\(\int -(d x)^m (a+b \log (c x^n)) \log (1-e x^q) \, dx\) [226]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx=-\text {Int}\left ((d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ),x\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(266\) vs. \(2(30)=60\).

Time = 0.32 (sec) , antiderivative size = 266, normalized size of antiderivative = 10.23 \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx=-\frac {x (d x)^m \left (-a q-a m q+2 b n q-b n q \, _3F_2\left (1,\frac {1}{q}+\frac {m}{q},\frac {1}{q}+\frac {m}{q};1+\frac {1}{q}+\frac {m}{q},1+\frac {1}{q}+\frac {m}{q};e x^q\right )-b q \log \left (c x^n\right )-b m q \log \left (c x^n\right )+q \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{q},\frac {1+m+q}{q},e x^q\right ) \left (a+a m-b n+b (1+m) \log \left (c x^n\right )\right )+a \log \left (1-e x^q\right )+2 a m \log \left (1-e x^q\right )+a m^2 \log \left (1-e x^q\right )-b n \log \left (1-e x^q\right )-b m n \log \left (1-e x^q\right )+b \log \left (c x^n\right ) \log \left (1-e x^q\right )+2 b m \log \left (c x^n\right ) \log \left (1-e x^q\right )+b m^2 \log \left (c x^n\right ) \log \left (1-e x^q\right )\right )}{(1+m)^3} \] Input:

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

Output:

-((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)
 

Rubi [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {25, 2826}

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.

\(\displaystyle \int -(d x)^m \log \left (1-e x^q\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx\)

\(\Big \downarrow \) 2826

\(\displaystyle -\int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right )dx\)

Input:

Int[-((d*x)^m*(a + b*Log[c*x^n])*Log[1 - 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 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]
 

Maple [N/A] (verified)

Time = 104.70 (sec) , antiderivative size = 844, normalized size of antiderivative = 32.46

method	result	size
meijerg	\(-\frac {\left (d x \right )^{m} x^{-m} \left (-e \right )^{-\frac {m}{q}-\frac {1}{q}} a \left (\frac {q \,x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{1+m}-\frac {q \,x^{1+m +q} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \left (-q -m -1\right ) \operatorname {LerchPhi}\left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m +q \right ) \left (1+m \right )}\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 \,x^{1+m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{1+m}-\frac {q \,x^{1+m +q} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \left (-q -m -1\right ) \operatorname {LerchPhi}\left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m +q \right ) \left (1+m \right )}\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 \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (1-e \,x^{q}\right )}{1+m}-\frac {q \,x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \left (-q -m -1\right ) \operatorname {LerchPhi}\left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m +q \right ) \left (1+m \right )}\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 \,x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (x \right ) \ln \left (1-e \,x^{q}\right )}{1+m}+\frac {x^{m} \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \ln \left (1-e \,x^{q}\right )}{1+m}-\frac {q \,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^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \left (-q -m -1\right ) \operatorname {LerchPhi}\left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m +q \right )^{2} \left (1+m \right )}-\frac {q \,x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (x \right ) \left (-q -m -1\right ) \operatorname {LerchPhi}\left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m +q \right ) \left (1+m \right )}-\frac {x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \ln \left (-e \right ) \left (-q -m -1\right ) \operatorname {LerchPhi}\left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m +q \right ) \left (1+m \right )}+\frac {q \,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 +q \right ) \left (1+m \right )}+\frac {q \,x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \left (-q -m -1\right ) \operatorname {LerchPhi}\left (e \,x^{q}, 1, \frac {1+m +q}{q}\right )}{\left (1+m +q \right ) \left (1+m \right )^{2}}+\frac {x^{q +m} e \left (-e \right )^{\frac {m}{q}+\frac {1}{q}} \left (-q -m -1\right ) \operatorname {LerchPhi}\left (e \,x^{q}, 2, \frac {1+m +q}{q}\right )}{\left (1+m +q \right ) \left (1+m \right )}\right )}{q}\right ) x\)	\(844\)

Input:

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

Output:

