Optimal. Leaf size=61 \[ \frac{1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{140} b^3 x^7 \]
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Rubi [A] time = 0.04446, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 30} \[ \frac{1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{140} b^3 x^7 \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^3 \tanh ^{-1}(\tanh (a+b x))^3 \, dx &=\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{4} (3 b) \int x^4 \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=-\frac{3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{10} \left (3 b^2\right ) \int x^5 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=\frac{1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3-\frac{1}{20} b^3 \int x^6 \, dx\\ &=-\frac{1}{140} b^3 x^7+\frac{1}{20} b^2 x^6 \tanh ^{-1}(\tanh (a+b x))-\frac{3}{20} b x^5 \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^3\\ \end{align*}
Mathematica [A] time = 0.0232976, size = 54, normalized size = 0.89 \[ -\frac{1}{140} x^4 \left (-7 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+21 b x \tanh ^{-1}(\tanh (a+b x))^2-35 \tanh ^{-1}(\tanh (a+b x))^3+b^3 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 56, normalized size = 0.9 \begin{align*}{\frac{{x}^{4} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{4}}-{\frac{3\,b}{4} \left ({\frac{{x}^{5} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{5}}-{\frac{2\,b}{5} \left ({\frac{{x}^{6}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{6}}-{\frac{{x}^{7}b}{42}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56287, size = 73, normalized size = 1.2 \begin{align*} -\frac{3}{20} \, b x^{5} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} + \frac{1}{4} \, x^{4} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} - \frac{1}{140} \,{\left (b^{2} x^{7} - 7 \, b x^{6} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42188, size = 80, normalized size = 1.31 \begin{align*} \frac{1}{7} \, b^{3} x^{7} + \frac{1}{2} \, a b^{2} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{4} \, a^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.78503, size = 80, normalized size = 1.31 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{4 b} - \frac{3 x^{2} \operatorname{atanh}^{5}{\left (\tanh{\left (a + b x \right )} \right )}}{20 b^{2}} + \frac{x \operatorname{atanh}^{6}{\left (\tanh{\left (a + b x \right )} \right )}}{20 b^{3}} - \frac{\operatorname{atanh}^{7}{\left (\tanh{\left (a + b x \right )} \right )}}{140 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \operatorname{atanh}^{3}{\left (\tanh{\left (a \right )} \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14231, size = 47, normalized size = 0.77 \begin{align*} \frac{1}{7} \, b^{3} x^{7} + \frac{1}{2} \, a b^{2} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{4} \, a^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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