Optimal. Leaf size=177 \[ -\frac {16}{35 a \sqrt {1-a^2 x^2}}-\frac {8}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x \tanh ^{-1}(a x)}{35 \sqrt {1-a^2 x^2}}+\frac {8 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5960, 5958} \[ -\frac {16}{35 a \sqrt {1-a^2 x^2}}-\frac {8}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x \tanh ^{-1}(a x)}{35 \sqrt {1-a^2 x^2}}+\frac {8 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 5958
Rule 5960
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx &=-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {6}{175 a \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {24}{35} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8}{105 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16}{35} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16}{35 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x \tanh ^{-1}(a x)}{35 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 81, normalized size = 0.46 \[ \frac {1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-105 a x \left (16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right ) \tanh ^{-1}(a x)-2161}{3675 a \left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 121, normalized size = 0.68 \[ \frac {{\left (3360 \, a^{6} x^{6} - 10640 \, a^{4} x^{4} + 11452 \, a^{2} x^{2} - 105 \, {\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 4322\right )} \sqrt {-a^{2} x^{2} + 1}}{7350 \, {\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 138, normalized size = 0.78 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} {\left (2 \, {\left (4 \, {\left (2 \, a^{6} x^{2} - 7 \, a^{4}\right )} x^{2} + 35 \, a^{2}\right )} x^{2} - 35\right )} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{70 \, {\left (a^{2} x^{2} - 1\right )}^{4}} - \frac {126 \, a^{2} x^{2} + 1680 \, {\left (a^{2} x^{2} - 1\right )}^{3} - 280 \, {\left (a^{2} x^{2} - 1\right )}^{2} - 201}{3675 \, {\left (a^{2} x^{2} - 1\right )}^{3} \sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 99, normalized size = 0.56 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (1680 \arctanh \left (a x \right ) x^{7} a^{7}-1680 x^{6} a^{6}-5880 \arctanh \left (a x \right ) x^{5} a^{5}+5320 x^{4} a^{4}+7350 a^{3} x^{3} \arctanh \left (a x \right )-5726 a^{2} x^{2}-3675 a x \arctanh \left (a x \right )+2161\right )}{3675 a \left (a^{2} x^{2}-1\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 140, normalized size = 0.79 \[ -\frac {1}{3675} \, a {\left (\frac {1680}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {280}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {126}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}} + \frac {75}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}} a^{2}}\right )} + \frac {1}{35} \, {\left (\frac {16 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {6 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} + \frac {5 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}}}\right )} \operatorname {artanh}\left (a x\right ) \]
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 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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