### 3.15 $$\int \frac{\log (c x^n)}{x (a x^m+b \log ^2(c x^n))^3} \, dx$$

Optimal. Leaf size=67 $-\frac{a m \text{CannotIntegrate}\left (\frac{x^{m-1}}{\left (a x^m+b \log ^2\left (c x^n\right )\right )^3},x\right )}{2 b n}-\frac{1}{4 b n \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}$

[Out]

-(a*m*CannotIntegrate[x^(-1 + m)/(a*x^m + b*Log[c*x^n]^2)^3, x])/(2*b*n) - 1/(4*b*n*(a*x^m + b*Log[c*x^n]^2)^2
)

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Rubi [A]  time = 0.177491, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $\int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^3} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[Log[c*x^n]/(x*(a*x^m + b*Log[c*x^n]^2)^3),x]

[Out]

-1/(4*b*n*(a*x^m + b*Log[c*x^n]^2)^2) - (a*m*Defer[Int][x^(-1 + m)/(a*x^m + b*Log[c*x^n]^2)^3, x])/(2*b*n)

Rubi steps

\begin{align*} \int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^3} \, dx &=-\frac{1}{4 b n \left (a x^m+b \log ^2\left (c x^n\right )\right )^2}-\frac{(a m) \int \frac{x^{-1+m}}{\left (a x^m+b \log ^2\left (c x^n\right )\right )^3} \, dx}{2 b n}\\ \end{align*}

Mathematica [A]  time = 2.8723, size = 0, normalized size = 0. $\int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^3} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[Log[c*x^n]/(x*(a*x^m + b*Log[c*x^n]^2)^3),x]

[Out]

Integrate[Log[c*x^n]/(x*(a*x^m + b*Log[c*x^n]^2)^3), x]

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Maple [A]  time = 56.374, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c{x}^{n} \right ) }{x \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2} \right ) ^{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x^n)/x/(a*x^m+b*ln(c*x^n)^2)^3,x)

[Out]

int(ln(c*x^n)/x/(a*x^m+b*ln(c*x^n)^2)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)/x/(a*x^m+b*log(c*x^n)^2)^3,x, algorithm="maxima")

[Out]

