Optimal. Leaf size=191 \[ -\frac {2}{27 a \left (1-a^2 x^2\right )^{3/2}}-\frac {40}{9 a \sqrt {1-a^2 x^2}}+\frac {2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x \tanh ^{-1}(a x)}{9 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^3}{3 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 191, 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,
6105, 6107} \begin {gather*} -\frac {40}{9 a \sqrt {1-a^2 x^2}}-\frac {2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^3}{3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x \tanh ^{-1}(a x)}{9 \sqrt {1-a^2 x^2}}+\frac {2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6105
Rule 6107
Rule 6109
Rule 6111
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {2}{3} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^3}{3 \sqrt {1-a^2 x^2}}+\frac {4}{9} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+4 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {2}{27 a \left (1-a^2 x^2\right )^{3/2}}-\frac {40}{9 a \sqrt {1-a^2 x^2}}+\frac {2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x \tanh ^{-1}(a x)}{9 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^3}{3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 87, normalized size = 0.46 \begin {gather*} \frac {-122+120 a^2 x^2-6 a x \left (-21+20 a^2 x^2\right ) \tanh ^{-1}(a x)+9 \left (-7+6 a^2 x^2\right ) \tanh ^{-1}(a x)^2-9 a x \left (-3+2 a^2 x^2\right ) \tanh ^{-1}(a x)^3}{27 a \left (1-a^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 105, normalized size = 0.55
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (18 \arctanh \left (a x \right )^{3} a^{3} x^{3}+120 a^{3} x^{3} \arctanh \left (a x \right )-54 a^{2} x^{2} \arctanh \left (a x \right )^{2}-27 \arctanh \left (a x \right )^{3} a x -120 a^{2} x^{2}-126 a x \arctanh \left (a x \right )+63 \arctanh \left (a x \right )^{2}+122\right )}{27 a \left (a^{2} x^{2}-1\right )^{2}}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 134, normalized size = 0.70 \begin {gather*} \frac {{\left (960 \, a^{2} x^{2} - 9 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 18 \, {\left (6 \, a^{2} x^{2} - 7\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 24 \, {\left (20 \, a^{3} x^{3} - 21 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 976\right )} \sqrt {-a^{2} x^{2} + 1}}{216 \, {\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}^{3}{\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 )}^3}{{\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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