Optimal. Leaf size=92 \[ \frac {6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^5}+\frac {6 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac {2 x^3}{b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {3 x^2}{b^3} \]
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Rubi [A] time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2168, 2159, 2158, 2157, 29} \[ -\frac {2 x^3}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {6 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac {6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^5}-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {3 x^2}{b^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 2157
Rule 2158
Rule 2159
Rule 2168
Rubi steps
\begin {align*} \int \frac {x^4}{\tanh ^{-1}(\tanh (a+b x))^3} \, dx &=-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {2 \int \frac {x^3}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx}{b}\\ &=-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {2 x^3}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {6 \int \frac {x^2}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac {3 x^2}{b^3}-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {2 x^3}{b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {\left (6 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {x}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^3}\\ &=\frac {3 x^2}{b^3}+\frac {6 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {2 x^3}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {\left (6 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {1}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^4}\\ &=\frac {3 x^2}{b^3}+\frac {6 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {2 x^3}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {\left (6 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^5}\\ &=\frac {3 x^2}{b^3}+\frac {6 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {2 x^3}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 114, normalized size = 1.24 \[ -\frac {\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^4}{2 b^5 \tanh ^{-1}(\tanh (a+b x))^2}+\frac {4 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^3}{b^5 \tanh ^{-1}(\tanh (a+b x))}+\frac {6 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^5}-\frac {3 x \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )}{b^4}+\frac {x^2}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 95, normalized size = 1.03 \[ \frac {b^{4} x^{4} - 4 \, a b^{3} x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x + 7 \, a^{4} + 12 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 61, normalized size = 0.66 \[ \frac {6 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} x^{2} - 6 \, a b^{2} x}{2 \, b^{6}} + \frac {8 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b x + a\right )}^{2} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 371, normalized size = 4.03 \[ \frac {x^{2}}{2 b^{3}}-\frac {3 a x}{b^{4}}-\frac {3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) x}{b^{4}}+\frac {6 \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) a^{2}}{b^{5}}+\frac {12 \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{5}}+\frac {6 \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{5}}-\frac {a^{4}}{2 b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {2 a^{3} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {3 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{4}}{2 b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}+\frac {4 a^{3}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {12 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {12 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.38, size = 81, normalized size = 0.88 \[ \frac {b^{4} x^{4} - 4 \, a b^{3} x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x + 7 \, a^{4}}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {6 \, a^{2} \log \left (b x + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 867, normalized size = 9.42 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{4}}{\operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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