Optimal. Leaf size=139 \[ \frac {2 x}{27 \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x}{27 \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac {4 \tanh ^{-1}(a x)}{3 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^2}{3 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {40 x}{27 \sqrt {1-a^2 x^2}}+\frac {2 x}{27 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^2}{3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {4 \tanh ^{-1}(a x)}{3 a \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}} \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 )^{5/2}} \, dx &=-\frac {2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {2}{3} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{27 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac {4 \tanh ^{-1}(a x)}{3 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^2}{3 \sqrt {1-a^2 x^2}}+\frac {4}{27} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {4}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{27 \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x}{27 \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac {4 \tanh ^{-1}(a x)}{3 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^2}{3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 70, normalized size = 0.50 \begin {gather*} \frac {42 a x-40 a^3 x^3+6 \left (-7+6 a^2 x^2\right ) \tanh ^{-1}(a x)-9 a x \left (-3+2 a^2 x^2\right ) \tanh ^{-1}(a x)^2}{27 a \left (1-a^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 84, normalized size = 0.60
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (18 \arctanh \left (a x \right )^{2} a^{3} x^{3}+40 a^{3} x^{3}-36 a^{2} x^{2} \arctanh \left (a x \right )-27 \arctanh \left (a x \right )^{2} a x -42 a x +42 \arctanh \left (a x \right )\right )}{27 a \left (a^{2} x^{2}-1\right )^{2}}\) | \(84\) |
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. 304 vs.
\(2 (115) = 230\).
time = 0.52, size = 304, normalized size = 2.19 \begin {gather*} \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {1}{27} \, a {\left (\frac {\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}}{a} + \frac {\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}}{a} - \frac {18 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x + a\right )} a} - \frac {18 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x - a\right )} a} - \frac {18 \, \log \left (a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {18 \, \log \left (-a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {3 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {3 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{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 = 105, normalized size = 0.76 \begin {gather*} -\frac {{\left (160 \, a^{3} x^{3} + 9 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 168 \, a x - 12 \, {\left (6 \, a^{2} x^{2} - 7\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{108 \, {\left (a^{5} x^{4} - 2 \, 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 {5}{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 = -0.01 \begin {gather*} \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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