3.3 Test file Number [58]

3.3.1 Mathematica

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

[B]   time = 0.205894 (sec), size = 304 ,normalized size = 9.81 $$\frac{x(gx{)}^q(-akm+2bkmn-akmq-bkmn{}_3{F}_2(1,\frac{1}{m}+\frac{q}{m},\frac{1}{m}+\frac{q}{m};1+\frac{1}{m}+\frac{q}{m},1+\frac{1}{m}+\frac{q}{m};-\frac{f{x}^m}{e})-bkm\log \left(c{x}^n\right)-bkmq\log \left(c{x}^n\right)+km{}_2{F}_1(1,\frac{1+q}{m};\frac{1+m+q}{m};-\frac{f{x}^m}{e})(a-bn+aq+b(1+q)\log \left(c{x}^n\right))+a\log \left(d{(e+f{x}^m)}^k\right)-bn\log \left(d{(e+f{x}^m)}^k\right)+2aq\log \left(d{(e+f{x}^m)}^k\right)-bnq\log \left(d{(e+f{x}^m)}^k\right)+a{q}^2\log \left(d{(e+f{x}^m)}^k\right)+b\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)+2bq\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)+b{q}^2\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right))}{(1+q{)}^3}$$

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

[B]   time = 0.133774 (sec), size = 292 ,normalized size = 10.07 $$-\frac{x^3(-6bekmn-2bek{m}^{2}n+9afkm{x}^m{}_2{F}_1(1,\frac{3+m}{m};2+\frac{3}{m};-\frac{f{x}^m}{e})+bekm(3+m)n{}_3{F}_2(1,\frac{3}{m},\frac{3}{m};1+\frac{3}{m},1+\frac{3}{m};-\frac{f{x}^m}{e})+bekm(3+m){}_2{F}_1(1,\frac{3}{m};\frac{3+m}{m};-\frac{f{x}^m}{e})(n-3\log \left(c{x}^n\right))+9bekm\log \left(c{x}^n\right)+3bek{m}^2\log \left(c{x}^n\right)-27ae\log \left(d{(e+f{x}^m)}^k\right)-9aem\log \left(d{(e+f{x}^m)}^k\right)+9ben\log \left(d{(e+f{x}^m)}^k\right)+3bemn\log \left(d{(e+f{x}^m)}^k\right)-27be\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)-9bem\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right))}{27e(3+m)}$$

[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] $$\int x(a+b\log \left(c{x}^n\right))\log \left(d{(e+f{x}^m)}^k\right)dx$$

[B]   time = 0.125026 (sec), size = 292 ,normalized size = 10.81 $$-\frac{x^2(-4bekmn-2bek{m}^{2}n+4afkm{x}^m{}_2{F}_1(1,\frac{2+m}{m};2+\frac{2}{m};-\frac{f{x}^m}{e})+bekm(2+m)n{}_3{F}_2(1,\frac{2}{m},\frac{2}{m};1+\frac{2}{m},1+\frac{2}{m};-\frac{f{x}^m}{e})+bekm(2+m){}_2{F}_1(1,\frac{2}{m};\frac{2+m}{m};-\frac{f{x}^m}{e})(n-2\log \left(c{x}^n\right))+4bekm\log \left(c{x}^n\right)+2bek{m}^2\log \left(c{x}^n\right)-8ae\log \left(d{(e+f{x}^m)}^k\right)-4aem\log \left(d{(e+f{x}^m)}^k\right)+4ben\log \left(d{(e+f{x}^m)}^k\right)+2bemn\log \left(d{(e+f{x}^m)}^k\right)-8be\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)-4bem\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right))}{8e(2+m)}$$

[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] $$\int (a+b\log \left(c{x}^n\right))\log \left(d{(e+f{x}^m)}^k\right)dx$$

[B]   time = 0.118708 (sec), size = 165 ,normalized size = 6.35 $$bkmnx-kmx(a+b(-n\log (x)+\log \left(c{x}^n\right)))+x(bkmn-bkmn{}_3{F}_2(1,\frac{1}{m},\frac{1}{m};1+\frac{1}{m},1+\frac{1}{m};-\frac{f{x}^m}{e})-bkmn\log (x)+km{}_2{F}_1(1,\frac{1}{m};1+\frac{1}{m};-\frac{f{x}^m}{e})(a-bn+b\log \left(c{x}^n\right))+a\log \left(d{(e+f{x}^m)}^k\right)-bn\log \left(d{(e+f{x}^m)}^k\right)+b\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right))$$

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

[B]   time = 0.117701 (sec), size = 282 ,normalized size = 9.72 $$\frac{2bekmn-2bek{m}^{2}n+afkm{x}^m{}_2{F}_1(1,\frac{-1+m}{m};2-\frac{1}{m};-\frac{f{x}^m}{e})+bek(-1+m)mn{}_3{F}_2(1,-\frac{1}{m},-\frac{1}{m};1-\frac{1}{m},1-\frac{1}{m};-\frac{f{x}^m}{e})+bekm\log \left(c{x}^n\right)-bek{m}^2\log \left(c{x}^n\right)+bek(-1+m)m{}_2{F}_1(1,-\frac{1}{m};\frac{-1+m}{m};-\frac{f{x}^m}{e})(n+\log \left(c{x}^n\right))+ae\log \left(d{(e+f{x}^m)}^k\right)-aem\log \left(d{(e+f{x}^m)}^k\right)+ben\log \left(d{(e+f{x}^m)}^k\right)-bemn\log \left(d{(e+f{x}^m)}^k\right)+be\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)-bem\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)}{e(-1+m)x}$$