-1/2*(24*b^3*m*n^4*log(c)^3 - 5*a^3*m^4*n*x^(3*m) - (m^5*log(c)^3 + 7*m^4*n*log(c)^2 - 18*m^3*n^2*log(c) - 4*m
^2*n^3)*a^2*b*x^(2*m) + 2*(5*m^3*n^2*log(c)^3 - 6*m^2*n^3*log(c)^2 + 20*m*n^4*log(c) + 16*n^5)*a*b^2*x^m - (a^
2*b*m^5*x^(2*m) - 10*a*b^2*m^3*n^2*x^m - 24*b^3*m*n^4)*log(x^n)^3 + (72*b^3*m*n^4*log(c) - (3*m^5*log(c) + 7*m
^4*n)*a^2*b*x^(2*m) + 6*(5*m^3*n^2*log(c) - 2*m^2*n^3)*a*b^2*x^m)*log(x^n)^2 + (72*b^3*m*n^4*log(c)^2 - (3*m^5
*log(c)^2 + 14*m^4*n*log(c) - 18*m^3*n^2)*a^2*b*x^(2*m) + 2*(15*m^3*n^2*log(c)^2 - 12*m^2*n^3*log(c) + 20*m*n^
4)*a*b^2*x^m)*log(x^n))/(64*a*b^5*n^6*x^m*log(c)^4 + a^6*m^6*x^(6*m) + 2*(m^6*log(c)^2 + 6*m^4*n^2)*a^5*b*x^(5
*m) + (m^6*log(c)^4 + 24*m^4*n^2*log(c)^2 + 48*m^2*n^4)*a^4*b^2*x^(4*m) + 4*(3*m^4*n^2*log(c)^4 + 24*m^2*n^4*l
og(c)^2 + 16*n^6)*a^3*b^3*x^(3*m) + 16*(3*m^2*n^4*log(c)^4 + 8*n^6*log(c)^2)*a^2*b^4*x^(2*m) + (a^4*b^2*m^6*x^
(4*m) + 12*a^3*b^3*m^4*n^2*x^(3*m) + 48*a^2*b^4*m^2*n^4*x^(2*m) + 64*a*b^5*n^6*x^m)*log(x^n)^4 + 4*(a^4*b^2*m^
6*x^(4*m)*log(c) + 12*a^3*b^3*m^4*n^2*x^(3*m)*log(c) + 48*a^2*b^4*m^2*n^4*x^(2*m)*log(c) + 64*a*b^5*n^6*x^m*lo
g(c))*log(x^n)^3 + 2*(192*a*b^5*n^6*x^m*log(c)^2 + a^5*b*m^6*x^(5*m) + 3*(m^6*log(c)^2 + 4*m^4*n^2)*a^4*b^2*x^
(4*m) + 12*(3*m^4*n^2*log(c)^2 + 4*m^2*n^4)*a^3*b^3*x^(3*m) + 16*(9*m^2*n^4*log(c)^2 + 4*n^6)*a^2*b^4*x^(2*m))
*log(x^n)^2 + 4*(64*a*b^5*n^6*x^m*log(c)^3 + a^5*b*m^6*x^(5*m)*log(c) + (m^6*log(c)^3 + 12*m^4*n^2*log(c))*a^4
*b^2*x^(4*m) + 12*(m^4*n^2*log(c)^3 + 4*m^2*n^4*log(c))*a^3*b^3*x^(3*m) + 16*(3*m^2*n^4*log(c)^3 + 4*n^6*log(c
))*a^2*b^4*x^(2*m))*log(x^n)) + integrate(1/2*((2*m^8*log(c) + 15*m^7*n)*a^3*x^(3*m) - 2*(17*m^6*n^2*log(c) -
m^5*n^3)*a^2*b*x^(2*m) - 32*(3*m^4*n^4*log(c) + 2*m^3*n^5)*a*b^2*x^m - 96*(m^2*n^6*log(c) + m*n^7)*b^3 + 2*(a^
3*m^8*x^(3*m) - 17*a^2*b*m^6*n^2*x^(2*m) - 48*a*b^2*m^4*n^4*x^m - 48*b^3*m^2*n^6)*log(x^n))/(256*a*b^5*n^8*x*x
^m*log(c)^2 + a^6*m^8*x*x^(6*m) + (m^8*log(c)^2 + 16*m^6*n^2)*a^5*b*x*x^(5*m) + 16*(m^6*n^2*log(c)^2 + 6*m^4*n
^4)*a^4*b^2*x*x^(4*m) + 32*(3*m^4*n^4*log(c)^2 + 8*m^2*n^6)*a^3*b^3*x*x^(3*m) + 256*(m^2*n^6*log(c)^2 + n^8)*a
^2*b^4*x*x^(2*m) + (a^5*b*m^8*x*x^(5*m) + 16*a^4*b^2*m^6*n^2*x*x^(4*m) + 96*a^3*b^3*m^4*n^4*x*x^(3*m) + 256*a^
2*b^4*m^2*n^6*x*x^(2*m) + 256*a*b^5*n^8*x*x^m)*log(x^n)^2 + 2*(a^5*b*m^8*x*x^(5*m)*log(c) + 16*a^4*b^2*m^6*n^2
*x*x^(4*m)*log(c) + 96*a^3*b^3*m^4*n^4*x*x^(3*m)*log(c) + 256*a^2*b^4*m^2*n^6*x*x^(2*m)*log(c) + 256*a*b^5*n^8
*x*x^m*log(c))*log(x^n)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (c x^{n}\right )}{b^{3} x \log \left (c x^{n}\right )^{6} + 3 \, a b^{2} x x^{m} \log \left (c x^{n}\right )^{4} + 3 \, a^{2} b x x^{2 \, m} \log \left (c x^{n}\right )^{2} + a^{3} x x^{3 \, m}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)/x/(a*x^m+b*log(c*x^n)^2)^3,x, algorithm="fricas")

[Out]

integral(log(c*x^n)/(b^3*x*log(c*x^n)^6 + 3*a*b^2*x*x^m*log(c*x^n)^4 + 3*a^2*b*x*x^(2*m)*log(c*x^n)^2 + a^3*x*
x^(3*m)), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x**n)/x/(a*x**m+b*ln(c*x**n)**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (c x^{n}\right )}{{\left (b \log \left (c x^{n}\right )^{2} + a x^{m}\right )}^{3} x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)/x/(a*x^m+b*log(c*x^n)^2)^3,x, algorithm="giac")

[Out]

integrate(log(c*x^n)/((b*log(c*x^n)^2 + a*x^m)^3*x), x)