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

Optimal. Leaf size=30 \[ -\text {Int}\left ((d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ),x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int -(d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx &=-\int (d x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (1-e x^q\right ) \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(266\) vs. \(2(30)=60\).
time = 0.14, size = 266, normalized size = 8.87 \begin {gather*} -\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 \, _2F_1\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} \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

Maple [A] Leaf count of result is larger than twice the leaf count of optimal. \(843\) vs. \(2(29)=58\).
time = 0.40, size = 844, normalized size = 28.13

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 ) \Phi \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 ) \Phi \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 ) \Phi \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 ) \Phi \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 ) \Phi \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 ) \Phi \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}} \Phi \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 ) \Phi \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 ) \Phi \left (e \,x^{q}, 2, \frac {1+m +q}{q}\right )}{\left (1+m +q \right ) \left (1+m \right )}\right )}{q}\right ) x\)	\(844\)

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________