Optimal. Leaf size=215 \[ -\frac {\tanh ^{-1}(a x)^4}{32 a^3}-\frac {3 \tanh ^{-1}(a x)^2}{128 a^3}-\frac {x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}+\frac {3}{128 a^3 \left (1-a^2 x^2\right )}-\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2} \]
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Rubi [A] time = 0.36, antiderivative size = 215, 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 = {6028, 5956, 5994, 261, 5964, 5960} \[ \frac {3}{128 a^3 \left (1-a^2 x^2\right )}-\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^4}{32 a^3}-\frac {3 \tanh ^{-1}(a x)^2}{128 a^3} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5956
Rule 5960
Rule 5964
Rule 5994
Rule 6028
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx &=\frac {\int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx}{a^2}-\frac {\int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx}{a^2}\\ &=-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^3}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^4}{8 a^3}+\frac {3 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx}{8 a^2}+\frac {3 \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx}{4 a^2}+\frac {3 \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{2 a}\\ &=-\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^4}{32 a^3}+\frac {9 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{32 a^2}-\frac {3 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}-\frac {9 \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{8 a}\\ &=-\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {39 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac {39 \tanh ^{-1}(a x)^2}{128 a^3}-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^4}{32 a^3}+\frac {9 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{8 a^2}-\frac {9 \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx}{64 a}+\frac {3 \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx}{4 a}\\ &=-\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac {39}{128 a^3 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{128 a^3}-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^4}{32 a^3}-\frac {9 \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx}{16 a}\\ &=-\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac {3}{128 a^3 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{128 a^3}-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^4}{32 a^3}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 107, normalized size = 0.50 \[ \frac {16 \left (a^3 x^3+a x\right ) \tanh ^{-1}(a x)^3+6 \left (a^3 x^3+a x\right ) \tanh ^{-1}(a x)-3 a^2 x^2-4 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4-3 \left (a^4 x^4+6 a^2 x^2+1\right ) \tanh ^{-1}(a x)^2}{128 a^3 \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 161, normalized size = 0.75 \[ -\frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 12 \, a^{2} x^{2} - 8 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 3 \, {\left (a^{4} x^{4} + 6 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (a^{3} x^{3} + a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{512 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )}} \]
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 \[ \int -\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.85, size = 2646, normalized size = 12.31 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 657, normalized size = 3.06 \[ \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 )^{3} - \frac {3 \, {\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 )^{2}}{64 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} + \frac {1}{512} \, {\left (\frac {{\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{4} - 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{4} - 12 \, a^{2} x^{2} + 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 )^{2} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 2 \, {\left (2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{3} + 3 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )\right )} a^{2}}{a^{10} x^{4} - 2 \, a^{8} x^{2} + a^{6}} + \frac {4 \, {\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 \operatorname {artanh}\left (a x\right )}{a^{9} x^{4} - 2 \, a^{7} x^{2} + a^{5}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.19, size = 831, normalized size = 3.87 \[ \frac {3\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )}-\frac {3\,{\ln \left (1-a\,x\right )}^2}{512\,a^3}-\frac {{\ln \left (a\,x+1\right )}^4}{512\,a^3}-\frac {{\ln \left (1-a\,x\right )}^4}{512\,a^3}-\frac {3\,x^2}{2\,\left (64\,a^5\,x^4-128\,a^3\,x^2+64\,a\right )}-\frac {x\,{\ln \left (1-a\,x\right )}^3}{8\,\left (8\,a^6\,x^4-16\,a^4\,x^2+8\,a^2\right )}-\frac {6\,x^2\,{\ln \left (1-a\,x\right )}^2}{128\,a^5\,x^4-256\,a^3\,x^2+128\,a}-\frac {3\,{\ln \left (a\,x+1\right )}^2}{512\,a^3}+\frac {x^3\,{\ln \left (a\,x+1\right )}^3}{64\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}-\frac {x^3\,{\ln \left (1-a\,x\right )}^3}{8\,\left (8\,a^4\,x^4-16\,a^2\,x^2+8\right )}+\frac {3\,x\,\ln \left (a\,x+1\right )}{128\,\left (a^6\,x^4-2\,a^4\,x^2+a^2\right )}+\frac {\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^3}{128\,a^3}+\frac {{\ln \left (a\,x+1\right )}^3\,\ln \left (1-a\,x\right )}{128\,a^3}-\frac {3\,x\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {3\,x^2\,{\ln \left (a\,x+1\right )}^2}{64\,\left (a^5\,x^4-2\,a^3\,x^2+a\right )}+\frac {x\,{\ln \left (a\,x+1\right )}^3}{64\,\left (a^6\,x^4-2\,a^4\,x^2+a^2\right )}-\frac {3\,{\ln \left (a\,x+1\right )}^2\,{\ln \left (1-a\,x\right )}^2}{256\,a^3}+\frac {3\,x^3\,\ln \left (a\,x+1\right )}{128\,\left (a^4\,x^4-2\,a^2\,x^2+1\right )}-\frac {3\,a\,x^3\,\ln \left (1-a\,x\right )}{128\,a^5\,x^4-256\,a^3\,x^2+128\,a}+\frac {6\,x\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {6\,x\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}+\frac {6\,x^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{64\,a^5\,x^4-128\,a^3\,x^2+64\,a}+\frac {6\,a^2\,x^3\,\ln \left (a\,x+1\right )\,{\ln \left (1-a\,x\right )}^2}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {6\,a^2\,x^3\,{\ln \left (a\,x+1\right )}^2\,\ln \left (1-a\,x\right )}{128\,a^6\,x^4-256\,a^4\,x^2+128\,a^2}-\frac {3\,a^2\,x^2\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{2\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )}+\frac {3\,a^4\,x^4\,\ln \left (a\,x+1\right )\,\ln \left (1-a\,x\right )}{4\,\left (64\,a^7\,x^4-128\,a^5\,x^2+64\,a^3\right )} \]
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 \[ - \int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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