Optimal. Leaf size=16 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^5}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0046208, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2157, 30} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^5}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2157
Rule 30
Rubi steps
\begin{align*} \int \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int x^4 \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b}\\ &=\frac{\tanh ^{-1}(\tanh (a+b x))^5}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0054625, size = 16, normalized size = 1. \[ \frac{\tanh ^{-1}(\tanh (a+b x))^5}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 15, normalized size = 0.9 \begin{align*}{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{5}}{5\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.7329, size = 93, normalized size = 5.81 \begin{align*} -2 \, b x^{2} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3} + x \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4} + \frac{1}{5} \,{\left (10 \, b x^{3} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} +{\left (b^{2} x^{5} - 5 \, b x^{4} \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 [B] time = 1.53508, size = 85, normalized size = 5.31 \begin{align*} \frac{1}{5} \, b^{4} x^{5} + a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{3} + 2 \, a^{3} b x^{2} + a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.01654, size = 20, normalized size = 1.25 \begin{align*} \begin{cases} \frac{\operatorname{atanh}^{5}{\left (\tanh{\left (a + b x \right )} \right )}}{5 b} & \text{for}\: b \neq 0 \\x \operatorname{atanh}^{4}{\left (\tanh{\left (a \right )} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.13025, size = 57, normalized size = 3.56 \begin{align*} \frac{1}{5} \, b^{4} x^{5} + a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{3} + 2 \, a^{3} b x^{2} + a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]