\(\int \frac {(a+b \log (c x^n))^2}{x^3} \, dx\) [56]

Optimal result

Integrand size = 16, antiderivative size = 52 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {b^2 n^2}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2} \]

[Out]

-1/4*b^2*n^2/x^2-1/2*b*n*(a+b*ln(c*x^n))/x^2-1/2*(a+b*ln(c*x^n))^2/x^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2342, 2341} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {b^2 n^2}{4 x^2} \]

[In]

Int[(a + b*Log[c*x^n])^2/x^3,x]

[Out]

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{4 x^2} \]

[In]

Integrate[(a + b*Log[c*x^n])^2/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13

method	result	size
parallelrisch	\(-\frac {2 b^{2} \ln \left (c \,x^{n}\right )^{2}+2 \ln \left (c \,x^{n}\right ) b^{2} n +b^{2} n^{2}+4 a b \ln \left (c \,x^{n}\right )+2 a b n +2 a^{2}}{4 x^{2}}\)	\(59\)
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 x^{2}}-\frac {\left (-i \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b^{2} \ln \left (c \right )+b^{2} n +2 a b \right ) \ln \left (x^{n}\right )}{2 x^{2}}-\frac {4 a^{2}-2 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 b^{2} n^{2}+8 \ln \left (c \right ) a b +4 \ln \left (c \right )^{2} b^{2}+4 b^{2} \ln \left (c \right ) n +4 a b n -\pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+4 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8 x^{2}}\)	\(703\)

[In]

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

[Out]

-1/4/x^2*(2*b^2*ln(c*x^n)^2+2*ln(c*x^n)*b^2*n+b^2*n^2+4*a*b*ln(c*x^n)+2*a*b*n+2*a^2)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {2 \, b^{2} n^{2} \log \left (x\right )^{2} + b^{2} n^{2} + 2 \, b^{2} \log \left (c\right )^{2} + 2 \, a b n + 2 \, a^{2} + 2 \, {\left (b^{2} n + 2 \, a b\right )} \log \left (c\right ) + 2 \, {\left (b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, a b n\right )} \log \left (x\right )}{4 \, x^{2}} \]

[In]

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

[Out]

-1/4*(2*b^2*n^2*log(x)^2 + b^2*n^2 + 2*b^2*log(c)^2 + 2*a*b*n + 2*a^2 + 2*(b^2*n + 2*a*b)*log(c) + 2*(b^2*n^2
+ 2*b^2*n*log(c) + 2*a*b*n)*log(x))/x^2

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=- \frac {a^{2}}{2 x^{2}} - \frac {a b n}{2 x^{2}} - \frac {a b \log {\left (c x^{n} \right )}}{x^{2}} - \frac {b^{2} n^{2}}{4 x^{2}} - \frac {b^{2} n \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b^{2} \log {\left (c x^{n} \right )}^{2}}{2 x^{2}} \]

[In]

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

[Out]

-a**2/(2*x**2) - a*b*n/(2*x**2) - a*b*log(c*x**n)/x**2 - b**2*n**2/(4*x**2) - b**2*n*log(c*x**n)/(2*x**2) - b*
*2*log(c*x**n)**2/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {1}{4} \, b^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {b^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b n}{2 \, x^{2}} - \frac {a b \log \left (c x^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]

[In]

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

[Out]

-1/4*b^2*(n^2/x^2 + 2*n*log(c*x^n)/x^2) - 1/2*b^2*log(c*x^n)^2/x^2 - 1/2*a*b*n/x^2 - a*b*log(c*x^n)/x^2 - 1/2*
a^2/x^2

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {b^{2} n^{2} \log \left (x\right )^{2}}{2 \, x^{2}} - \frac {{\left (b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, a b n\right )} \log \left (x\right )}{2 \, x^{2}} - \frac {b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, b^{2} \log \left (c\right )^{2} + 2 \, a b n + 4 \, a b \log \left (c\right ) + 2 \, a^{2}}{4 \, x^{2}} \]

[In]

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

[Out]

-1/2*b^2*n^2*log(x)^2/x^2 - 1/2*(b^2*n^2 + 2*b^2*n*log(c) + 2*a*b*n)*log(x)/x^2 - 1/4*(b^2*n^2 + 2*b^2*n*log(c
) + 2*b^2*log(c)^2 + 2*a*b*n + 4*a*b*log(c) + 2*a^2)/x^2

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {\frac {a^2}{2}+\frac {a\,b\,n}{2}+\frac {b^2\,n^2}{4}}{x^2}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {n\,b^2}{2}+a\,b\right )}{x^2}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2}{2\,x^2} \]

[In]

int((a + b*log(c*x^n))^2/x^3,x)

[Out]

- (a^2/2 + (b^2*n^2)/4 + (a*b*n)/2)/x^2 - (log(c*x^n)*(a*b + (b^2*n)/2))/x^2 - (b^2*log(c*x^n)^2)/(2*x^2)