3.3.22 \(\int (d x)^m (a+b \log (c x^n)) \text {Li}_3(e x^q) \, dx\) [222]

Optimal. Leaf size=245 \[ \frac {2 b e n q^3 x^{1+q} (d x)^m \, _2F_1\left (1,\frac {1+m+q}{q};\frac {1+m+2 q}{q};e x^q\right )}{(1+m)^4 (1+m+q)}+\frac {2 b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}+\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right )}{d (1+m)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\left (e x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \text {Int}\left ((d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ),x\right )}{(1+m)^2} \]

[Out]

2*b*e*n*q^3*x^(1+q)*(d*x)^m*hypergeom([1, (1+m+q)/q],[(1+m+2*q)/q],e*x^q)/(1+m)^4/(1+m+q)+2*b*n*q^2*(d*x)^(1+m
)*ln(1-e*x^q)/d/(1+m)^4+2*b*n*q*(d*x)^(1+m)*polylog(2,e*x^q)/d/(1+m)^3-q*(d*x)^(1+m)*(a+b*ln(c*x^n))*polylog(2
,e*x^q)/d/(1+m)^2-b*n*(d*x)^(1+m)*polylog(3,e*x^q)/d/(1+m)^2+(d*x)^(1+m)*(a+b*ln(c*x^n))*polylog(3,e*x^q)/d/(1
+m)-q^2*Unintegrable((d*x)^m*(a+b*ln(c*x^n))*ln(1-e*x^q),x)/(1+m)^2

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Rubi [A]
time = 0.15, 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 (3,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[3, e*x^q],x]

[Out]

(2*b*e*n*q^3*x^(1 + q)*(d*x)^m*Hypergeometric2F1[1, (1 + m + q)/q, (1 + m + 2*q)/q, e*x^q])/((1 + m)^4*(1 + m
+ q)) + (2*b*n*q^2*(d*x)^(1 + m)*Log[1 - e*x^q])/(d*(1 + m)^4) + (2*b*n*q*(d*x)^(1 + m)*PolyLog[2, e*x^q])/(d*
(1 + m)^3) - (q*(d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[2, e*x^q])/(d*(1 + m)^2) - (b*n*(d*x)^(1 + m)*PolyLog
[3, e*x^q])/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])*PolyLog[3, e*x^q])/(d*(1 + m)) - (q^2*Defer[Int]
[(d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q], x])/(1 + m)^2

Rubi steps

\begin {align*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (e x^q\right ) \, dx &=-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (e x^q\right ) \, dx}{1+m}+\frac {(b n q) \int (d x)^m \text {Li}_2\left (e x^q\right ) \, dx}{(1+m)^2}\\ &=\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (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)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \frac {\left (b n q^2\right ) \int (d x)^m \log \left (1-e x^q\right ) \, dx}{(1+m)^3}\\ &=\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (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)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \left (\frac {b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}+\frac {\left (b e n q^3\right ) \int \frac {x^{-1+q} (d x)^{1+m}}{1-e x^q} \, dx}{d (1+m)^4}\right )\\ &=\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (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)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}+2 \left (\frac {b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}+\frac {\left (b e n q^3 x^{-m} (d x)^m\right ) \int \frac {x^{m+q}}{1-e x^q} \, dx}{(1+m)^4}\right )\\ &=2 \left (\frac {b e n q^3 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)^4 (1+m+q)}+\frac {b n q^2 (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^4}\right )+\frac {2 b n q (d x)^{1+m} \text {Li}_2\left (e x^q\right )}{d (1+m)^3}-\frac {q (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)^2}-\frac {b n (d x)^{1+m} \text {Li}_3\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}_3\left (e x^q\right )}{d (1+m)}-\frac {q^2 \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx}{(1+m)^2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\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[3, e*x^q],x]

[Out]

Integrate[(d*x)^m*(a + b*Log[c*x^n])*PolyLog[3, e*x^q], x]

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Maple [A] Leaf count of result is larger than twice the leaf count of optimal. \(1064\) vs. \(2(246)=492\).
time = 0.86, size = 1065, normalized size = 4.35

method	result	size
meijerg	\(\text {Expression too large to display}\)	\(1065\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*ln(c*x^n))*polylog(3,e*x^q),x,method=_RETURNVERBOSE)

[Out]

