Optimal. Leaf size=94 \[ -\frac {3}{16 a \left (1-a^2 x^2\right )}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5960, 5956, 261} \[ -\frac {3}{16 a \left (1-a^2 x^2\right )}-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 5956
Rule 5960
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3}{4} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^2}{16 a}-\frac {1}{8} (3 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{16 a \left (1-a^2 x^2\right )^2}-\frac {3}{16 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)^2}{16 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 65, normalized size = 0.69 \[ \frac {\left (10 a x-6 a^3 x^3\right ) \tanh ^{-1}(a x)+3 a^2 x^2+3 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2-4}{16 a \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 97, normalized size = 1.03 \[ \frac {12 \, a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 4 \, {\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 16}{64 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 225, normalized size = 2.39 \[ \frac {\arctanh \left (a x \right )}{16 a \left (a x -1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 a \left (a x -1\right )}-\frac {3 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{16 a}-\frac {\arctanh \left (a x \right )}{16 a \left (a x +1\right )^{2}}-\frac {3 \arctanh \left (a x \right )}{16 a \left (a x +1\right )}+\frac {3 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{16 a}-\frac {3 \ln \left (a x -1\right )^{2}}{64 a}+\frac {3 \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{32 a}-\frac {3 \ln \left (a x +1\right )^{2}}{64 a}-\frac {3 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{32 a}+\frac {3 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{32 a}-\frac {1}{64 a \left (a x -1\right )^{2}}+\frac {7}{64 a \left (a x -1\right )}-\frac {1}{64 a \left (a x +1\right )^{2}}-\frac {7}{64 a \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.32, size = 182, normalized size = 1.94 \[ -\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac {3 \, \log \left (a x + 1\right )}{a} + \frac {3 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right ) + \frac {{\left (12 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a}{64 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.37, size = 154, normalized size = 1.64 \[ \frac {\frac {3\,a\,x^2}{2}-\frac {2}{a}}{8\,a^4\,x^4-16\,a^2\,x^2+8}-\ln \left (1-a\,x\right )\,\left (\frac {3\,\ln \left (a\,x+1\right )}{32\,a}+\frac {\frac {5\,x}{8}-\frac {3\,a^2\,x^3}{8}}{2\,a^4\,x^4-4\,a^2\,x^2+2}\right )+\frac {3\,{\ln \left (a\,x+1\right )}^2}{64\,a}+\frac {3\,{\ln \left (1-a\,x\right )}^2}{64\,a}+\frac {\ln \left (a\,x+1\right )\,\left (\frac {5\,x}{16\,a}-\frac {3\,a\,x^3}{16}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {atanh}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________