\(\int (d x)^m (a+b \log (c x^n)) \operatorname {PolyLog}(2,e x^q) \, dx\) [227]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx=-\frac {b e n q^2 x^{1+q} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+q}{q},\frac {1+m+2 q}{q},e x^q\right )}{(1+m)^3 (1+m+q)}-\frac {b n q (d x)^{1+m} \log \left (1-e x^q\right )}{d (1+m)^3}-\frac {b n (d x)^{1+m} \operatorname {PolyLog}\left (2,e x^q\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right )}{d (1+m)}+\frac {q \text {Int}\left ((d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ),x\right )}{1+m} \] Output:

-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)*pol 
ylog(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*Defer(Int)((d*x)^m*(a+b*ln(c*x^n))*ln(1-e*x^q),x)/(1+m)
 

Mathematica [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx=\int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {2832, 25, 2826, 2905, 30, 888}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d x)^m \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2832

\(\displaystyle -\frac {q \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {b n q \int -(d x)^m \log \left (1-e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}-\frac {b n q \int (d x)^m \log \left (1-e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\)

\(\Big \downarrow \) 2826

\(\displaystyle \frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}-\frac {b n q \int (d x)^m \log \left (1-e x^q\right )dx}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\)

\(\Big \downarrow \) 2905

\(\displaystyle \frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}-\frac {b n q \left (\frac {e q \int \frac {x^{q-1} (d x)^{m+1}}{1-e x^q}dx}{d (m+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\)

\(\Big \downarrow \) 30

\(\displaystyle \frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}-\frac {b n q \left (\frac {e q x^{-m} (d x)^m \int \frac {x^{m+q}}{1-e x^q}dx}{m+1}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {q \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx}{m+1}+\frac {(d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right ) \left (a+b \log \left (c x^n\right )\right )}{d (m+1)}-\frac {b n q \left (\frac {e q x^{q+1} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {m+q+1}{q},\frac {m+2 q+1}{q},e x^q\right )}{(m+1) (m+q+1)}+\frac {(d x)^{m+1} \log \left (1-e x^q\right )}{d (m+1)}\right )}{(m+1)^2}-\frac {b n (d x)^{m+1} \operatorname {PolyLog}\left (2,e x^q\right )}{d (m+1)^2}\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 2826

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> Unintegrable[(g*x)^q*(a + 
b*Log[c*x^n])^p*Log[d*(e + f*x^m)^r], x] /; FreeQ[{a, b, c, d, e, f, g, r, 
m, n, p, q}, x]
 

rule 2832

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.)*PolyLog[k_, (e 
_.)*(x_)^(q_.)], x_Symbol] :> Simp[(-b)*n*(d*x)^(m + 1)*(PolyLog[k, e*x^q]/ 
(d*(m + 1)^2)), x] + (Simp[(d*x)^(m + 1)*PolyLog[k, e*x^q]*((a + b*Log[c*x^ 
n])/(d*(m + 1))), x] - Simp[q/(m + 1)   Int[(d*x)^m*PolyLog[k - 1, e*x^q]*( 
a + b*Log[c*x^n]), x], x] + Simp[b*n*(q/(m + 1)^2)   Int[(d*x)^m*PolyLog[k 
- 1, e*x^q], x], x]) /; FreeQ[{a, b, c, d, e, m, n, q}, x] && IGtQ[k, 0]
 

rule 2905

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^ 
(m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m 
+ 1))), x] - Simp[b*e*n*(p/(f*(m + 1)))   Int[x^(n - 1)*((f*x)^(m + 1)/(d + 
 e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(866\) vs. \(2(179)=358\).

Time = 0.07 (sec) , antiderivative size = 867, normalized size of antiderivative = 37.70

\[-\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}} \operatorname {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}} \operatorname {LerchPhi}\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}} \operatorname {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}} \operatorname {LerchPhi}\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}} \operatorname {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}} \operatorname {LerchPhi}\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 ) \operatorname {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 ) \operatorname {polylog}\left (2, e \,x^{q}\right )}{1+m}+\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \operatorname {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 ) \operatorname {LerchPhi}\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 ) \operatorname {LerchPhi}\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}} \operatorname {LerchPhi}\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}} \operatorname {LerchPhi}\left (e \,x^{q}, 2, \frac {1+m +q}{q}\right )}{\left (1+m \right )^{2}}\right )}{q}\right ) x\]

Input:

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

Output:

-(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*l 
n(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*LerchP 
hi(e*x^q,2,(1+m+q)/q)))*x
 

Fricas [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} {\rm Li}_2\left (e x^{q}\right ) \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 302, normalized size of antiderivative = 13.13 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} {\rm Li}_2\left (e x^{q}\right ) \,d x } \] Input:

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

Output:

(((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)
 

Giac [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} {\rm Li}_2\left (e x^{q}\right ) \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 25.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx=\int {\left (d\,x\right )}^m\,\mathrm {polylog}\left (2,e\,x^q\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,e x^q\right ) \, dx=d^{m} \left (\left (\int x^{m} \mathrm {log}\left (x^{n} c \right ) \mathit {polylog}\left (2, x^{q} e \right )d x \right ) b +\left (\int x^{m} \mathit {polylog}\left (2, x^{q} e \right )d x \right ) a \right ) \] Input:

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

Output:

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