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

#### 3.3.1 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 = 0.348144 (sec), size = 304 ,normalized size = 9.81 $\frac {x (g x)^q \left (-b k m n \, _3F_2\left (1,\frac {q}{m}+\frac {1}{m},\frac {q}{m}+\frac {1}{m};\frac {q}{m}+\frac {1}{m}+1,\frac {q}{m}+\frac {1}{m}+1;-\frac {f x^m}{e}\right )+k m \, _2F_1\left (1,\frac {q+1}{m};\frac {m+q+1}{m};-\frac {f x^m}{e}\right ) \left (a q+a+b (q+1) \log \left (c x^n\right )-b n\right )+a q^2 \log \left (d \left (e+f x^m\right )^k\right )+2 a q \log \left (d \left (e+f x^m\right )^k\right )+a \log \left (d \left (e+f x^m\right )^k\right )-a k m q-a k m+b q^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+2 b q \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b k m q \log \left (c x^n\right )-b k m \log \left (c x^n\right )-b n q \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )+2 b k m n\right )}{(q+1)^3}$

[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 = 0.188693 (sec), size = 292 ,normalized size = 10.07 $-\frac {x^3 \left (b e k m (m+3) n \, _3F_2\left (1,\frac {3}{m},\frac {3}{m};1+\frac {3}{m},1+\frac {3}{m};-\frac {f x^m}{e}\right )-27 a e \log \left (d \left (e+f x^m\right )^k\right )-9 a e m \log \left (d \left (e+f x^m\right )^k\right )+9 a f k m x^m \, _2F_1\left (1,\frac {m+3}{m};2+\frac {3}{m};-\frac {f x^m}{e}\right )-27 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-9 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k m (m+3) \left (n-3 \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac {3}{m};\frac {m+3}{m};-\frac {f x^m}{e}\right )+3 b e k m^2 \log \left (c x^n\right )+9 b e k m \log \left (c x^n\right )+9 b e n \log \left (d \left (e+f x^m\right )^k\right )+3 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n-6 b e k m n\right )}{27 e (m+3)}$

[In]

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

[Out]

-1/27*(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)*Hyp
ergeometric2F1[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*L
og[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]))/(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 = 0.176377 (sec), size = 292 ,normalized size = 10.81 $-\frac {x^2 \left (b e k m (m+2) n \, _3F_2\left (1,\frac {2}{m},\frac {2}{m};1+\frac {2}{m},1+\frac {2}{m};-\frac {f x^m}{e}\right )-8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 a f k m x^m \, _2F_1\left (1,\frac {m+2}{m};2+\frac {2}{m};-\frac {f x^m}{e}\right )-8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k m (m+2) \left (n-2 \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac {2}{m};\frac {m+2}{m};-\frac {f x^m}{e}\right )+2 b e k m^2 \log \left (c x^n\right )+4 b e k m \log \left (c x^n\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )+2 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n-4 b e k m n\right )}{8 e (m+2)}$

[In]

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

[Out]

-1/8*(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)*Hype
rgeometric2F1[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]))/(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 = 0.181192 (sec), size = 165 ,normalized size = 6.35 $x \left (-b k m n \, _3F_2\left (1,\frac {1}{m},\frac {1}{m};1+\frac {1}{m},1+\frac {1}{m};-\frac {f x^m}{e}\right )+k m \, _2F_1\left (1,\frac {1}{m};1+\frac {1}{m};-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )-b n\right )+a \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )-b k m n \log (x)+b k m n\right )-k m x \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b k m n x$

[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 = 0.17089 (sec), size = 282 ,normalized size = 9.72 $\frac {b e k (m-1) m n \, _3F_2\left (1,-\frac {1}{m},-\frac {1}{m};1-\frac {1}{m},1-\frac {1}{m};-\frac {f x^m}{e}\right )+a e \log \left (d \left (e+f x^m\right )^k\right )-a e m \log \left (d \left (e+f x^m\right )^k\right )+a f k m x^m \, _2F_1\left (1,\frac {m-1}{m};2-\frac {1}{m};-\frac {f x^m}{e}\right )+b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k (m-1) m \left (\log \left (c x^n\right )+n\right ) \, _2F_1\left (1,-\frac {1}{m};\frac {m-1}{m};-\frac {f x^m}{e}\right )-b e k m^2 \log \left (c x^n\right )+b e k m \log \left (c x^n\right )+b e n \log \left (d \left (e+f x^m\right )^k\right )-b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n+2 b e k m n}{e (m-1) x}$

