3.3.21 \(\int (d x)^m (a+b \log (c x^n)) \text {Li}_2(e x^q) \, dx\) [221]

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

[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 = 0.08, 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 = {} \begin {gather*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,e x^q\right ) \, dx \end {gather*}

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 = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right ) \, dx \end {gather*}

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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Maple [A] Leaf count of result is larger than twice the leaf count of optimal. \(866\) vs. \(2(179)=358\).
time = 0.21, size = 867, normalized size = 4.87

method	result	size
meijerg	\(-\frac {\left (d x \right )^{m} x^{-m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} a \left (-\frac {q^{2} x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right )}{1+m}-\frac {q^{2} x^{1+m +q} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q}-\frac {\left (d x \right )^{m} x^{-m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} b \ln \left (c \right ) \left (-\frac {q^{2} x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right )}{1+m}-\frac {q^{2} x^{1+m +q} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q}+\left (\frac {\left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} \ln \left (-e \right ) \left (d x \right )^{m} x^{-m} b n \left (-\frac {q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right )}{1+m}-\frac {q^{2} x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q^{2}}-\frac {\left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} \left (d x \right )^{m} x^{-m} b n \left (-\frac {q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (x \right ) \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{2}}+\frac {2 q^{2} x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{\left (1+m \right )^{3}}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (x \right ) \polylog \left (2, e \,x^{q}\right )}{1+m}-\frac {x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \polylog \left (2, e \,x^{q}\right )}{1+m}+\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \polylog \left (2, e \,x^{q}\right )}{\left (1+m \right )^{2}}-\frac {q^{2} x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (x \right ) \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}-\frac {q \,x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}+\frac {2 q^{2} x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{3}}+\frac {q \,x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \Phi \left (e \,x^{q}, 2, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q}\right ) x\)	\(867\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*a/q*(-q^2*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^(1+m)*(-e)^(m/q+1/q)/
(1+m)*polylog(2,e*x^q)-q^2*x^(1+m+q)*e*(-e)^(m/q+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(-m)*(-e)
^(-m/q-1/q)*b*ln(c)/q*(-q^2*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^2*ln(1-e*x^q)-q*x^(1+m)*(-e)^(m/q+1/q)/(1+m)*polylog(
2,e*x^q)-q^2*x^(1+m+q)*e*(-e)^(m/q+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/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)/(1+m)^2*ln(1-e*x^q)-q*x^m*(-e)^(m/q+1/q)/(1+m)*polylog(2,e*x^q)-q^2*x^(
q+m)*e*(-e)^(m/q+1/q)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/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)/(1+m)^2*ln(1-e*x^q)-q*x^m*(-e)^(m/q+1/q)*ln(-e)/(1+m)^2*ln(1-e*x^q)+2*q^2*x^m*(-e)^(m/q+1/q)/
(1+m)^3*ln(1-e*x^q)-q*x^m*(-e)^(m/q+1/q)*ln(x)/(1+m)*polylog(2,e*x^q)-x^m*(-e)^(m/q+1/q)*ln(-e)/(1+m)*polylog(
2,e*x^q)+q*x^m*(-e)^(m/q+1/q)/(1+m)^2*polylog(2,e*x^q)-q^2*x^(q+m)*e*(-e)^(m/q+1/q)*ln(x)/(1+m)^2*LerchPhi(e*x
^q,1,(1+m+q)/q)-q*x^(q+m)*e*(-e)^(m/q+1/q)*ln(-e)/(1+m)^2*LerchPhi(e*x^q,1,(1+m+q)/q)+2*q^2*x^(q+m)*e*(-e)^(m/
q+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q*x^(q+m)*e*(-e)^(m/q+1/q)/(1+m)^2*LerchPhi(e*x^q,2,(1+m+q)/q)))*x

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Maxima [A]
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]

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

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Fricas [A]
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]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [A]
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]

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(x^q*e), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

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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