Optimal. Leaf size=42 \[ -\frac {1}{6} b x^4 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2+\frac {b^2 x^5}{30} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 30} \[ -\frac {1}{6} b x^4 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2+\frac {b^2 x^5}{30} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2168
Rubi steps
\begin {align*} \int x^2 \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2-\frac {1}{3} (2 b) \int x^3 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac {1}{6} b x^4 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2+\frac {1}{6} b^2 \int x^4 \, dx\\ &=\frac {b^2 x^5}{30}-\frac {1}{6} b x^4 \tanh ^{-1}(\tanh (a+b x))+\frac {1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 37, normalized size = 0.88 \[ \frac {1}{30} x^3 \left (-5 b x \tanh ^{-1}(\tanh (a+b x))+10 \tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 24, normalized size = 0.57 \[ \frac {1}{5} \, b^{2} x^{5} + \frac {1}{2} \, a b x^{4} + \frac {1}{3} \, a^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 24, normalized size = 0.57 \[ \frac {1}{5} \, b^{2} x^{5} + \frac {1}{2} \, a b x^{4} + \frac {1}{3} \, a^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.14, size = 38, normalized size = 0.90 \[ \frac {x^{3} \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{3}-\frac {2 b \left (\frac {x^{4} \arctanh \left (\tanh \left (b x +a \right )\right )}{4}-\frac {b \,x^{5}}{20}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.46, size = 36, normalized size = 0.86 \[ \frac {1}{30} \, b^{2} x^{5} - \frac {1}{6} \, b x^{4} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) + \frac {1}{3} \, x^{3} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.97, size = 36, normalized size = 0.86 \[ \frac {b^2\,x^5}{30}-\frac {b\,x^4\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{6}+\frac {x^3\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.23, size = 60, normalized size = 1.43 \[ \begin {cases} \frac {x^{2} \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{3 b} - \frac {x \operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{6 b^{2}} + \frac {\operatorname {atanh}^{5}{\left (\tanh {\left (a + b x \right )} \right )}}{30 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {atanh}^{2}{\left (\tanh {\relax (a )} \right )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________