[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 = 0.162371 (sec), size = 292 ,normalized size = 10.07 $\frac {b e k (m-2) m n \, _3F_2\left (1,-\frac {2}{m},-\frac {2}{m};1-\frac {2}{m},1-\frac {2}{m};-\frac {f x^m}{e}\right )+8 a e \log \left (d \left (e+f x^m\right )^k\right )-4 a e m \log \left (d \left (e+f x^m\right )^k\right )+4 a f k m x^m \, _2F_1\left (1,\frac {m-2}{m};2-\frac {2}{m};-\frac {f x^m}{e}\right )+8 b e \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-4 b e m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k (m-2) m \left (2 \log \left (c x^n\right )+n\right ) \, _2F_1\left (1,-\frac {2}{m};\frac {m-2}{m};-\frac {f x^m}{e}\right )-2 b e k m^2 \log \left (c x^n\right )+4 b e k m \log \left (c x^n\right )+4 b e n \log \left (d \left (e+f x^m\right )^k\right )-2 b e m n \log \left (d \left (e+f x^m\right )^k\right )-2 b e k m^2 n+4 b e k m n}{8 e (m-2) x^2}$

[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 = 0.243344 (sec), size = 266 ,normalized size = 8.87 $-\frac {x (d x)^m \left (-b n q \, _3F_2\left (1,\frac {m}{q}+\frac {1}{q},\frac {m}{q}+\frac {1}{q};\frac {m}{q}+\frac {1}{q}+1,\frac {m}{q}+\frac {1}{q}+1;e x^q\right )+q \, _2F_1\left (1,\frac {m+1}{q};\frac {m+q+1}{q};e x^q\right ) \left (a m+a+b (m+1) \log \left (c x^n\right )-b n\right )+a m^2 \log \left (1-e x^q\right )+2 a m \log \left (1-e x^q\right )+a \log \left (1-e x^q\right )-a m q-a q+b m^2 \log \left (c x^n\right ) \log \left (1-e x^q\right )+2 b m \log \left (c x^n\right ) \log \left (1-e x^q\right )+b \log \left (c x^n\right ) \log \left (1-e x^q\right )-b m q \log \left (c x^n\right )-b q \log \left (c x^n\right )-b m n \log \left (1-e x^q\right )-b n \log \left (1-e x^q\right )+2 b n q\right )}{(m+1)^3}$

[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)

#### 3.3.2 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 = 0.866 (sec), size = 844 ,normalized size = 28.13 $-\frac {\left (-\frac {\left (-m -q -1\right ) e q \,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 +q +1\right ) \left (m +1\right )}+\frac {q \,x^{m +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\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 {\left (-m -q -1\right ) e q \,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 +q +1\right ) \left (m +1\right )}+\frac {q \,x^{m +1} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\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 {\left (-m -q -1\right ) e q \,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 +q +1\right ) \left (m +1\right )}+\frac {\left (-m -q -1\right ) e q \,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 +q +1\right )^{2} \left (m +1\right )}+\frac {e q \,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 +q +1\right ) \left (m +1\right )}+\frac {\left (-m -q -1\right ) e q \,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 +q +1\right ) \left (m +1\right )^{2}}-\frac {\left (-m -q -1\right ) e \,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 +q +1\right ) \left (m +1\right )}+\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \relax (x ) \ln \left (-e \,x^{q}+1\right )}{m +1}+\frac {\left (-m -q -1\right ) e \,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 +q +1\right ) \left (m +1\right )}-\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\right )}{\left (m +1\right )^{2}}+\frac {x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \ln \left (-e \,x^{q}+1\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 {\left (-m -q -1\right ) e q \,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 +q +1\right ) \left (m +1\right )}+\frac {q \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \,x^{q}+1\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$

[In]

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

[Out]

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

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

[B]   time = 0.352 (sec), 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$

[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))

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

[B]   time = 2.282 (sec), size = 1065 ,normalized size = 4.35 $\text {result too large to display}$

[In]

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

[Out]

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