Optimal. Leaf size=34 \[ \frac {x \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^6}{30 b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2168, 2157, 30} \[ \frac {x \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^6}{30 b^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int x \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac {x \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {\int \tanh ^{-1}(\tanh (a+b x))^5 \, dx}{5 b}\\ &=\frac {x \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {\operatorname {Subst}\left (\int x^5 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{5 b^2}\\ &=\frac {x \tanh ^{-1}(\tanh (a+b x))^5}{5 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^6}{30 b^2}\\ \end {align*}
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Mathematica [B] time = 0.10, size = 125, normalized size = 3.68 \[ -\frac {(a+b x) \left (-20 \left (2 a^2+a b x-b^2 x^2\right ) \tanh ^{-1}(\tanh (a+b x))^3+(5 a-b x) (a+b x)^4-6 (4 a-b x) (a+b x)^3 \tanh ^{-1}(\tanh (a+b x))+15 (3 a-b x) (a+b x)^2 \tanh ^{-1}(\tanh (a+b x))^2+15 (a-b x) \tanh ^{-1}(\tanh (a+b x))^4\right )}{30 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 46, normalized size = 1.35 \[ \frac {1}{6} \, b^{4} x^{6} + \frac {4}{5} \, a b^{3} x^{5} + \frac {3}{2} \, a^{2} b^{2} x^{4} + \frac {4}{3} \, a^{3} b x^{3} + \frac {1}{2} \, a^{4} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 46, normalized size = 1.35 \[ \frac {1}{6} \, b^{4} x^{6} + \frac {4}{5} \, a b^{3} x^{5} + \frac {3}{2} \, a^{2} b^{2} x^{4} + \frac {4}{3} \, a^{3} b x^{3} + \frac {1}{2} \, a^{4} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 74, normalized size = 2.18 \[ \frac {x^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{2}-2 b \left (\frac {x^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{3}-b \left (\frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{4}-\frac {b \left (\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )}{5}-\frac {x^{6} b}{30}\right )}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 72, normalized size = 2.12 \[ -\frac {2}{3} \, b x^{3} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} + \frac {1}{2} \, x^{2} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac {1}{30} \, {\left (15 \, b x^{4} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + {\left (b^{2} x^{6} - 6 \, b x^{5} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 70, normalized size = 2.06 \[ \frac {b^4\,x^6}{30}-\frac {b^3\,x^5\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{5}+\frac {b^2\,x^4\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{2}-\frac {2\,b\,x^3\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{3}+\frac {x^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.33, size = 41, normalized size = 1.21 \[ \begin {cases} \frac {x \operatorname {atanh}^{5}{\left (\tanh {\left (a + b x \right )} \right )}}{5 b} - \frac {\operatorname {atanh}^{6}{\left (\tanh {\left (a + b x \right )} \right )}}{30 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {atanh}^{4}{\left (\tanh {\relax (a )} \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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