-(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)*LerchP 
hi(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*Lerch 
Phi(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)*L 
erchPhi(e*x^q,2,(1+m+q)/q)))*x
 

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx=\int { -{\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} \log \left (-e x^{q} + 1\right ) \,d x } \] Input:

integrate(-(d*x)^m*(a+b*log(c*x^n))*log(1-e*x^q),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 6.62 \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx=\int { -{\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} \log \left (-e x^{q} + 1\right ) \,d x } \] Input:

integrate(-(d*x)^m*(a+b*log(c*x^n))*log(1-e*x^q),x, algorithm="maxima")
 

Output:

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

Giac [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx=\int { -{\left (b \log \left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} \log \left (-e x^{q} + 1\right ) \,d x } \] Input:

integrate(-(d*x)^m*(a+b*log(c*x^n))*log(1-e*x^q),x, algorithm="giac")
 

Output:

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

Mupad [N/A]

Not integrable

Time = 25.70 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx=\int -\ln \left (1-e\,x^q\right )\,{\left (d\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 871, normalized size of antiderivative = 33.50 \[ \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(d**m*( - x**m*log( - x**q*e + 1)*log(x**n*c)*b*m**2*x - 2*x**m*log( - x** 
q*e + 1)*log(x**n*c)*b*m*x - x**m*log( - x**q*e + 1)*log(x**n*c)*b*x - x** 
m*log( - x**q*e + 1)*a*m**2*x - 2*x**m*log( - x**q*e + 1)*a*m*x - x**m*log 
( - x**q*e + 1)*a*x + x**m*log( - x**q*e + 1)*b*m*n*x + x**m*log( - x**q*e 
 + 1)*b*n*x + x**m*log(x**n*c)*b*m*q*x + x**m*log(x**n*c)*b*q*x + x**m*a*m 
*q*x + x**m*a*q*x - 2*x**m*b*n*q*x + int(x**m/(x**q*e*m**2 + 2*x**q*e*m + 
x**q*e - m**2 - 2*m - 1),x)*a*m**4*q + 4*int(x**m/(x**q*e*m**2 + 2*x**q*e* 
m + x**q*e - m**2 - 2*m - 1),x)*a*m**3*q + 6*int(x**m/(x**q*e*m**2 + 2*x** 
q*e*m + x**q*e - m**2 - 2*m - 1),x)*a*m**2*q + 4*int(x**m/(x**q*e*m**2 + 2 
*x**q*e*m + x**q*e - m**2 - 2*m - 1),x)*a*m*q + int(x**m/(x**q*e*m**2 + 2* 
x**q*e*m + x**q*e - m**2 - 2*m - 1),x)*a*q - int(x**m/(x**q*e*m**2 + 2*x** 
q*e*m + x**q*e - m**2 - 2*m - 1),x)*b*m**3*n*q - 3*int(x**m/(x**q*e*m**2 + 
 2*x**q*e*m + x**q*e - m**2 - 2*m - 1),x)*b*m**2*n*q - 3*int(x**m/(x**q*e* 
m**2 + 2*x**q*e*m + x**q*e - m**2 - 2*m - 1),x)*b*m*n*q - int(x**m/(x**q*e 
*m**2 + 2*x**q*e*m + x**q*e - m**2 - 2*m - 1),x)*b*n*q + int((x**m*log(x** 
n*c))/(x**q*e*m**2 + 2*x**q*e*m + x**q*e - m**2 - 2*m - 1),x)*b*m**4*q + 4 
*int((x**m*log(x**n*c))/(x**q*e*m**2 + 2*x**q*e*m + x**q*e - m**2 - 2*m - 
1),x)*b*m**3*q + 6*int((x**m*log(x**n*c))/(x**q*e*m**2 + 2*x**q*e*m + x**q 
*e - m**2 - 2*m - 1),x)*b*m**2*q + 4*int((x**m*log(x**n*c))/(x**q*e*m**2 + 
 2*x**q*e*m + x**q*e - m**2 - 2*m - 1),x)*b*m*q + int((x**m*log(x**n*c)...