Optimal. Leaf size=80 \[ -\frac{1}{126} b^3 x^9 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^{10}}{1260} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0564928, 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} \[ -\frac{1}{126} b^3 x^9 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4+\frac{b^4 x^{10}}{1260} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^5 \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{3} (2 b) \int x^6 \tanh ^{-1}(\tanh (a+b x))^3 \, dx\\ &=-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{7} \left (2 b^2\right ) \int x^7 \tanh ^{-1}(\tanh (a+b x))^2 \, dx\\ &=\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4-\frac{1}{14} b^3 \int x^8 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{126} b^3 x^9 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4+\frac{1}{126} b^4 \int x^9 \, dx\\ &=\frac{b^4 x^{10}}{1260}-\frac{1}{126} b^3 x^9 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{28} b^2 x^8 \tanh ^{-1}(\tanh (a+b x))^2-\frac{2}{21} b x^7 \tanh ^{-1}(\tanh (a+b x))^3+\frac{1}{6} x^6 \tanh ^{-1}(\tanh (a+b x))^4\\ \end{align*}
Mathematica [A] time = 0.0308633, size = 71, normalized size = 0.89 \[ \frac{x^6 \left (-10 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+45 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-120 b x \tanh ^{-1}(\tanh (a+b x))^3+210 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4\right )}{1260} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 74, normalized size = 0.9 \begin{align*}{\frac{{x}^{6} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{6}}-{\frac{2\,b}{3} \left ({\frac{{x}^{7} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{7}}-{\frac{3\,b}{7} \left ({\frac{{x}^{8} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{8}}-{\frac{b}{4} \left ({\frac{{x}^{9}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{9}}-{\frac{{x}^{10}b}{90}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.73772, size = 97, normalized size = 1.21 \begin{align*} -\frac{2}{21} \, b x^{7} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} + \frac{1}{6} \, x^{6} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac{1}{1260} \,{\left (45 \, b x^{8} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} +{\left (b^{2} x^{10} - 10 \, b x^{9} \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.51244, size = 107, normalized size = 1.34 \begin{align*} \frac{1}{10} \, b^{4} x^{10} + \frac{4}{9} \, a b^{3} x^{9} + \frac{3}{4} \, a^{2} b^{2} x^{8} + \frac{4}{7} \, a^{3} b x^{7} + \frac{1}{6} \, a^{4} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 18.9303, size = 76, normalized size = 0.95 \begin{align*} \frac{b^{4} x^{10}}{1260} - \frac{b^{3} x^{9} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{126} + \frac{b^{2} x^{8} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{28} - \frac{2 b x^{7} \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{21} + \frac{x^{6} \operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12196, size = 62, normalized size = 0.78 \begin{align*} \frac{1}{10} \, b^{4} x^{10} + \frac{4}{9} \, a b^{3} x^{9} + \frac{3}{4} \, a^{2} b^{2} x^{8} + \frac{4}{7} \, a^{3} b x^{7} + \frac{1}{6} \, a^{4} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]