Optimal. Leaf size=178 \[ \frac {q \text {Int}\left ((d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ),x\right )}{m+1}+\frac {(d x)^{m+1} \text {Li}_2\left (e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b e n q^2 x^{q+1} (d x)^m \, _2F_1\left (1,\frac {m+q+1}{q};\frac {m+2 q+1}{q};e x^q\right )}{(m+1)^3 (m+q+1)}-\frac {b n (d x)^{m+1} \text {Li}_2\left (e x^q\right )}{d (m+1)^2}-\frac {b n q (d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)^3} \]
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Rubi [A] time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,e x^q\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right ) \, dx &=-\frac {b n (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right )}{d (1+m)}+\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac {(b n q) \int (d x)^m \log \left (1-e x^q\right ) \, dx}{(1+m)^2}\\ &=-\frac {b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac {b n (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right )}{d (1+m)}+\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac {\left (b e n q^2\right ) \int \frac {x^{-1+q} (d x)^{1+m}}{1-e x^q} \, dx}{d (1+m)^3}\\ &=-\frac {b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac {b n (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right )}{d (1+m)}+\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}-\frac {\left (b e n q^2 x^{-m} (d x)^m\right ) \int \frac {x^{m+q}}{1-e x^q} \, dx}{(1+m)^3}\\ &=-\frac {b e n q^2 x^{1+q} (d x)^m \, _2F_1\left (1,\frac {1+m+q}{q};\frac {1+m+2 q}{q};e x^q\right )}{(1+m)^3 (1+m+q)}-\frac {b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac {b n (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right )}{d (1+m)}+\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x\right )^{m} b {\rm Li}_2\left (e x^{q}\right ) \log \left (c x^{n}\right ) + \left (d x\right )^{m} a {\rm Li}_2\left (e x^{q}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} {\rm Li}_2\left (e x^{q}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 867, normalized size = 4.87 \[ -\frac {\left (-\frac {e \,q^{2} x^{m +q +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +1\right )^{2}}-\frac {q^{2} x^{m +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{\left (m +1\right )^{2}}-\frac {q \,x^{m +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right )}{m +1}\right ) b \,x^{-m} \left (d x \right )^{m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} \ln \relax (c )}{q}-\frac {\left (-\frac {e \,q^{2} x^{m +q +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +1\right )^{2}}-\frac {q^{2} x^{m +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{\left (m +1\right )^{2}}-\frac {q \,x^{m +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right )}{m +1}\right ) a \,x^{-m} \left (d x \right )^{m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}}}{q}+\left (-\frac {\left (-\frac {e \,q^{2} x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \relax (x ) \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +1\right )^{2}}+\frac {2 e \,q^{2} x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +1\right )^{3}}-\frac {e q \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +1\right )^{2}}-\frac {q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \relax (x ) \ln \left (-e \,x^{q}+1\right )}{\left (m +1\right )^{2}}+\frac {e q \,x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 2, \frac {m +q +1}{q}\right )}{\left (m +1\right )^{2}}+\frac {2 q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{\left (m +1\right )^{3}}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right ) \ln \relax (x )}{m +1}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \ln \left (-e \,x^{q}+1\right )}{\left (m +1\right )^{2}}+\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right )}{\left (m +1\right )^{2}}-\frac {x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right ) \ln \left (-e \right )}{m +1}\right ) b n \,x^{-m} \left (d x \right )^{m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}}}{q}+\frac {\left (-\frac {e \,q^{2} x^{m +q} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {m +q +1}{q}\right )}{\left (m +1\right )^{2}}-\frac {q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{\left (m +1\right )^{2}}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right )}{m +1}\right ) b n \,x^{-m} \left (d x \right )^{m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} \ln \left (-e \right )}{q^{2}}\right ) x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left ({\left (b d^{m} m^{2} + 2 \, b d^{m} m + b d^{m}\right )} x x^{m} \log \left (x^{n}\right ) + {\left ({\left (b \log \relax (c) + a\right )} d^{m} m^{2} + 2 \, {\left (b \log \relax (c) + a\right )} d^{m} m + {\left (b \log \relax (c) + a\right )} d^{m} - {\left (b d^{m} m + b d^{m}\right )} n\right )} x x^{m}\right )} {\rm Li}_2\left (e x^{q}\right ) + {\left ({\left (b d^{m} m + b d^{m}\right )} q x x^{m} \log \left (x^{n}\right ) + {\left ({\left (b \log \relax (c) + a\right )} d^{m} m - 2 \, b d^{m} n + {\left (b \log \relax (c) + a\right )} d^{m}\right )} q x x^{m}\right )} \log \left (-e x^{q} + 1\right )}{m^{3} + 3 \, m^{2} + 3 \, m + 1} - \int -\frac {{\left (b d^{m} e m + b d^{m} e\right )} q^{2} e^{\left (m \log \relax (x) + q \log \relax (x)\right )} \log \left (x^{n}\right ) + {\left ({\left (b \log \relax (c) + a\right )} d^{m} e m - 2 \, b d^{m} e n + {\left (b \log \relax (c) + a\right )} d^{m} e\right )} q^{2} e^{\left (m \log \relax (x) + q \log \relax (x)\right )}}{m^{3} + 3 \, m^{2} - {\left (e m^{3} + 3 \, e m^{2} + 3 \, e m + e\right )} x^{q} + 3 \, m + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,\mathrm {polylog}\left (2,e\,x^q\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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