Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2167} \[ \frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2167
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^4} \, dx &=\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 34, normalized size = 1.10 \[ -\frac {b x \tanh ^{-1}(\tanh (a+b x))+\tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 22, normalized size = 0.71 \[ -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.81, size = 22, normalized size = 0.71 \[ -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.14, size = 38, normalized size = 1.23 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{3 x^{3}}+\frac {2 b \left (-\frac {b}{2 x}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{2 x^{2}}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 36, normalized size = 1.16 \[ -\frac {b^{2}}{3 \, x} - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{3 \, x^{2}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.94, size = 32, normalized size = 1.03 \[ -\frac {b^2\,x^2+b\,x\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.80, size = 37, normalized size = 1.19 \[ - \frac {b^{2}}{3 x} - \frac {b \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{2}} - \frac {\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________