Optimal. Leaf size=68 \[ \frac {8 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \text {Li}_2(a x)}{d \sqrt {d x}}+\frac {4 \log (1-a x)}{d \sqrt {d x}} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6591, 2395, 63, 206} \[ -\frac {2 \text {PolyLog}(2,a x)}{d \sqrt {d x}}+\frac {8 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {4 \log (1-a x)}{d \sqrt {d x}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 2395
Rule 6591
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(a x)}{(d x)^{3/2}} \, dx &=-\frac {2 \text {Li}_2(a x)}{d \sqrt {d x}}-2 \int \frac {\log (1-a x)}{(d x)^{3/2}} \, dx\\ &=\frac {4 \log (1-a x)}{d \sqrt {d x}}-\frac {2 \text {Li}_2(a x)}{d \sqrt {d x}}+\frac {(4 a) \int \frac {1}{\sqrt {d x} (1-a x)} \, dx}{d}\\ &=\frac {4 \log (1-a x)}{d \sqrt {d x}}-\frac {2 \text {Li}_2(a x)}{d \sqrt {d x}}+\frac {(8 a) \operatorname {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{d^2}\\ &=\frac {8 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {4 \log (1-a x)}{d \sqrt {d x}}-\frac {2 \text {Li}_2(a x)}{d \sqrt {d x}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 51, normalized size = 0.75 \[ \frac {2 x \left (-\text {Li}_2(a x)+2 \log (1-a x)+4 \sqrt {a} \sqrt {x} \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 132, normalized size = 1.94 \[ \left [\frac {2 \, {\left (2 \, d x \sqrt {\frac {a}{d}} \log \left (\frac {a x + 2 \, \sqrt {d x} \sqrt {\frac {a}{d}} + 1}{a x - 1}\right ) - \sqrt {d x} {\left ({\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )}\right )}}{d^{2} x}, -\frac {2 \, {\left (4 \, d x \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + \sqrt {d x} {\left ({\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )}\right )}}{d^{2} x}\right ] \]
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 \[ \int \frac {{\rm Li}_2\left (a x\right )}{\left (d x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.93 \[ -\frac {2 \polylog \left (2, a x \right )}{d \sqrt {d x}}+\frac {4 \ln \left (\frac {-a d x +d}{d}\right )}{d \sqrt {d x}}+\frac {8 a \arctanh \left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{d \sqrt {a d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 71, normalized size = 1.04 \[ -\frac {2 \, {\left (\frac {2 \, a \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d}} + \frac {{\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a d x + d\right ) + 2 \, \log \relax (d)}{\sqrt {d x}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {polylog}\left (2,a\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {Li}_{2}\left (a x\right )}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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