Optimal. Leaf size=97 \[ \frac {16 \sqrt {d x}}{d}-\frac {16 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {a} \sqrt {d}}-\frac {8 \sqrt {d x} \log (1-a x)}{d}-\frac {4 \sqrt {d x} \text {PolyLog}(2,a x)}{d}+\frac {2 \sqrt {d x} \text {PolyLog}(3,a x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6726, 2442, 52,
65, 212} \begin {gather*} -\frac {4 \sqrt {d x} \text {Li}_2(a x)}{d}+\frac {2 \sqrt {d x} \text {Li}_3(a x)}{d}-\frac {8 \sqrt {d x} \log (1-a x)}{d}-\frac {16 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {a} \sqrt {d}}+\frac {16 \sqrt {d x}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 212
Rule 2442
Rule 6726
Rubi steps
\begin {align*} \int \frac {\text {Li}_3(a x)}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \text {Li}_3(a x)}{d}-2 \int \frac {\text {Li}_2(a x)}{\sqrt {d x}} \, dx\\ &=-\frac {4 \sqrt {d x} \text {Li}_2(a x)}{d}+\frac {2 \sqrt {d x} \text {Li}_3(a x)}{d}-4 \int \frac {\log (1-a x)}{\sqrt {d x}} \, dx\\ &=-\frac {8 \sqrt {d x} \log (1-a x)}{d}-\frac {4 \sqrt {d x} \text {Li}_2(a x)}{d}+\frac {2 \sqrt {d x} \text {Li}_3(a x)}{d}-\frac {(8 a) \int \frac {\sqrt {d x}}{1-a x} \, dx}{d}\\ &=\frac {16 \sqrt {d x}}{d}-\frac {8 \sqrt {d x} \log (1-a x)}{d}-\frac {4 \sqrt {d x} \text {Li}_2(a x)}{d}+\frac {2 \sqrt {d x} \text {Li}_3(a x)}{d}-8 \int \frac {1}{\sqrt {d x} (1-a x)} \, dx\\ &=\frac {16 \sqrt {d x}}{d}-\frac {8 \sqrt {d x} \log (1-a x)}{d}-\frac {4 \sqrt {d x} \text {Li}_2(a x)}{d}+\frac {2 \sqrt {d x} \text {Li}_3(a x)}{d}-\frac {16 \text {Subst}\left (\int \frac {1}{1-\frac {a x^2}{d}} \, dx,x,\sqrt {d x}\right )}{d}\\ &=\frac {16 \sqrt {d x}}{d}-\frac {16 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {a} \sqrt {d}}-\frac {8 \sqrt {d x} \log (1-a x)}{d}-\frac {4 \sqrt {d x} \text {Li}_2(a x)}{d}+\frac {2 \sqrt {d x} \text {Li}_3(a x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 57, normalized size = 0.59 \begin {gather*} \frac {2 x \left (8-\frac {8 \tanh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a} \sqrt {x}}-4 \log (1-a x)-2 \text {PolyLog}(2,a x)+\text {PolyLog}(3,a x)\right )}{\sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 127, normalized size = 1.31
method | result | size |
meijerg | \(\frac {\sqrt {x}\, \sqrt {-a}\, \left (\frac {16 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}}}{a}+\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{a \sqrt {a x}}-\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \ln \left (-a x +1\right )}{a}-\frac {4 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \polylog \left (2, a x \right )}{a}+\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \polylog \left (3, a x \right )}{a}\right )}{\sqrt {d x}\, a}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 94, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (4 \, \sqrt {d x} {\left (\log \left (d\right ) + 2\right )} - 2 \, \sqrt {d x} {\rm Li}_2\left (a x\right ) - 4 \, \sqrt {d x} \log \left (-a d x + d\right ) + \frac {4 \, d \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d}} + \sqrt {d x} {\rm Li}_{3}(a x)\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 161, normalized size = 1.66 \begin {gather*} \left [\frac {2 \, {\left (\sqrt {d x} a {\rm polylog}\left (3, a x\right ) - 2 \, \sqrt {d x} {\left (a {\rm Li}_2\left (a x\right ) + 2 \, a \log \left (-a x + 1\right ) - 4 \, a\right )} + 4 \, \sqrt {a d} \log \left (\frac {a d x - 2 \, \sqrt {a d} \sqrt {d x} + d}{a x - 1}\right )\right )}}{a d}, \frac {2 \, {\left (\sqrt {d x} a {\rm polylog}\left (3, a x\right ) - 2 \, \sqrt {d x} {\left (a {\rm Li}_2\left (a x\right ) + 2 \, a \log \left (-a x + 1\right ) - 4 \, a\right )} + 8 \, \sqrt {-a d} \arctan \left (\frac {\sqrt {-a d} \sqrt {d x}}{a d x}\right )\right )}}{a d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{3}\left (a x\right )}{\sqrt {d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {polylog}\left (3,a\,x\right )}{\sqrt {d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________