4.3 Test file Number [58] 3-Logarithms/3.1.5-u-a+b-log-c-x^n-^p

4.3.1 Mathematica

Integral number [138] \[ \int (g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.329968 (sec), size = 304 ,normalized size = 10.13 \[ \frac{x (g x)^q \left (-b k m n \, _3F_2\left (1,\frac{q}{m}+\frac{1}{m},\frac{q}{m}+\frac{1}{m};\frac{q}{m}+\frac{1}{m}+1,\frac{q}{m}+\frac{1}{m}+1;-\frac{f x^m}{e}\right )+k m \, _2F_1\left (1,\frac{q+1}{m};\frac{m+q+1}{m};-\frac{f x^m}{e}\right ) \left (a q+a+b (q+1) \log \left (c x^n\right )-b n\right )+a q^2 \log \left (d \left (e+f x^m\right )^k\right )+2 a q \log \left (d \left (e+f x^m\right )^k\right )+a \log \left (d \left (e+f x^m\right )^k\right )-a k m q-a k m+b q^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+2 b q \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b k m q \log \left (c x^n\right )-b k m \log \left (c x^n\right )-b n q \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )+2 b k m n\right )}{(q+1)^3} \]

[In]

Integrate[(g*x)^q*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

(x*(g*x)^q*(-(a*k*m) + 2*b*k*m*n - a*k*m*q - b*k*m*n*HypergeometricPFQ[{1, m^(-1) + q/m, m^(-1) + q/m}, {1 + m
^(-1) + q/m, 1 + m^(-1) + q/m}, -((f*x^m)/e)] - b*k*m*Log[c*x^n] - b*k*m*q*Log[c*x^n] + k*m*Hypergeometric2F1[
1, (1 + q)/m, (1 + m + q)/m, -((f*x^m)/e)]*(a - b*n + a*q + b*(1 + q)*Log[c*x^n]) + a*Log[d*(e + f*x^m)^k] - b
*n*Log[d*(e + f*x^m)^k] + 2*a*q*Log[d*(e + f*x^m)^k] - b*n*q*Log[d*(e + f*x^m)^k] + a*q^2*Log[d*(e + f*x^m)^k]
+ b*Log[c*x^n]*Log[d*(e + f*x^m)^k] + 2*b*q*Log[c*x^n]*Log[d*(e + f*x^m)^k] + b*q^2*Log[c*x^n]*Log[d*(e + f*x
^m)^k]))/(1 + q)^3

Integral number [144] \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.178429 (sec), size = 292 ,normalized size = 10.43 \[ -\frac{x^3 \left (b e k m (m+3) n \, _3F_2\left (1,\frac{3}{m},\frac{3}{m};1+\frac{3}{m},1+\frac{3}{m};-\frac{f x^m}{e}\right )-27 a e \log \left (d \left (e+f x^m\right )^k\right )-9 a e m \log \left (d \left (e+f x^m\right )^k\right )+9 a f k m x^m \, _2F_1\left (1,\frac{m+3}{m};2+\frac{3}{m};-\frac{f x^m}{e}\right )-27 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-9 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k m (m+3) \left (n-3 \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac{3}{m};\frac{m+3}{m};-\frac{f x^m}{e}\right )+3 b e k m^2 \log \left (c x^n\right )+9 b e k m \log \left (c x^n\right )+9 b e n \log \left (d \left (e+f x^m\right )^k\right )+3 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n-6 b e k m n\right )}{27 e (m+3)} \]

[In]

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

[Out]

-(x^3*(-6*b*e*k*m*n - 2*b*e*k*m^2*n + 9*a*f*k*m*x^m*Hypergeometric2F1[1, (3 + m)/m, 2 + 3/m, -((f*x^m)/e)] + b
*e*k*m*(3 + m)*n*HypergeometricPFQ[{1, 3/m, 3/m}, {1 + 3/m, 1 + 3/m}, -((f*x^m)/e)] + b*e*k*m*(3 + m)*Hypergeo
metric2F1[1, 3/m, (3 + m)/m, -((f*x^m)/e)]*(n - 3*Log[c*x^n]) + 9*b*e*k*m*Log[c*x^n] + 3*b*e*k*m^2*Log[c*x^n]
- 27*a*e*Log[d*(e + f*x^m)^k] - 9*a*e*m*Log[d*(e + f*x^m)^k] + 9*b*e*n*Log[d*(e + f*x^m)^k] + 3*b*e*m*n*Log[d*
(e + f*x^m)^k] - 27*b*e*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 9*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k]))/(27*e*(3 +
m))

