Optimal. Leaf size=215 \[ -\frac {3}{128 a^3 \left (1-a^2 x^2\right )^2}+\frac {3}{128 a^3 \left (1-a^2 x^2\right )}+\frac {3 x \tanh ^{-1}(a x)}{32 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 x \tanh ^{-1}(a x)}{64 a^2 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)^2}{128 a^3}-\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)^2}{16 a^3 \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {x \tanh ^{-1}(a x)^3}{8 a^2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^4}{32 a^3} \]
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Rubi [A]
time = 0.26, 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 = {6175, 6103,
6141, 267, 6111, 6107} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6103
Rule 6107
Rule 6111
Rule 6141
Rule 6175
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 = 0.05, size = 107, normalized size = 0.50 \begin {gather*} \frac {-3 a^2 x^2+6 \left (a x+a^3 x^3\right ) \tanh ^{-1}(a x)-3 \left (1+6 a^2 x^2+a^4 x^4\right ) \tanh ^{-1}(a x)^2+16 \left (a x+a^3 x^3\right ) \tanh ^{-1}(a x)^3-4 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}{128 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 = 387.63, size = 2059, normalized size = 9.58
method | result | size |
risch | \(-\frac {\ln \left (a x +1\right )^{4}}{512 a^{3}}+\frac {\left (x^{4} \ln \left (-a x +1\right ) a^{4}+2 a^{3} x^{3}-2 x^{2} \ln \left (-a x +1\right ) a^{2}+2 a x +\ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )^{3}}{128 a^{3} \left (a^{2} x^{2}-1\right )^{2}}-\frac {3 \left (2 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+a^{4} x^{4}+8 a^{3} x^{3} \ln \left (-a x +1\right )-4 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+6 a^{2} x^{2}+8 a x \ln \left (-a x +1\right )+2 \ln \left (-a x +1\right )^{2}+1\right ) \ln \left (a x +1\right )^{2}}{512 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}+\frac {\left (2 a^{4} x^{4} \ln \left (-a x +1\right )^{3}+3 x^{4} \ln \left (-a x +1\right ) a^{4}+12 a^{3} x^{3} \ln \left (-a x +1\right )^{2}-4 a^{2} x^{2} \ln \left (-a x +1\right )^{3}+6 a^{3} x^{3}+18 x^{2} \ln \left (-a x +1\right ) a^{2}+12 a \ln \left (-a x +1\right )^{2} x +2 \ln \left (-a x +1\right )^{3}+6 a x +3 \ln \left (-a x +1\right )\right ) \ln \left (a x +1\right )}{256 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {a^{4} x^{4} \ln \left (-a x +1\right )^{4}+3 a^{4} x^{4} \ln \left (-a x +1\right )^{2}+8 a^{3} x^{3} \ln \left (-a x +1\right )^{3}-2 a^{2} x^{2} \ln \left (-a x +1\right )^{4}+12 a^{3} x^{3} \ln \left (-a x +1\right )+18 a^{2} x^{2} \ln \left (-a x +1\right )^{2}+8 a x \ln \left (-a x +1\right )^{3}+12 a^{2} x^{2}+\ln \left (-a x +1\right )^{4}+12 a x \ln \left (-a x +1\right )+3 \ln \left (-a x +1\right )^{2}}{512 a^{3} \left (a x +1\right ) \left (a x -1\right ) \left (a^{2} x^{2}-1\right )}\) | \(559\) |
derivativedivides | \(\text {Expression too large to display}\) | \(2059\) |
default | \(\text {Expression too large to display}\) | \(2059\) |
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. 657 vs.
\(2 (187) = 374\).
time = 0.28, size = 657, normalized size = 3.06 \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 )^{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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 161, normalized size = 0.75 \begin {gather*} -\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 )}} \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}^{3}{\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 = 3.19, size = 831, normalized size = 3.87 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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