Optimal. Leaf size=90 \[ -\frac {a x}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {\sqrt {1-a^2 x^2}}{a x \tanh ^{-1}(a x)}+\text {Chi}\left (\tanh ^{-1}(a x)\right )-\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)},x\right )}{a} \]
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Rubi [A]
time = 0.40, 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 = {}
\begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \end {gather*}
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}+\text {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 = 4.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 2.67, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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