Optimal. Leaf size=41 \[ \frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a}+\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{8 a}+\frac {3 \log \left (\tanh ^{-1}(a x)\right )}{8 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6115, 3393,
3382} \begin {gather*} \frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a}+\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{8 a}+\frac {3 \log \left (\tanh ^{-1}(a x)\right )}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 6115
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cosh (2 x)}{2 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {3 \log \left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}+\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a}+\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{8 a}+\frac {3 \log \left (\tanh ^{-1}(a x)\right )}{8 a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 33, normalized size = 0.80 \begin {gather*} -\frac {-4 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )-\text {Chi}\left (4 \tanh ^{-1}(a x)\right )-3 \log \left (\tanh ^{-1}(a x)\right )}{8 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.59, size = 31, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {\frac {3 \ln \left (\arctanh \left (a x \right )\right )}{8}+\frac {\hyperbolicCosineIntegral \left (2 \arctanh \left (a x \right )\right )}{2}+\frac {\hyperbolicCosineIntegral \left (4 \arctanh \left (a x \right )\right )}{8}}{a}\) | \(31\) |
default | \(\frac {\frac {3 \ln \left (\arctanh \left (a x \right )\right )}{8}+\frac {\hyperbolicCosineIntegral \left (2 \arctanh \left (a x \right )\right )}{2}+\frac {\hyperbolicCosineIntegral \left (4 \arctanh \left (a x \right )\right )}{8}}{a}\) | \(31\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs.
\(2 (35) = 70\).
time = 0.41, size = 118, normalized size = 2.88 \begin {gather*} \frac {6 \, \log \left (\log \left (-\frac {a x + 1}{a x - 1}\right )\right ) + \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 4 \, \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + 4 \, \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )}{16 \, a} \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 {1}{a^{6} x^{6} \operatorname {atanh}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )} - \operatorname {atanh}{\left (a x \right )}}\, 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 {1}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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