Optimal. Leaf size=80 \[ -\frac{1}{210} b^3 x^{10} \tanh ^{-1}(\tanh (a+b x))+\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^{11}}{2310} \]
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Rubi [A] time = 0.0632376, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 30} \[ -\frac{1}{210} b^3 x^{10} \tanh ^{-1}(\tanh (a+b x))+\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^{11}}{2310} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^6 \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{7} (4 b) \int x^7 \tanh ^{-1}(\tanh (a+b x))^3 \, dx\\ &=-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{14} \left (3 b^2\right ) \int x^8 \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{21} b^3 \int x^9 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{210} b^3 x^{10} \tanh ^{-1}(\tanh (a+b x))+\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{210} b^4 \int x^{10} \, dx\\ &=\frac{b^4 x^{11}}{2310}-\frac{1}{210} b^3 x^{10} \tanh ^{-1}(\tanh (a+b x))+\frac{1}{42} b^2 x^9 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{14} b x^8 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{7} x^7 \tanh ^{-1}(\tanh (a+b x))^4\\ \end{align*}
Mathematica [A] time = 0.0622043, size = 71, normalized size = 0.89 \[ \frac{x^7 \left (-11 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+55 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-165 b x \tanh ^{-1}(\tanh (a+b x))^3+330 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4\right )}{2310} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 74, normalized size = 0.9 \begin{align*}{\frac{{x}^{7} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{7}}-{\frac{4\,b}{7} \left ({\frac{{x}^{8} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{8}}-{\frac{3\,b}{8} \left ({\frac{{x}^{9} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{9}}-{\frac{2\,b}{9} \left ({\frac{{x}^{10}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{10}}-{\frac{{x}^{11}b}{110}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73779, size = 97, normalized size = 1.21 \begin{align*} -\frac{1}{14} \, b x^{8} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} + \frac{1}{7} \, x^{7} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac{1}{2310} \,{\left (55 \, b x^{9} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} +{\left (b^{2} x^{11} - 11 \, b x^{10} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )\right )} b\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55855, size = 108, normalized size = 1.35 \begin{align*} \frac{1}{11} \, b^{4} x^{11} + \frac{2}{5} \, a b^{3} x^{10} + \frac{2}{3} \, a^{2} b^{2} x^{9} + \frac{1}{2} \, a^{3} b x^{8} + \frac{1}{7} \, a^{4} x^{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.5487, size = 75, normalized size = 0.94 \begin{align*} \frac{b^{4} x^{11}}{2310} - \frac{b^{3} x^{10} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{210} + \frac{b^{2} x^{9} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{42} - \frac{b x^{8} \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{14} + \frac{x^{7} \operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12612, size = 62, normalized size = 0.78 \begin{align*} \frac{1}{11} \, b^{4} x^{11} + \frac{2}{5} \, a b^{3} x^{10} + \frac{2}{3} \, a^{2} b^{2} x^{9} + \frac{1}{2} \, a^{3} b x^{8} + \frac{1}{7} \, a^{4} x^{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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