Optimal. Leaf size=211 \[ -\frac {2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {2 \left (a^2 x^2+1\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {a^2 x^2+1}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {a^2 x^2+1}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}+\frac {4 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{315 a} \]
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Rubi [A] time = 0.26, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5966, 5996, 6034, 5448, 12, 3298} \[ -\frac {2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {2 \left (a^2 x^2+1\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {a^2 x^2+1}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {a^2 x^2+1}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}+\frac {4 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{315 a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 5448
Rule 5966
Rule 5996
Rule 6034
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^8} \, dx &=-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}+\frac {1}{7} (2 a) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^7} \, dx\\ &=-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}+\frac {1}{105} (4 a) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5} \, dx\\ &=-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac {x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac {1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}+\frac {1}{315} (4 a) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac {x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac {1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {1}{315} (8 a) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac {x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac {1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {8 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{315 a}\\ &=-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac {x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac {1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {8 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{315 a}\\ &=-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac {x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac {1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {4 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{315 a}\\ &=-\frac {1}{7 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^7}-\frac {x}{21 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac {1+a^2 x^2}{105 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac {x}{105 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac {1+a^2 x^2}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {2 x}{315 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2 \left (1+a^2 x^2\right )}{315 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {4 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{315 a}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 128, normalized size = 0.61 \[ \frac {4 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^7 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+2 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^6+\left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^4+3 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^2+2 a x \tanh ^{-1}(a x)^5+3 a x \tanh ^{-1}(a x)^3+15 a x \tanh ^{-1}(a x)+45}{315 a \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 249, normalized size = 1.18 \[ \frac {2 \, {\left ({\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{7} + 4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{5} + 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{6} + 24 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 4 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 480 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + 48 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 2880\right )}}{315 \, {\left (a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{7}} \]
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 \[ \int \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 128, normalized size = 0.61 \[ \frac {-\frac {1}{14 \arctanh \left (a x \right )^{7}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{14 \arctanh \left (a x \right )^{7}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{42 \arctanh \left (a x \right )^{6}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{105 \arctanh \left (a x \right )^{5}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{210 \arctanh \left (a x \right )^{4}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{315 \arctanh \left (a x \right )^{3}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{315 \arctanh \left (a x \right )^{2}}-\frac {2 \cosh \left (2 \arctanh \left (a x \right )\right )}{315 \arctanh \left (a x \right )}+\frac {4 \Shi \left (2 \arctanh \left (a x \right )\right )}{315}}{a} \]
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 \[ -16 \, a \int -\frac {x}{315 \, {\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )\right )}}\,{d x} + \frac {4 \, {\left (2 \, a x \log \left (a x + 1\right )^{5} + {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{6} + {\left (a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )^{6} - 2 \, {\left (a x + 3 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{5} + 12 \, a x \log \left (a x + 1\right )^{3} + 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{4} + {\left (2 \, a^{2} x^{2} + 10 \, a x \log \left (a x + 1\right ) + 15 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2\right )} \log \left (-a x + 1\right )^{4} - 4 \, {\left (5 \, a x \log \left (a x + 1\right )^{2} + 5 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 3 \, a x + 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{3} + 240 \, a x \log \left (a x + 1\right ) + 24 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + {\left (20 \, a x \log \left (a x + 1\right )^{3} + 15 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{4} + 24 \, a^{2} x^{2} + 36 \, a x \log \left (a x + 1\right ) + 12 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 24\right )} \log \left (-a x + 1\right )^{2} - 2 \, {\left (5 \, a x \log \left (a x + 1\right )^{4} + 3 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{5} + 18 \, a x \log \left (a x + 1\right )^{2} + 4 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 120 \, a x + 24 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right ) + 1440\right )}}{315 \, {\left ({\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{7} - 7 \, {\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{6} \log \left (-a x + 1\right ) + 21 \, {\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{5} \log \left (-a x + 1\right )^{2} - 35 \, {\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{4} \log \left (-a x + 1\right )^{3} + 35 \, {\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{3} \log \left (-a x + 1\right )^{4} - 21 \, {\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right )^{5} + 7 \, {\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{6} - {\left (a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )^{7}\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 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^8\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{8}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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