Optimal. Leaf size=87 \[ -6 b^3 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+3 b^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+6 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2199, 2190,
2189, 29} \begin {gather*} -6 b^3 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+3 b^2 \tanh ^{-1}(\tanh (a+b x))^2+6 b^2 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 2189
Rule 2190
Rule 2199
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^3} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+(2 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^2} \, dx\\ &=-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+\left (6 b^2\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=3 b^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}-\left (6 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-6 b^3 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+3 b^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+\left (6 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac {1}{x} \, dx\\ &=-6 b^3 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+3 b^2 \tanh ^{-1}(\tanh (a+b x))^2-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+6 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 81, normalized size = 0.93 \begin {gather*} -\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{2 x^2}+6 b^4 x^2 \log (x)-6 b^3 x \tanh ^{-1}(\tanh (a+b x)) (1+2 \log (x))+3 b^2 \tanh ^{-1}(\tanh (a+b x))^2 (3+2 \log (x)) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.67, size = 114, normalized size = 1.31
method | result | size |
default | \(-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{2 x^{2}}+2 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{x}+3 b \left (\ln \left (x \right ) \arctanh \left (\tanh \left (b x +a \right )\right )^{2}-2 b \left (\frac {b \,x^{2} \ln \left (x \right )}{2}-\frac {b \,x^{2}}{4}+\ln \left (x \right ) x a -x a +\ln \left (x \right ) x \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )-x \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )\right )\right )\right )\) | \(114\) |
risch | \(\text {Expression too large to display}\) | \(15946\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.74, size = 83, normalized size = 0.95 \begin {gather*} -\frac {2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{x} + 3 \, {\left (2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \log \left (x\right ) + {\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right ) - 2 \, \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2} \log \left (x\right )\right )} b\right )} b - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 47, normalized size = 0.54 \begin {gather*} \frac {b^{4} x^{4} + 8 \, a b^{3} x^{3} + 12 \, a^{2} b^{2} x^{2} \log \left (x\right ) - 8 \, a^{3} b x - a^{4}}{2 \, x^{2}} \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 {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.38, size = 43, normalized size = 0.49 \begin {gather*} \frac {1}{2} \, b^{4} x^{2} + 4 \, a b^{3} x + 6 \, a^{2} b^{2} \log \left ({\left | x \right |}\right ) - \frac {8 \, a^{3} b x + a^{4}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.54, size = 672, normalized size = 7.72 \begin {gather*} \frac {9\,b^2\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{4}-\frac {{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^4}{32\,x^2}-\frac {{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^4}{32\,x^2}+\frac {9\,b^2\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{4}-3\,b^3\,x\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\frac {b\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{4\,x}+\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{8\,x^2}+\frac {{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{8\,x^2}-\frac {b\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^3}{4\,x}+\frac {3\,b^2\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,\ln \left (x\right )}{2}-\frac {3\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{16\,x^2}+\frac {3\,b^2\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,\ln \left (x\right )}{2}+6\,b^4\,x^2\,\ln \left (x\right )-\frac {9\,b^2\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}+3\,b^3\,x\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\frac {3\,b\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,{\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2}{4\,x}-\frac {3\,b\,{\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}^2\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{4\,x}-3\,b^2\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (x\right )+6\,b^3\,x\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (x\right )-6\,b^3\,x\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________