Optimal. Leaf size=39 \[ 2 b^2 x-\frac {\coth ^{-1}(\tanh (a+b x))^2}{x}-2 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2199, 2189, 29}
\begin {gather*} -\frac {\coth ^{-1}(\tanh (a+b x))^2}{x}-2 b \log (x) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )+2 b^2 x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 2189
Rule 2199
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^2}{x^2} \, dx &=-\frac {\coth ^{-1}(\tanh (a+b x))^2}{x}+(2 b) \int \frac {\coth ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=2 b^2 x-\frac {\coth ^{-1}(\tanh (a+b x))^2}{x}-\left (2 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {1}{x} \, dx\\ &=2 b^2 x-\frac {\coth ^{-1}(\tanh (a+b x))^2}{x}-2 b \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 37, normalized size = 0.95 \begin {gather*} -\frac {\coth ^{-1}(\tanh (a+b x))^2}{x}-2 b^2 x \log (x)+2 b \coth ^{-1}(\tanh (a+b x)) (1+\log (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.16, size = 1095, normalized size = 28.08
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1095\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 54, normalized size = 1.38 \begin {gather*} 2 \, b \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right ) \log \left (x\right ) - 2 \, {\left (b {\left (x + \frac {a}{b}\right )} \log \left (x\right ) - b {\left (x + \frac {a \log \left (x\right )}{b}\right )}\right )} b - \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 29, normalized size = 0.74 \begin {gather*} \frac {4 \, b^{2} x^{2} + 8 \, a b x \log \left (x\right ) + \pi ^{2} - 4 \, a^{2}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acoth}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 0.40, size = 36, normalized size = 0.92 \begin {gather*} b^{2} x + {\left (i \, \pi b + 2 \, a b\right )} \log \left (x\right ) + \frac {\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.25, size = 207, normalized size = 5.31 \begin {gather*} b\,\ln \left (\frac {{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\frac {{\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2}{4\,x}-b\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\frac {{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}^2}{4\,x}+b\,\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\ln \left (x\right )-b\,\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\ln \left (x\right )+\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )\,\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2\,x}-2\,b^2\,x\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________