Integral number [145] \[ \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.170561 (sec), size = 292 ,normalized size = 11.23 \[ -\frac{x^2 \left (b e k m (m+2) n \, _3F_2\left (1,\frac{2}{m},\frac{2}{m};1+\frac{2}{m},1+\frac{2}{m};-\frac{f x^m}{e}\right )-8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 a f k m x^m \, _2F_1\left (1,\frac{m+2}{m};2+\frac{2}{m};-\frac{f x^m}{e}\right )-8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k m (m+2) \left (n-2 \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac{2}{m};\frac{m+2}{m};-\frac{f x^m}{e}\right )+2 b e k m^2 \log \left (c x^n\right )+4 b e k m \log \left (c x^n\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )+2 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n-4 b e k m n\right )}{8 e (m+2)} \]

[In]

Integrate[x*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

-(x^2*(-4*b*e*k*m*n - 2*b*e*k*m^2*n + 4*a*f*k*m*x^m*Hypergeometric2F1[1, (2 + m)/m, 2 + 2/m, -((f*x^m)/e)] + b
*e*k*m*(2 + m)*n*HypergeometricPFQ[{1, 2/m, 2/m}, {1 + 2/m, 1 + 2/m}, -((f*x^m)/e)] + b*e*k*m*(2 + m)*Hypergeo
metric2F1[1, 2/m, (2 + m)/m, -((f*x^m)/e)]*(n - 2*Log[c*x^n]) + 4*b*e*k*m*Log[c*x^n] + 2*b*e*k*m^2*Log[c*x^n]
- 8*a*e*Log[d*(e + f*x^m)^k] - 4*a*e*m*Log[d*(e + f*x^m)^k] + 4*b*e*n*Log[d*(e + f*x^m)^k] + 2*b*e*m*n*Log[d*(
e + f*x^m)^k] - 8*b*e*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 4*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k]))/(8*e*(2 + m)
)

Integral number [146] \[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

[B]   time = 0.169122 (sec), size = 165 ,normalized size = 6.6 \[ x \left (-b k m n \, _3F_2\left (1,\frac{1}{m},\frac{1}{m};1+\frac{1}{m},1+\frac{1}{m};-\frac{f x^m}{e}\right )+k m \, _2F_1\left (1,\frac{1}{m};1+\frac{1}{m};-\frac{f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )-b n\right )+a \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )-b k m n \log (x)+b k m n\right )-k m x \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b k m n x \]

[In]

Integrate[(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

b*k*m*n*x - k*m*x*(a + b*(-(n*Log[x]) + Log[c*x^n])) + x*(b*k*m*n - b*k*m*n*HypergeometricPFQ[{1, m^(-1), m^(-
1)}, {1 + m^(-1), 1 + m^(-1)}, -((f*x^m)/e)] - b*k*m*n*Log[x] + k*m*Hypergeometric2F1[1, m^(-1), 1 + m^(-1), -
((f*x^m)/e)]*(a - b*n + b*Log[c*x^n]) + a*Log[d*(e + f*x^m)^k] - b*n*Log[d*(e + f*x^m)^k] + b*Log[c*x^n]*Log[d
*(e + f*x^m)^k])

Integral number [148] \[ \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^2} \, dx \]

[B]   time = 0.164843 (sec), size = 282 ,normalized size = 10.07 \[ \frac{b e k (m-1) m n \, _3F_2\left (1,-\frac{1}{m},-\frac{1}{m};1-\frac{1}{m},1-\frac{1}{m};-\frac{f x^m}{e}\right )+a e \log \left (d \left (e+f x^m\right )^k\right )-a e m \log \left (d \left (e+f x^m\right )^k\right )+a f k m x^m \, _2F_1\left (1,\frac{m-1}{m};2-\frac{1}{m};-\frac{f x^m}{e}\right )+b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k (m-1) m \left (\log \left (c x^n\right )+n\right ) \, _2F_1\left (1,-\frac{1}{m};\frac{m-1}{m};-\frac{f x^m}{e}\right )-b e k m^2 \log \left (c x^n\right )+b e k m \log \left (c x^n\right )+b e n \log \left (d \left (e+f x^m\right )^k\right )-b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n+2 b e k m n}{e (m-1) x} \]

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^2,x]

[Out]

