Optimal. Leaf size=63 \[ \frac {2 x}{\sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6109, 197}
\begin {gather*} \frac {2 x}{\sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 6109
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=-\frac {2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{\sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.60 \begin {gather*} \frac {2 a x-2 \tanh ^{-1}(a x)+a x \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 49, normalized size = 0.78
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\arctanh \left (a x \right )^{2} a x +2 a x -2 \arctanh \left (a x \right )\right )}{a \left (a^{2} x^{2}-1\right )}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 57, normalized size = 0.90 \begin {gather*} \frac {x \operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {2 \, \operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 69, normalized size = 1.10 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 8 \, a x - 4 \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )}}{4 \, {\left (a^{3} x^{2} - a\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\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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