3.221 \(\int (d x)^m (a+b \log (c x^n)) \text {Li}_2(e x^q) \, dx\)

Optimal. Leaf size=178 \[ \frac {q \text {Int}\left ((d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ),x\right )}{m+1}+\frac {(d x)^{m+1} \text {Li}_2\left (e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b e n q^2 x^{q+1} (d x)^m \, _2F_1\left (1,\frac {m+q+1}{q};\frac {m+2 q+1}{q};e x^q\right )}{(m+1)^3 (m+q+1)}-\frac {b n (d x)^{m+1} \text {Li}_2\left (e x^q\right )}{d (m+1)^2}-\frac {b n q (d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)^3} \]

[Out]

-b*e*n*q^2*x^(1+q)*(d*x)^m*hypergeom([1, (1+m+q)/q],[(1+m+2*q)/q],e*x^q)/(1+m)^3/(1+m+q)-b*n*q*(d*x)^(1+m)*ln(
1-e*x^q)/d/(1+m)^3-b*n*(d*x)^(1+m)*polylog(2,e*x^q)/d/(1+m)^2+(d*x)^(1+m)*(a+b*ln(c*x^n))*polylog(2,e*x^q)/d/(
1+m)+q*Unintegrable((d*x)^m*(a+b*ln(c*x^n))*ln(1-e*x^q),x)/(1+m)

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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.

[In]

Int[(d*x)^m*(a + b*Log[c*x^n])*PolyLog[2, e*x^q],x]

[Out]

-((b*e*n*q^2*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, e*x^q])/((1 + m)^3*(1 + m
+ q))) - (b*n*q*(d*x)^(1 + m)*Log[1 - e*x^q])/(d*(1 + m)^3) - (b*n*(d*x)^(1 + m)*PolyLog[2, e*x^q])/(d*(1 + m)
^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[2, e*x^q])/(d*(1 + m)) + (q*Defer[Int][(d*x)^m*(a + b*Log[c*x^
n])*Log[1 - e*x^q], x])/(1 + m)

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.

[In]

Integrate[(d*x)^m*(a + b*Log[c*x^n])*PolyLog[2, e*x^q],x]

[Out]

Integrate[(d*x)^m*(a + b*Log[c*x^n])*PolyLog[2, e*x^q], x]

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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.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))*polylog(2,e*x^q),x, algorithm="fricas")

[Out]

integral((d*x)^m*b*dilog(e*x^q)*log(c*x^n) + (d*x)^m*a*dilog(e*x^q), x)

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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.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))*polylog(2,e*x^q),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*(d*x)^m*dilog(e*x^q), x)

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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.

[In]

int((d*x)^m*(b*ln(c*x^n)+a)*polylog(2,e*x^q),x)

[Out]

-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*b*ln(c)/q*(-q^2*x^(m+1)*(-e)^(m/q+1/q)/(m+1)^2*ln(-e*x^q+1)-q*x^(m+1)*(-e)^(m/
q+1/q)/(m+1)*polylog(2,e*x^q)-q^2*x^(m+q+1)*e*(-e)^(m/q+1/q)/(m+1)^2*LerchPhi(e*x^q,1,(m+q+1)/q))+((-e)^(-m/q-
1/q)*ln(-e)/q^2*(d*x)^m*x^(-m)*b*n*(-q^2*x^m*(-e)^(m/q+1/q)/(m+1)^2*ln(-e*x^q+1)-q*x^m*(-e)^(m/q+1/q)/(m+1)*po
lylog(2,e*x^q)-q^2*x^(m+q)*e*(-e)^(m/q+1/q)/(m+1)^2*LerchPhi(e*x^q,1,(m+q+1)/q))-(-e)^(-m/q-1/q)*(d*x)^m*x^(-m
)*b*n/q*(-q^2*x^m*(-e)^(m/q+1/q)*ln(x)/(m+1)^2*ln(-e*x^q+1)-q*x^m*(-e)^(m/q+1/q)*ln(-e)/(m+1)^2*ln(-e*x^q+1)+2
*q^2*x^m*(-e)^(m/q+1/q)/(m+1)^3*ln(-e*x^q+1)-q*x^m*(-e)^(m/q+1/q)*ln(x)/(m+1)*polylog(2,e*x^q)-x^m*(-e)^(m/q+1
/q)*ln(-e)/(m+1)*polylog(2,e*x^q)+q*x^m*(-e)^(m/q+1/q)/(m+1)^2*polylog(2,e*x^q)-q^2*x^(m+q)*e*(-e)^(m/q+1/q)*l
n(x)/(m+1)^2*LerchPhi(e*x^q,1,(m+q+1)/q)-q*x^(m+q)*e*(-e)^(m/q+1/q)*ln(-e)/(m+1)^2*LerchPhi(e*x^q,1,(m+q+1)/q)
+2*q^2*x^(m+q)*e*(-e)^(m/q+1/q)/(m+1)^3*LerchPhi(e*x^q,1,(m+q+1)/q)+q*x^(m+q)*e*(-e)^(m/q+1/q)/(m+1)^2*LerchPh
i(e*x^q,2,(m+q+1)/q)))*x-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*a/q*(-q^2*x^(m+1)*(-e)^(m/q+1/q)/(m+1)^2*ln(-e*x^q+1)-
q*x^(m+1)*(-e)^(m/q+1/q)/(m+1)*polylog(2,e*x^q)-q^2*x^(m+q+1)*e*(-e)^(m/q+1/q)/(m+1)^2*LerchPhi(e*x^q,1,(m+q+1
)/q))

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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.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))*polylog(2,e*x^q),x, algorithm="maxima")

[Out]

(((b*d^m*m^2 + 2*b*d^m*m + b*d^m)*x*x^m*log(x^n) + ((b*log(c) + a)*d^m*m^2 + 2*(b*log(c) + a)*d^m*m + (b*log(c
) + a)*d^m - (b*d^m*m + b*d^m)*n)*x*x^m)*dilog(e*x^q) + ((b*d^m*m + b*d^m)*q*x*x^m*log(x^n) + ((b*log(c) + a)*
d^m*m - 2*b*d^m*n + (b*log(c) + a)*d^m)*q*x*x^m)*log(-e*x^q + 1))/(m^3 + 3*m^2 + 3*m + 1) - integrate(-((b*d^m
*e*m + b*d^m*e)*q^2*e^(m*log(x) + q*log(x))*log(x^n) + ((b*log(c) + a)*d^m*e*m - 2*b*d^m*e*n + (b*log(c) + a)*
d^m*e)*q^2*e^(m*log(x) + q*log(x)))/(m^3 + 3*m^2 - (e*m^3 + 3*e*m^2 + 3*e*m + e)*x^q + 3*m + 1), x)

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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.

[In]

int((d*x)^m*polylog(2, e*x^q)*(a + b*log(c*x^n)),x)

[Out]

int((d*x)^m*polylog(2, e*x^q)*(a + b*log(c*x^n)), x)

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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.

[In]

integrate((d*x)**m*(a+b*ln(c*x**n))*polylog(2,e*x**q),x)

[Out]

Timed out

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