(2*b*e*k*m*n - 2*b*e*k*m^2*n + a*f*k*m*x^m*Hypergeometric2F1[1, (-1 + m)/m, 2 - m^(-1), -((f*x^m)/e)] + b*e*k*
(-1 + m)*m*n*HypergeometricPFQ[{1, -m^(-1), -m^(-1)}, {1 - m^(-1), 1 - m^(-1)}, -((f*x^m)/e)] + b*e*k*m*Log[c*
x^n] - b*e*k*m^2*Log[c*x^n] + b*e*k*(-1 + m)*m*Hypergeometric2F1[1, -m^(-1), (-1 + m)/m, -((f*x^m)/e)]*(n + Lo
g[c*x^n]) + a*e*Log[d*(e + f*x^m)^k] - a*e*m*Log[d*(e + f*x^m)^k] + b*e*n*Log[d*(e + f*x^m)^k] - b*e*m*n*Log[d
*(e + f*x^m)^k] + b*e*Log[c*x^n]*Log[d*(e + f*x^m)^k] - b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(e*(-1 + m)*x)

Integral number [149] \[ \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{x^3} \, dx \]

[B]   time = 0.152721 (sec), size = 292 ,normalized size = 10.43 \[ \frac{b e k (m-2) m n \, _3F_2\left (1,-\frac{2}{m},-\frac{2}{m};1-\frac{2}{m},1-\frac{2}{m};-\frac{f x^m}{e}\right )+8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 a f k m x^m \, _2F_1\left (1,\frac{m-2}{m};2-\frac{2}{m};-\frac{f x^m}{e}\right )+8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k (m-2) m \left (2 \log \left (c x^n\right )+n\right ) \, _2F_1\left (1,-\frac{2}{m};\frac{m-2}{m};-\frac{f x^m}{e}\right )-2 b e k m^2 \log \left (c x^n\right )+4 b e k m \log \left (c x^n\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )-2 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n+4 b e k m n}{8 e (m-2) x^2} \]

[In]

Integrate[((a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k])/x^3,x]

[Out]

(4*b*e*k*m*n - 2*b*e*k*m^2*n + 4*a*f*k*m*x^m*Hypergeometric2F1[1, (-2 + m)/m, 2 - 2/m, -((f*x^m)/e)] + b*e*k*(
-2 + m)*m*n*HypergeometricPFQ[{1, -2/m, -2/m}, {1 - 2/m, 1 - 2/m}, -((f*x^m)/e)] + 4*b*e*k*m*Log[c*x^n] - 2*b*
e*k*m^2*Log[c*x^n] + b*e*k*(-2 + m)*m*Hypergeometric2F1[1, -2/m, (-2 + m)/m, -((f*x^m)/e)]*(n + 2*Log[c*x^n])
+ 8*a*e*Log[d*(e + f*x^m)^k] - 4*a*e*m*Log[d*(e + f*x^m)^k] + 4*b*e*n*Log[d*(e + f*x^m)^k] - 2*b*e*m*n*Log[d*(
e + f*x^m)^k] + 8*b*e*Log[c*x^n]*Log[d*(e + f*x^m)^k] - 4*b*e*m*Log[c*x^n]*Log[d*(e + f*x^m)^k])/(8*e*(-2 + m)
*x^2)

Integral number [220] \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx \]

[B]   time = 0.232955 (sec), size = 266 ,normalized size = 9.17 \[ -\frac{x (d x)^m \left (-b n q \, _3F_2\left (1,\frac{m}{q}+\frac{1}{q},\frac{m}{q}+\frac{1}{q};\frac{m}{q}+\frac{1}{q}+1,\frac{m}{q}+\frac{1}{q}+1;e x^q\right )+q \, _2F_1\left (1,\frac{m+1}{q};\frac{m+q+1}{q};e x^q\right ) \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )+a m^2 \log \left (1-e x^q\right )+2 a m \log \left (1-e x^q\right )+a \log \left (1-e x^q\right )-a m q-a q+b m^2 \log \left (c x^n\right ) \log \left (1-e x^q\right )+2 b m \log \left (c x^n\right ) \log \left (1-e x^q\right )+b \log \left (c x^n\right ) \log \left (1-e x^q\right )-b m q \log \left (c x^n\right )-b q \log \left (c x^n\right )-b m n \log \left (1-e x^q\right )-b n \log \left (1-e x^q\right )+2 b n q\right )}{(m+1)^3} \]

[In]

Integrate[-((d*x)^m*(a + b*Log[c*x^n])*Log[1 - e*x^q]),x]

[Out]

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

4.3.2 Maple

Integral number [220] \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx \]

