Optimal. Leaf size=277 \[ \frac {413312 x}{385875 \sqrt {1-a^2 x^2}}+\frac {30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac {888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x \tanh ^{-1}(a x)^2}{35 \sqrt {1-a^2 x^2}}+\frac {8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {32 \tanh ^{-1}(a x)}{35 a \sqrt {1-a^2 x^2}}-\frac {16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5964, 5962, 191, 192} \[ \frac {413312 x}{385875 \sqrt {1-a^2 x^2}}+\frac {30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac {888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x \tanh ^{-1}(a x)^2}{35 \sqrt {1-a^2 x^2}}+\frac {8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac {32 \tanh ^{-1}(a x)}{35 a \sqrt {1-a^2 x^2}}-\frac {16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 5962
Rule 5964
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{9/2}} \, dx &=-\frac {2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac {x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {2}{49} \int \frac {1}{\left (1-a^2 x^2\right )^{9/2}} \, dx+\frac {6}{7} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx\\ &=\frac {2 x}{343 \left (1-a^2 x^2\right )^{7/2}}-\frac {2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {12}{343} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac {12}{175} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac {24}{35} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {48 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{1715}+\frac {48}{875} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {16}{105} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {16}{35} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {32 \tanh ^{-1}(a x)}{35 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x \tanh ^{-1}(a x)^2}{35 \sqrt {1-a^2 x^2}}+\frac {32 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{1715}+\frac {32}{875} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {32}{315} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {32}{35} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac {888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac {30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac {413312 x}{385875 \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac {12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac {16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac {32 \tanh ^{-1}(a x)}{35 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x \tanh ^{-1}(a x)^2}{35 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 120, normalized size = 0.43 \[ \frac {2 a x \left (-206656 a^6 x^6+635096 a^4 x^4-654220 a^2 x^2+226905\right )-11025 a x \left (16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right ) \tanh ^{-1}(a x)^2+210 \left (1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-2161\right ) \tanh ^{-1}(a x)}{385875 a \left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 169, normalized size = 0.61 \[ -\frac {{\left (1653248 \, a^{7} x^{7} - 5080768 \, a^{5} x^{5} + 5233760 \, a^{3} x^{3} + 11025 \, {\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 1815240 \, a x - 420 \, {\left (1680 \, a^{6} x^{6} - 5320 \, a^{4} x^{4} + 5726 \, a^{2} x^{2} - 2161\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{1543500 \, {\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 152, normalized size = 0.55 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (176400 \arctanh \left (a x \right )^{2} x^{7} a^{7}+413312 x^{7} a^{7}-352800 \arctanh \left (a x \right ) x^{6} a^{6}-617400 \arctanh \left (a x \right )^{2} x^{5} a^{5}-1270192 x^{5} a^{5}+1117200 a^{4} x^{4} \arctanh \left (a x \right )+771750 \arctanh \left (a x \right )^{2} x^{3} a^{3}+1308440 x^{3} a^{3}-1202460 a^{2} x^{2} \arctanh \left (a x \right )-385875 \arctanh \left (a x \right )^{2} a x -453810 a x +453810 \arctanh \left (a x \right )\right )}{385875 a \left (a^{2} x^{2}-1\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 751, normalized size = 2.71 \[ \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 )^{2} + \frac {1}{385875} \, a {\left (\frac {225 \, {\left (\frac {16 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {5}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2} x + {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a} + \frac {6 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\right )}}{a} + \frac {225 \, {\left (\frac {16 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {8 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {5}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2} x - {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a} + \frac {6 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\right )}}{a} + \frac {882 \, {\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 {882 \, {\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 {9800 \, {\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 {9800 \, {\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 {176400 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x + a\right )} a} - \frac {176400 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x - a\right )} a} - \frac {176400 \, \log \left (a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {176400 \, \log \left (-a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {29400 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {29400 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} - \frac {13230 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}} + \frac {13230 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}} - \frac {7875 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}} a^{2}} + \frac {7875 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}} a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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