### 3.25 $$\int (a+\frac{2 b n \log (c x^n)}{x}) \, dx$$

Optimal. Leaf size=14 $a x+b \log ^2\left (c x^n\right )$

[Out]

a*x + b*Log[c*x^n]^2

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Rubi [A]  time = 0.0092727, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {2301} $a x+b \log ^2\left (c x^n\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[a + (2*b*n*Log[c*x^n])/x,x]

[Out]

a*x + b*Log[c*x^n]^2

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0018801, size = 14, normalized size = 1. $a x+b \log ^2\left (c x^n\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[a + (2*b*n*Log[c*x^n])/x,x]

[Out]

a*x + b*Log[c*x^n]^2

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Maple [A]  time = 0.003, size = 15, normalized size = 1.1 \begin{align*} ax+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01534, size = 19, normalized size = 1.36 \begin{align*} b \log \left (c x^{n}\right )^{2} + a x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(c*x^n)^2 + a*x

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Fricas [A]  time = 1.88183, size = 59, normalized size = 4.21 \begin{align*} b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + a x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.67615, size = 60, normalized size = 4.29 \begin{align*} a x + 2 b n \left (\begin{cases} \frac{\log{\left (c x^{n} \right )}^{2}}{2 n} & \text{for}\: \left |{c x^{n}}\right | < 1 \\\frac{\log{\left (\frac{x^{- n}}{c} \right )}^{2}}{2 n} & \text{for}\: \frac{1}{\left |{c x^{n}}\right |} < 1 \\\frac{{G_{3, 3}^{3, 0}\left (\begin{matrix} & 1, 1, 1 \\0, 0, 0 & \end{matrix} \middle |{c x^{n}} \right )}}{n} + \frac{{G_{3, 3}^{0, 3}\left (\begin{matrix} 1, 1, 1 & \\ & 0, 0, 0 \end{matrix} \middle |{c x^{n}} \right )}}{n} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + 2*b*n*Piecewise((log(c*x**n)**2/(2*n), Abs(c*x**n) < 1), (log(x**(-n)/c)**2/(2*n), 1/Abs(c*x**n) < 1), (
meijerg(((), (1, 1, 1)), ((0, 0, 0), ()), c*x**n)/n + meijerg(((1, 1, 1), ()), ((), (0, 0, 0)), c*x**n)/n, Tru
e))

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Giac [A]  time = 1.24135, size = 27, normalized size = 1.93 \begin{align*}{\left (n \log \left (x\right )^{2} + 2 \, \log \left (c\right ) \log \left (x\right )\right )} b n + a x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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