[B]   time = 0.468 (sec), size = 844 ,normalized size = 29.1 \[ -{\frac{ \left ( dx \right ) ^{m}{x}^{-m}a}{q} \left ( -e \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{q{x}^{1+m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{1+m+q}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) } \right ) }-{\frac{ \left ( dx \right ) ^{m}{x}^{-m}b\ln \left ( c \right ) }{q} \left ( -e \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{q{x}^{1+m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{1+m+q}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) } \right ) }+ \left ({\frac{\ln \left ( -e \right ) \left ( dx \right ) ^{m}{x}^{-m}bn}{{q}^{2}} \left ( -e \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{q{x}^{m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{q+m}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) } \right ) }-{\frac{ \left ( dx \right ) ^{m}{x}^{-m}bn}{q} \left ( -e \right ) ^{-{\frac{m}{q}}-{q}^{-1}} \left ({\frac{q\ln \left ( x \right ){x}^{m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}+{\frac{\ln \left ( -e \right ){x}^{m}\ln \left ( 1-e{x}^{q} \right ) }{1+m} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}-{\frac{q{x}^{m}\ln \left ( 1-e{x}^{q} \right ) }{ \left ( 1+m \right ) ^{2}} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}}+{\frac{q{x}^{q+m}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) ^{2}} \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }-{\frac{q{x}^{q+m}e\ln \left ( x \right ) \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }-{\frac{{x}^{q+m}e\ln \left ( -e \right ) \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }+{\frac{q{x}^{q+m}e}{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }+{\frac{q{x}^{q+m}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) ^{2} \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},1,{\frac{1+m+q}{q}} \right ) }+{\frac{{x}^{q+m}e \left ( -q-m-1 \right ) }{ \left ( 1+m \right ) \left ( 1+m+q \right ) } \left ( -e \right ) ^{{\frac{m}{q}}+{q}^{-1}}{\it LerchPhi} \left ( e{x}^{q},2,{\frac{1+m+q}{q}} \right ) } \right ) } \right ) x \]

[In]

int(-(d*x)^m*(a+b*ln(c*x^n))*ln(1-e*x^q),x)

[Out]

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

Integral number [221] \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,e x^q\right ) \, dx \]

[B]   time = 0.267 (sec), size = 867 ,normalized size = 4.9 \[ \text{result too large to display} \]

[In]

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

[Out]

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

Integral number [222] \[ \int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,e x^q\right ) \, dx \]

[B]   time = 1.089 (sec), size = 1065 ,normalized size = 4.36 \[ \text{result too large to display} \]

[In]

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

[Out]

-(d*x)^m*x^(-m)*(-e)^(-1/q*m-1/q)*a/q*(q^3*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^(1+m)*(-e)^(1/q*
m+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(1+m+q)*e*(-e)^(1/q*m+
1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q))-(d*x)^m*x^(-m)*(-e)^(-1/q*m-1/q)*b*ln(c)/q*(q^3*x^(1+m)*(-e)^(1/q*m+
1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)-q*x^(1+m)*(-e)^(1/q*m+1/q)/(1+m
)*polylog(3,e*x^q)+q^3*x^(1+m+q)*e*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q))+(1/q^2*ln(-e)*(-e)^(-
1/q*m-1/q)*(d*x)^m*x^(-m)*b*n*(q^3*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)+q^2*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*p
olylog(2,e*x^q)-q*x^m*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q^3*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(
e*x^q,1,(1+m+q)/q))-(-e)^(-1/q*m-1/q)*(d*x)^m*x^(-m)*b*n/q*(q^3*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)
+q^2*ln(-e)*x^m*(-e)^(1/q*m+1/q)/(1+m)^3*ln(1-e*x^q)-3*q^3*x^m*(-e)^(1/q*m+1/q)/(1+m)^4*ln(1-e*x^q)+q^2*ln(x)*
x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)+q*ln(-e)*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(2,e*x^q)-2*q^2*x^m
*(-e)^(1/q*m+1/q)/(1+m)^3*polylog(2,e*x^q)-q*ln(x)*x^m*(-e)^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)-ln(-e)*x^m*(-e)
^(1/q*m+1/q)/(1+m)*polylog(3,e*x^q)+q*x^m*(-e)^(1/q*m+1/q)/(1+m)^2*polylog(3,e*x^q)+q^3*x^(q+m)*e*ln(x)*(-e)^(
1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,(1+m+q)/q)+q^2*x^(q+m)*e*ln(-e)*(-e)^(1/q*m+1/q)/(1+m)^3*LerchPhi(e*x^q,1,
(1+m+q)/q)-3*q^3*x^(q+m)*e*(-e)^(1/q*m+1/q)/(1+m)^4*LerchPhi(e*x^q,1,(1+m+q)/q)-q^2*x^(q+m)*e*(-e)^(1/q*m+1/q)
/(1+m)^3*LerchPhi(e*x^q,2,(1+m+q)/q)))*x