Optimal. Leaf size=107 \[ -\frac{7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}-\frac{1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}+\frac{35 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{128 a}+\frac{189 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{128 a}+\frac{175 \text{Chi}\left (5 \tanh ^{-1}(a x)\right )}{128 a}+\frac{49 \text{Chi}\left (7 \tanh ^{-1}(a x)\right )}{128 a} \]
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Rubi [A] time = 0.43919, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5966, 6032, 6034, 5448, 3301, 5968, 3312} \[ -\frac{7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}-\frac{1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}+\frac{35 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{128 a}+\frac{189 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{128 a}+\frac{175 \text{Chi}\left (5 \tanh ^{-1}(a x)\right )}{128 a}+\frac{49 \text{Chi}\left (7 \tanh ^{-1}(a x)\right )}{128 a} \]
Antiderivative was successfully verified.
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Rule 5966
Rule 6032
Rule 6034
Rule 5448
Rule 3301
Rule 5968
Rule 3312
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)^3} \, dx &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}+\frac{1}{2} (7 a) \int \frac{x}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac{7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac{7}{2} \int \frac{1}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)} \, dx+\left (21 a^2\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac{7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac{7 \operatorname{Subst}\left (\int \frac{\cosh ^7(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac{21 \operatorname{Subst}\left (\int \frac{\cosh ^5(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac{7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac{7 \operatorname{Subst}\left (\int \left (\frac{35 \cosh (x)}{64 x}+\frac{21 \cosh (3 x)}{64 x}+\frac{7 \cosh (5 x)}{64 x}+\frac{\cosh (7 x)}{64 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac{21 \operatorname{Subst}\left (\int \left (-\frac{5 \cosh (x)}{64 x}+\frac{\cosh (3 x)}{64 x}+\frac{3 \cosh (5 x)}{64 x}+\frac{\cosh (7 x)}{64 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac{7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac{7 \operatorname{Subst}\left (\int \frac{\cosh (7 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}+\frac{21 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac{21 \operatorname{Subst}\left (\int \frac{\cosh (7 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac{49 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}+\frac{63 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac{147 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}-\frac{105 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac{245 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}\\ &=-\frac{1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac{7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac{35 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{128 a}+\frac{189 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{128 a}+\frac{175 \text{Chi}\left (5 \tanh ^{-1}(a x)\right )}{128 a}+\frac{49 \text{Chi}\left (7 \tanh ^{-1}(a x)\right )}{128 a}\\ \end{align*}
Mathematica [A] time = 0.21183, size = 99, normalized size = 0.93 \[ \frac{1}{128} \left (-\frac{448 x}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}-\frac{64}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}+\frac{35 \text{Chi}\left (\tanh ^{-1}(a x)\right )}{a}+\frac{189 \text{Chi}\left (3 \tanh ^{-1}(a x)\right )}{a}+\frac{175 \text{Chi}\left (5 \tanh ^{-1}(a x)\right )}{a}+\frac{49 \text{Chi}\left (7 \tanh ^{-1}(a x)\right )}{a}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.186, size = 364, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{9}{2}} \operatorname{artanh}\left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{10} x^{10} - 5 \, a^{8} x^{8} + 10 \, a^{6} x^{6} - 10 \, a^{4} x^{4} + 5 \, a^{2} x^{2} - 1\right )} \operatorname{artanh}\left (a x\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{9}{2}} \operatorname{artanh}\left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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