Optimal. Leaf size=133 \[ -\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {4}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {8}{15 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \tanh ^{-1}(a x)}{15 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6107, 6105}
\begin {gather*} -\frac {8}{15 a \sqrt {1-a^2 x^2}}-\frac {4}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)}{15 \sqrt {1-a^2 x^2}}+\frac {4 x \tanh ^{-1}(a x)}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6105
Rule 6107
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {4}{45 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8}{15} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {4}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {8}{15 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \tanh ^{-1}(a x)}{15 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 65, normalized size = 0.49 \begin {gather*} \frac {-149+260 a^2 x^2-120 a^4 x^4+15 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \tanh ^{-1}(a x)}{225 a \left (1-a^2 x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.81, size = 79, normalized size = 0.59
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (120 \arctanh \left (a x \right ) a^{5} x^{5}-120 a^{4} x^{4}-300 a^{3} x^{3} \arctanh \left (a x \right )+260 a^{2} x^{2}+225 a x \arctanh \left (a x \right )-149\right )}{225 a \left (a^{2} x^{2}-1\right )^{3}}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 108, normalized size = 0.81 \begin {gather*} -\frac {1}{225} \, a {\left (\frac {120}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {20}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {9}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}}\right )} + \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 99, normalized size = 0.74 \begin {gather*} \frac {{\left (240 \, a^{4} x^{4} - 520 \, a^{2} x^{2} - 15 \, {\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 ) + 298\right )} \sqrt {-a^{2} x^{2} + 1}}{450 \, {\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}{\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 [A]
time = 0.44, size = 114, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} {\left (4 \, {\left (2 \, a^{4} x^{2} - 5 \, a^{2}\right )} x^{2} + 15\right )} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{30 \, {\left (a^{2} x^{2} - 1\right )}^{3}} + \frac {20 \, a^{2} x^{2} - 120 \, {\left (a^{2} x^{2} - 1\right )}^{2} - 29}{225 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} a} \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 )}{{\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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