3.55 \(\int \frac{(a+b \log (c x^n))^2}{x^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{2 b^2 n^2}{x} \]

[Out]

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

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Rubi [A]  time = 0.0348372, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2305, 2304} \[ -\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{2 b^2 n^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^2/x^2,x]

[Out]

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

Rule 2305

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]

Rule 2304

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
]

Rubi steps

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

Mathematica [A]  time = 0.0106017, size = 35, normalized size = 0.76 \[ -\frac{\left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (a+b \log \left (c x^n\right )+b n\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^2/x^2,x]

[Out]

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

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Maple [C]  time = 0.122, size = 704, normalized size = 15.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2/x*ln(x^n)^2-(I*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b^2*c
sgn(I*c*x^n)^3+I*Pi*b^2*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(c)*b^2+2*b^2*n+2*a*b)/x*ln(x^n)-1/4*(4*I*ln(c)*Pi*b^2*c
sgn(I*x^n)*csgn(I*c*x^n)^2+4*ln(c)^2*b^2-Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2+8*a*b*n+8*b^2*n^2+4*a^2-4*I*Pi*b
^2*n*csgn(I*c*x^n)^3+4*I*Pi*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)+2*Pi^2*b^
2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+2*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-Pi^2*b^2*csgn(I*x
^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-4*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+4*I*ln(c)*Pi*b^2*csgn(I*c*x
^n)^2*csgn(I*c)+4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)-4*I*Pi*a*b*csgn(I*
c*x^n)^3-4*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^3-4*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*b^2*n*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-Pi^2*b^2*csgn(I*c*x^n)^6+8*ln
(c)*a*b+8*ln(c)*b^2*n+2*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)+2*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^5-Pi^2*b^2*csg
n(I*x^n)^2*csgn(I*c*x^n)^4)/x

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Maxima [A]  time = 1.11781, size = 95, normalized size = 2.07 \begin{align*} -2 \, b^{2}{\left (\frac{n^{2}}{x} + \frac{n \log \left (c x^{n}\right )}{x}\right )} - \frac{b^{2} \log \left (c x^{n}\right )^{2}}{x} - \frac{2 \, a b n}{x} - \frac{2 \, a b \log \left (c x^{n}\right )}{x} - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.859764, size = 182, normalized size = 3.96 \begin{align*} -\frac{b^{2} n^{2} \log \left (x\right )^{2} + 2 \, b^{2} n^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b n + a^{2} + 2 \,{\left (b^{2} n + a b\right )} \log \left (c\right ) + 2 \,{\left (b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.07547, size = 110, normalized size = 2.39 \begin{align*} - \frac{a^{2}}{x} - \frac{2 a b n \log{\left (x \right )}}{x} - \frac{2 a b n}{x} - \frac{2 a b \log{\left (c \right )}}{x} - \frac{b^{2} n^{2} \log{\left (x \right )}^{2}}{x} - \frac{2 b^{2} n^{2} \log{\left (x \right )}}{x} - \frac{2 b^{2} n^{2}}{x} - \frac{2 b^{2} n \log{\left (c \right )} \log{\left (x \right )}}{x} - \frac{2 b^{2} n \log{\left (c \right )}}{x} - \frac{b^{2} \log{\left (c \right )}^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15635, size = 116, normalized size = 2.52 \begin{align*} -\frac{b^{2} n^{2} \log \left (x\right )^{2}}{x} - \frac{2 \,{\left (b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )}{x} - \frac{2 \, b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + b^{2} \log \left (c\right )^{2} + 2 \, a b n + 2 \, a b \log \left (c\right ) + a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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