Optimal. Leaf size=163 \[ \frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{64 a^3}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3} \]
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Rubi [A]
time = 0.18, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6175, 6103,
6141, 205, 212, 6111} \begin {gather*} -\frac {\tanh ^{-1}(a x)^3}{24 a^3}-\frac {\tanh ^{-1}(a x)}{64 a^3}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}+\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 212
Rule 6103
Rule 6111
Rule 6141
Rule 6175
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {\int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx}{a^2}-\frac {\int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{a^2}\\ &=-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{6 a^3}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3} \, dx}{8 a^2}+\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{4 a^2}+\frac {\int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{a}\\ &=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{2 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}+\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{32 a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}-\frac {3 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{4 a}\\ &=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {13 x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}+\frac {3 \int \frac {1}{1-a^2 x^2} \, dx}{64 a^2}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{4 a^2}+\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{8 a^2}\\ &=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {13 \tanh ^{-1}(a x)}{64 a^3}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}+\frac {3 \int \frac {1}{1-a^2 x^2} \, dx}{16 a^2}\\ &=\frac {x}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {x}{64 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)}{64 a^3}-\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)}{8 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^2}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^2}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{24 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 121, normalized size = 0.74 \begin {gather*} \frac {6 a x \left (1+a^2 x^2\right )-48 a^2 x^2 \tanh ^{-1}(a x)+48 \left (a x+a^3 x^3\right ) \tanh ^{-1}(a x)^2-16 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+3 \left (-1+a^2 x^2\right )^2 \log (1-a x)-3 \left (-1+a^2 x^2\right )^2 \log (1+a x)}{384 a^3 \left (-1+a^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 237.51, size = 2028, normalized size = 12.44
method | result | size |
risch | \(-\frac {\ln \left (a x +1\right )^{3}}{192 a^{3}}+\frac {\left (x^{4} \ln \left (-a x +1\right ) a^{4}+2 a^{3} x^{3}-2 x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{2}}{64 a^{3} \left (a^{2} x^{2}-1\right )^{2}}-\frac {\left (a^{4} x^{4} \ln \left (-a x +1\right )^{2}+4 a^{3} x^{3} \ln \left (-a x +1\right )-2 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+4 a^{2} x^{2}+4 a x \ln \left (-a x +1\right )+\ln \left (-a x +1\right )^{2}\right ) \ln \left (a x +1\right )}{64 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {-2 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+3 \ln \left (a x +1\right ) a^{4} x^{4}-3 x^{4} \ln \left (-a x +1\right ) a^{4}-12 a^{3} x^{3} \ln \left (-a x +1\right )^{2}+4 a^{2} x^{2} \ln \left (-a x +1\right )^{3}-6 a^{3} x^{3}-6 a^{2} x^{2} \ln \left (a x +1\right )-18 x^{2} \ln \left (-a x +1\right ) a^{2}-12 a \ln \left (-a x +1\right )^{2} x -2 \ln \left (-a x +1\right )^{3}-6 a x +3 \ln \left (a x +1\right )-3 \ln \left (-a x +1\right )}{384 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\) | \(394\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2028\) |
default | \(\text {Expression too large to display}\) | \(2028\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 388 vs.
\(2 (141) = 282\).
time = 0.29, size = 388, normalized size = 2.38 \begin {gather*} \frac {1}{16} \, {\left (\frac {2 \, {\left (a^{2} x^{3} + x\right )}}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {{\left (6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 6 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 6 \, a x - 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} + 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 )\right )} a^{2}}{384 \, {\left (a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}\right )}} - \frac {{\left (4 \, a^{2} x^{2} - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2}\right )} a \operatorname {artanh}\left (a x\right )}{32 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 136, normalized size = 0.83 \begin {gather*} \frac {6 \, a^{3} x^{3} - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 12 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 6 \, a x - 3 \, {\left (a^{4} x^{4} + 6 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{384 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.97, size = 350, normalized size = 2.15 \begin {gather*} \ln \left (1-a\,x\right )\,\left (\frac {\frac {3\,a\,x^3}{2}-\frac {x}{2\,a}+x^2}{32\,a^5\,x^4-64\,a^3\,x^2+32\,a}+\frac {\frac {x}{2\,a}-\frac {3\,a\,x^3}{2}+x^2}{32\,a^5\,x^4-64\,a^3\,x^2+32\,a}+\frac {{\ln \left (a\,x+1\right )}^2}{64\,a^3}-\frac {\ln \left (a\,x+1\right )\,\left (2\,a^2\,x^3+2\,x\right )}{32\,a^6\,x^4-64\,a^4\,x^2+32\,a^2}\right )+\frac {\frac {x}{8\,a^2}+\frac {x^3}{8}}{8\,a^4\,x^4-16\,a^2\,x^2+8}-{\ln \left (1-a\,x\right )}^2\,\left (\frac {\ln \left (a\,x+1\right )}{64\,a^3}-\frac {\frac {x}{8\,a^2}+\frac {x^3}{8}}{4\,a^4\,x^4-8\,a^2\,x^2+4}\right )-\frac {{\ln \left (a\,x+1\right )}^3}{192\,a^3}+\frac {{\ln \left (1-a\,x\right )}^3}{192\,a^3}+\frac {{\ln \left (a\,x+1\right )}^2\,\left (\frac {x}{32\,a^3}+\frac {x^3}{32\,a}\right )}{\frac {1}{a}-2\,a\,x^2+a^3\,x^4}-\frac {x^2\,\ln \left (a\,x+1\right )}{16\,a^2\,\left (\frac {1}{a}-2\,a\,x^2+a^3\,x^4\right )}+\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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