8 \, a^{3} x^{2}}, \frac {3 \, \sqrt {-a} b^{2} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b x - 2 \, a^{2}\right )} \sqrt {b x + a}}{4 \, a^{3} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 69, normalized size = 0.44 \[ \frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3} - 5 \, \sqrt {b x + a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 148, normalized size = 0.94 \[ 2 b^{2} \left (\frac {\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\left (-4 \arctanh \left (\tanh \left (b x +a \right )\right )+4 b x \right ) b^{2} x^{2}}+\frac {\frac {6 \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\left (-4 \arctanh \left (\tanh \left (b x +a \right )\right )+4 b x \right ) b x}-\frac {6 \arctanh \left (\frac {\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )-b x}}\right )}{\left (-4 \arctanh \left (\tanh \left (b x +a \right )\right )+4 b x \right ) \sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )-b x}}}{-4 \arctanh \left (\tanh \left (b x +a \right )\right )+4 b x}\right ) \]
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 \[ \int \frac {1}{x^{3} \sqrt {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.01, size = 802, normalized size = 5.08 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{x^{3} \sqrt {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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