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

[B]   time = 0.115654 (sec), size = 292 ,normalized size = 10.07 $$\frac{4bekmn-2bek{m}^{2}n+4afkm{x}^m{}_2{F}_1(1,\frac{-2+m}{m};2-\frac{2}{m};-\frac{f{x}^m}{e})+bek(-2+m)mn{}_3{F}_2(1,-\frac{2}{m},-\frac{2}{m};1-\frac{2}{m},1-\frac{2}{m};-\frac{f{x}^m}{e})+4bekm\log \left(c{x}^n\right)-2bek{m}^2\log \left(c{x}^n\right)+bek(-2+m)m{}_2{F}_1(1,-\frac{2}{m};\frac{-2+m}{m};-\frac{f{x}^m}{e})(n+2\log \left(c{x}^n\right))+8ae\log \left(d{(e+f{x}^m)}^k\right)-4aem\log \left(d{(e+f{x}^m)}^k\right)+4ben\log \left(d{(e+f{x}^m)}^k\right)-2bemn\log \left(d{(e+f{x}^m)}^k\right)+8be\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)-4bem\log \left(c{x}^n\right)\log \left(d{(e+f{x}^m)}^k\right)}{8e(-2+m)x^2}$$

[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] $$\int -(dx{)}^m(a+b\log \left(c{x}^n\right))\log (1-e{x}^q)dx$$

[B]   time = 0.14394 (sec), size = 266 ,normalized size = 8.87 $$-\frac{x(dx{)}^m(-aq-amq+2bnq-bnq{}_3{F}_2(1,\frac{1}{q}+\frac{m}{q},\frac{1}{q}+\frac{m}{q};1+\frac{1}{q}+\frac{m}{q},1+\frac{1}{q}+\frac{m}{q};e{x}^q)-bq\log \left(c{x}^n\right)-bmq\log \left(c{x}^n\right)+q{}_2{F}_1(1,\frac{1+m}{q};\frac{1+m+q}{q};e{x}^q)(a+am-bn+b(1+m)\log \left(c{x}^n\right))+a\log (1-e{x}^q)+2am\log (1-e{x}^q)+a{m}^2\log (1-e{x}^q)-bn\log (1-e{x}^q)-bmn\log (1-e{x}^q)+b\log \left(c{x}^n\right)\log (1-e{x}^q)+2bm\log \left(c{x}^n\right)\log (1-e{x}^q)+b{m}^2\log \left(c{x}^n\right)\log (1-e{x}^q))}{(1+m{)}^3}$$

[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] $$\int -(dx{)}^m(a+b\log \left(c{x}^n\right))\log (1-e{x}^q)dx$$

[B]   time = 0.399 (sec), size = 844 ,normalized size = 28.13

method	result	size
meijerg	$-\frac{{\left(dx\right)}^m{x}^{-m}{(-e)}^{-\frac{m}{q}-\frac{1}{q}}a(\frac{q{x}^{1+m}{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (1-e{x}^q)}{1+m}-\frac{q{x}^{1+m+q}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}(-q-m-1)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{(1+m+q)(1+m)})}{q}-\frac{{\left(dx\right)}^m{x}^{-m}{(-e)}^{-\frac{m}{q}-\frac{1}{q}}b\ln \left(c\right)(\frac{q{x}^{1+m}{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (1-e{x}^q)}{1+m}-\frac{q{x}^{1+m+q}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}(-q-m-1)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{(1+m+q)(1+m)})}{q}+(\frac{{(-e)}^{-\frac{m}{q}-\frac{1}{q}}\ln (-e){\left(dx\right)}^m{x}^{-m}bn(\frac{q{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (1-e{x}^q)}{1+m}-\frac{q{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}(-q-m-1)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{(1+m+q)(1+m)})}{q^2}-\frac{{(-e)}^{-\frac{m}{q}-\frac{1}{q}}{\left(dx\right)}^m{x}^{-m}bn(\frac{q{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln \left(x\right)\ln (1-e{x}^q)}{1+m}+\frac{x^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (-e)\ln (1-e{x}^q)}{1+m}-\frac{q{x}^m{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (1-e{x}^q)}{{(1+m)}^2}+\frac{q{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}(-q-m-1)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{{(1+m+q)}^2(1+m)}-\frac{q{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln \left(x\right)(-q-m-1)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{(1+m+q)(1+m)}-\frac{x^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\ln (-e)(-q-m-1)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{(1+m+q)(1+m)}+\frac{q{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{(1+m+q)(1+m)}+\frac{q{x}^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}(-q-m-1)\mathrm{\Phi }(e{x}^q,1,\frac{1+m+q}{q})}{(1+m+q){(1+m)}^2}+\frac{x^{q+m}e{(-e)}^{\frac{m}{q}+\frac{1}{q}}(-q-m-1)\mathrm{\Phi }(e{x}^q,2,\frac{1+m+q}{q})}{(1+m+q)(1+m)})}{q})x$	$844$

[In]

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

[Out]

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

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

[B]   time = 0.212 (sec), size = 867 ,normalized size = 4.87

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

[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

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

[B]   time = 0.86 (sec), size = 1065 ,normalized size = 4.35

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

[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