Optimal. Leaf size=69 \[ -\frac {a q^2 x^{-1+q} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}+\frac {q \log \left (1-a x^q\right )}{x}-\frac {\text {PolyLog}\left (2,a x^q\right )}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6726, 2505,
371} \begin {gather*} -\frac {a q^2 x^{q-1} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}-\frac {\text {Li}_2\left (a x^q\right )}{x}+\frac {q \log \left (1-a x^q\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 2505
Rule 6726
Rubi steps
\begin {align*} \int \frac {\text {Li}_2\left (a x^q\right )}{x^2} \, dx &=-\frac {\text {Li}_2\left (a x^q\right )}{x}-q \int \frac {\log \left (1-a x^q\right )}{x^2} \, dx\\ &=\frac {q \log \left (1-a x^q\right )}{x}-\frac {\text {Li}_2\left (a x^q\right )}{x}+\left (a q^2\right ) \int \frac {x^{-2+q}}{1-a x^q} \, dx\\ &=-\frac {a q^2 x^{-1+q} \, _2F_1\left (1,-\frac {1-q}{q};2-\frac {1}{q};a x^q\right )}{1-q}+\frac {q \log \left (1-a x^q\right )}{x}-\frac {\text {Li}_2\left (a x^q\right )}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 60, normalized size = 0.87 \begin {gather*} \frac {q \left (\frac {a q x^q \, _2F_1\left (1,\frac {-1+q}{q};2-\frac {1}{q};a x^q\right )}{-1+q}+\log \left (1-a x^q\right )\right )}{x}-\frac {\text {PolyLog}\left (2,a x^q\right )}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
5.
time = 0.12, size = 106, normalized size = 1.54
method | result | size |
meijerg | \(-\frac {\left (-a \right )^{\frac {1}{q}} \left (-\frac {q^{2} \left (-a \right )^{-\frac {1}{q}} \ln \left (1-a \,x^{q}\right )}{x}-\frac {q \left (-a \right )^{-\frac {1}{q}} \left (1-q \right ) \polylog \left (2, a \,x^{q}\right )}{\left (-1+q \right ) x}-q^{2} x^{-1+q} a \left (-a \right )^{-\frac {1}{q}} \Phi \left (a \,x^{q}, 1, \frac {-1+q}{q}\right )\right )}{q}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{2}\left (a x^{q}\right )}{x^{2}}\, 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 (2,a\,x^q\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________