Optimal. Leaf size=90 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)},x\right )}{a}-\frac {a x}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {\sqrt {1-a^2 x^2}}{a x \tanh ^{-1}(a x)}+\text {Chi}\left (\tanh ^{-1}(a x)\right ) \]
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Rubi [A] time = 0.42, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx &=a^2 \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx+\int \frac {1}{x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {a x}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {\sqrt {1-a^2 x^2}}{a x \tanh ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \, dx}{a}+a \int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {a x}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {\sqrt {1-a^2 x^2}}{a x \tanh ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \, dx}{a}+\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {a x}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {\sqrt {1-a^2 x^2}}{a x \tanh ^{-1}(a x)}+\text {Chi}\left (\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \, dx}{a}\\ \end {align*}
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Mathematica [A] time = 6.40, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{4} x^{5} - 2 \, a^{2} x^{3} + x\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctanh \left (a x \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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