Optimal. Leaf size=208 \[ \frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac {4144 x}{3375 \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 \tanh ^{-1}(a x)}{15 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{15 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6111, 6109,
197, 198} \begin {gather*} \frac {4144 x}{3375 \sqrt {1-a^2 x^2}}+\frac {272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{15 \sqrt {1-a^2 x^2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {16 \tanh ^{-1}(a x)}{15 a \sqrt {1-a^2 x^2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 6109
Rule 6111
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx &=-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {2}{25} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac {4}{5} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8}{125} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {8}{45} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {8}{15} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 \tanh ^{-1}(a x)}{15 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{15 \sqrt {1-a^2 x^2}}+\frac {16}{375} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {16}{135} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {16}{15} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac {4144 x}{3375 \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 \tanh ^{-1}(a x)}{15 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{15 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 94, normalized size = 0.45 \begin {gather*} \frac {4470 a x-8560 a^3 x^3+4144 a^5 x^5-30 \left (149-260 a^2 x^2+120 a^4 x^4\right ) \tanh ^{-1}(a x)+225 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \tanh ^{-1}(a x)^2}{3375 a \left (1-a^2 x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 118, normalized size = 0.57
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (1800 \arctanh \left (a x \right )^{2} a^{5} x^{5}+4144 a^{5} x^{5}-3600 a^{4} x^{4} \arctanh \left (a x \right )-4500 \arctanh \left (a x \right )^{2} a^{3} x^{3}-8560 a^{3} x^{3}+7800 a^{2} x^{2} \arctanh \left (a x \right )+3375 \arctanh \left (a x \right )^{2} a x +4470 a x -4470 \arctanh \left (a x \right )\right )}{3375 a \left (a^{2} x^{2}-1\right )^{3}}\) | \(118\) |
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. 514 vs.
\(2 (172) = 344\).
time = 0.52, size = 514, normalized size = 2.47 \begin {gather*} \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {3 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {1}{3375} \, a {\left (\frac {9 \, {\left (\frac {8 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} x + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a}\right )}}{a} + \frac {9 \, {\left (\frac {8 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} x - {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a}\right )}}{a} + \frac {100 \, {\left (\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} a^{2} x + \sqrt {-a^{2} x^{2} + 1} a}\right )}}{a} + \frac {100 \, {\left (\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} a^{2} x - \sqrt {-a^{2} x^{2} + 1} a}\right )}}{a} - \frac {1800 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x + a\right )} a} - \frac {1800 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x - a\right )} a} - \frac {1800 \, \log \left (a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {1800 \, \log \left (-a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {300 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {300 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} - \frac {135 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}} + \frac {135 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 139, normalized size = 0.67 \begin {gather*} -\frac {{\left (16576 \, a^{5} x^{5} - 34240 \, a^{3} x^{3} + 225 \, {\left (8 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 17880 \, a x - 60 \, {\left (120 \, a^{4} x^{4} - 260 \, a^{2} x^{2} + 149\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{13500 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\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 {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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