Optimal. Leaf size=92 \[ \frac {3 x^2}{b^3}+\frac {6 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^4}-\frac {x^4}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {2 x^3}{b^2 \tanh ^{-1}(\tanh (a+b x))}+\frac {6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^5} \]
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Rubi [A]
time = 0.09, 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 = {2199, 2190,
2189, 2188, 29} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2188
Rule 2189
Rule 2190
Rule 2199
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 ) \text {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 = 0.03, size = 114, normalized size = 1.24 \begin {gather*} \frac {x^2}{2 b^3}-\frac {3 x \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac {4 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^3}{b^5 \tanh ^{-1}(\tanh (a+b x))}-\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^4}{2 b^5 \tanh ^{-1}(\tanh (a+b x))^2}+\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 {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs.
\(2(90)=180\).
time = 0.83, size = 371, normalized size = 4.03
method | result | size |
default | \(\frac {x^{2}}{2 b^{3}}-\frac {3 x a}{b^{4}}-\frac {3 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right ) x}{b^{4}}+\frac {6 \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) a^{2}}{b^{5}}+\frac {12 \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{5}}+\frac {6 \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{5}}-\frac {a^{4}}{2 b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {2 a^{3} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {3 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}-\frac {\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{4}}{2 b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}+\frac {4 a^{3}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {12 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {12 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}+\frac {4 \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}}{b^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}\) | \(371\) |
risch | \(\text {Expression too large to display}\) | \(15090\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.85, size = 81, normalized size = 0.88 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 95, normalized size = 1.03 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \int \frac {x^{4}}{\operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 61, normalized size = 0.66 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \frac {\frac {7\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4+24\,a^2\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2+16\,a^4-8\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3-32\,a^3\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\right )}{4\,b}-x\,\left (4\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3-32\,a^3-24\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2+48\,a^2\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\right )}{2\,b^4\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2+x\,\left (16\,a\,b^5-8\,b^5\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\right )+8\,a^2\,b^4+8\,b^6\,x^2-8\,a\,b^4\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}+\frac {x^2}{2\,b^3}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )\,\left (3\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2-12\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )+12\,a^2\right )}{2\,b^5}+\frac {3\,x\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{2\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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