Optimal. Leaf size=34 \[ \frac {x \tanh ^{-1}(\tanh (a+b x))^7}{7 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^8}{56 b^2} \]
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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} \[ \frac {x \tanh ^{-1}(\tanh (a+b x))^7}{7 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^8}{56 b^2} \]
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))^6 \, dx &=\frac {x \tanh ^{-1}(\tanh (a+b x))^7}{7 b}-\frac {\int \tanh ^{-1}(\tanh (a+b x))^7 \, dx}{7 b}\\ &=\frac {x \tanh ^{-1}(\tanh (a+b x))^7}{7 b}-\frac {\operatorname {Subst}\left (\int x^7 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{7 b^2}\\ &=\frac {x \tanh ^{-1}(\tanh (a+b x))^7}{7 b}-\frac {\tanh ^{-1}(\tanh (a+b x))^8}{56 b^2}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 177, normalized size = 5.21 \[ -\frac {(a+b x) \left (-56 \left (2 a^2+a b x-b^2 x^2\right ) \tanh ^{-1}(\tanh (a+b x))^5+(7 a-b x) (a+b x)^6-8 (6 a-b x) (a+b x)^5 \tanh ^{-1}(\tanh (a+b x))+28 (5 a-b x) (a+b x)^4 \tanh ^{-1}(\tanh (a+b x))^2-56 (4 a-b x) (a+b x)^3 \tanh ^{-1}(\tanh (a+b x))^3+70 (3 a-b x) (a+b x)^2 \tanh ^{-1}(\tanh (a+b x))^4+28 (a-b x) \tanh ^{-1}(\tanh (a+b x))^6\right )}{56 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 68, normalized size = 2.00 \[ \frac {1}{8} \, b^{6} x^{8} + \frac {6}{7} \, a b^{5} x^{7} + \frac {5}{2} \, a^{2} b^{4} x^{6} + 4 \, a^{3} b^{3} x^{5} + \frac {15}{4} \, a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + \frac {1}{2} \, a^{6} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 68, normalized size = 2.00 \[ \frac {1}{8} \, b^{6} x^{8} + \frac {6}{7} \, a b^{5} x^{7} + \frac {5}{2} \, a^{2} b^{4} x^{6} + 4 \, a^{3} b^{3} x^{5} + \frac {15}{4} \, a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + \frac {1}{2} \, a^{6} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 110, normalized size = 3.24 \[ \frac {x^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{6}}{2}-3 b \left (\frac {x^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{5}}{3}-\frac {5 b \left (\frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{4}-b \left (\frac {x^{5} \arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{5}-\frac {3 b \left (\frac {x^{6} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{6}-\frac {b \left (\frac {x^{7} \arctanh \left (\tanh \left (b x +a \right )\right )}{7}-\frac {x^{8} b}{56}\right )}{3}\right )}{5}\right )\right )}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.72, size = 110, normalized size = 3.24 \[ -b x^{3} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{5} + \frac {1}{2} \, x^{2} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{6} + \frac {1}{56} \, {\left (70 \, b x^{4} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4} - {\left (56 \, b x^{5} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3} - {\left (28 \, b x^{6} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} + {\left (b^{2} x^{8} - 8 \, b x^{7} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )\right )} b\right )} b\right )} b\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 104, normalized size = 3.06 \[ \frac {b^6\,x^8}{56}-\frac {b^5\,x^7\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{7}+\frac {b^4\,x^6\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{2}-b^3\,x^5\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3+\frac {5\,b^2\,x^4\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{4}-b\,x^3\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^5+\frac {x^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^6}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.54, size = 41, normalized size = 1.21 \[ \begin {cases} \frac {x \operatorname {atanh}^{7}{\left (\tanh {\left (a + b x \right )} \right )}}{7 b} - \frac {\operatorname {atanh}^{8}{\left (\tanh {\left (a + b x \right )} \right )}}{56 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {atanh}^{6}{\left (\tanh {\relax (a )} \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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