-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*a/q*(q^3*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^(1+m)*(-e)^(m/q+1/q)
/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(m/q+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(1+m+q)*e*(-e)^(m/q+1/q)/(1+m)
^3*LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(-m)*(-e)^(-m/q-1/q)*b*ln(c)/q*(q^3*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^3*l
n(1-e*x^q)+q^2*x^(1+m)*(-e)^(m/q+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(m/q+1/q)/(1+m)*polylog(3,e*x^q)
+q^3*x^(1+m+q)*e*(-e)^(m/q+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q))+((-e)^(-m/q-1/q)/q^2*ln(-e)*(d*x)^m*x^(-m
)*b*n*(q^3*x^m*(-e)^(m/q+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^m*(-e)^(m/q+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^m*(-e)^(
m/q+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(q+m)*e*(-e)^(m/q+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q))-(-e)^(-m/q-1
/q)*(d*x)^m*x^(-m)*b*n/q*(q^3*x^m*(-e)^(m/q+1/q)*ln(x)/(1+m)^3*ln(1-e*x^q)+q^2*x^m*(-e)^(m/q+1/q)*ln(-e)/(1+m)
^3*ln(1-e*x^q)-3*q^3*x^m*(-e)^(m/q+1/q)/(1+m)^4*ln(1-e*x^q)+q^2*x^m*(-e)^(m/q+1/q)*ln(x)/(1+m)^2*polylog(2,e*x
^q)+q*x^m*(-e)^(m/q+1/q)*ln(-e)/(1+m)^2*polylog(2,e*x^q)-2*q^2*x^m*(-e)^(m/q+1/q)/(1+m)^3*polylog(2,e*x^q)-q*x
^m*(-e)^(m/q+1/q)*ln(x)/(1+m)*polylog(3,e*x^q)-x^m*(-e)^(m/q+1/q)*ln(-e)/(1+m)*polylog(3,e*x^q)+q*x^m*(-e)^(m/
q+1/q)/(1+m)^2*polylog(3,e*x^q)+q^3*x^(q+m)*e*(-e)^(m/q+1/q)*ln(x)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q^2*x^(
q+m)*e*(-e)^(m/q+1/q)*ln(-e)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)-3*q^3*x^(q+m)*e*(-e)^(m/q+1/q)/(1+m)^4*LerchP
hi(e*x^q,1,(1+m+q)/q)-q^2*x^(q+m)*e*(-e)^(m/q+1/q)/(1+m)^3*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(3,e*x^q),x, algorithm="maxima")

[Out]

-(((m^2*q + 2*m*q + q)*b*d^m*x*x^m*log(x^n) + ((m^2*q + 2*m*q + q)*a*d^m + ((m^2*q + 2*m*q + q)*d^m*log(c) - 2
*(m*n*q + n*q)*d^m)*b)*x*x^m)*dilog(e^(q*log(x) + 1)) + ((m*q^2 + q^2)*b*d^m*x*x^m*log(x^n) + ((m*q^2 + q^2)*a
*d^m - (3*d^m*n*q^2 - (m*q^2 + q^2)*d^m*log(c))*b)*x*x^m)*log(-e^(q*log(x) + 1) + 1) - ((m^3 + 3*m^2 + 3*m + 1
)*b*d^m*x*x^m*log(x^n) + ((m^3 + 3*m^2 + 3*m + 1)*a*d^m + ((m^3 + 3*m^2 + 3*m + 1)*d^m*log(c) - (m^2*n + 2*m*n
+ n)*d^m)*b)*x*x^m)*polylog(3, e^(q*log(x) + 1)))/(m^4 + 4*m^3 + 6*m^2 + 4*m + 1) + integrate(-((m*q^3 + q^3)
*b*d^m*e^(m*log(x) + q*log(x) + 1)*log(x^n) + ((m*q^3 + q^3)*a*d^m - (3*d^m*n*q^3 - (m*q^3 + q^3)*d^m*log(c))*
b)*e^(m*log(x) + q*log(x) + 1))/(m^4 + 4*m^3 + 6*m^2 - (m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*e^(q*log(x) + 1) + 4*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(3,e*x^q),x, algorithm="fricas")

[Out]

integral(((d*x)^m*b*log(c*x^n) + (d*x)^m*a)*polylog(3, x^q*e), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x^{q}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*ln(c*x**n))*polylog(3,e*x**q),x)

[Out]

Integral((d*x)**m*(a + b*log(c*x**n))*polylog(3, e*x**q), x)

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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(3,e*x^q),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*(d*x)^m*polylog(3, x^q*e), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d\,x\right )}^m\,\mathrm {polylog}\left (3,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(3, e*x^q)*(a + b*log(c*x^n)),x)

[Out]

int((d*x)^m*polylog(3, e*x^q)*(a + b*log(c*x^n)), x)

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