### 3.55 $$\int \log (c \log ^p(d x)) \, dx$$

Optimal. Leaf size=22 $x \log \left (c \log ^p(d x)\right )-\frac{p \text{li}(d x)}{d}$

[Out]

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Rubi [A]  time = 0.0067795, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {2520, 2298} $x \log \left (c \log ^p(d x)\right )-\frac{p \text{li}(d x)}{d}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[c*Log[d*x]^p],x]

[Out]

Rule 2520

Int[Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)], x_Symbol] :> Simp[x*Log[c*Log[d*x^n]^p], x] - Dist[n*p, Int[1/Log[
d*x^n], x], x] /; FreeQ[{c, d, n, p}, x]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rubi steps

\begin{align*} \int \log \left (c \log ^p(d x)\right ) \, dx &=x \log \left (c \log ^p(d x)\right )-p \int \frac{1}{\log (d x)} \, dx\\ &=x \log \left (c \log ^p(d x)\right )-\frac{p \text{li}(d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0140017, size = 22, normalized size = 1. $x \log \left (c \log ^p(d x)\right )-\frac{p \text{li}(d x)}{d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[c*Log[d*x]^p],x]

[Out]

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Maple [A]  time = 0.009, size = 26, normalized size = 1.2 \begin{align*} x\ln \left ( c \left ( \ln \left ( dx \right ) \right ) ^{p} \right ) +{\frac{p{\it Ei} \left ( 1,-\ln \left ( dx \right ) \right ) }{d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*ln(d*x)^p),x)

[Out]

x*ln(c*ln(d*x)^p)+p/d*Ei(1,-ln(d*x))

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Maxima [A]  time = 1.14172, size = 31, normalized size = 1.41 \begin{align*} x \log \left (c \log \left (d x\right )^{p}\right ) - \frac{p{\rm Ei}\left (\log \left (d x\right )\right )}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*log(d*x)^p),x, algorithm="maxima")

[Out]

x*log(c*log(d*x)^p) - p*Ei(log(d*x))/d

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Fricas [A]  time = 1.64578, size = 81, normalized size = 3.68 \begin{align*} \frac{d p x \log \left (\log \left (d x\right )\right ) + d x \log \left (c\right ) - p \logintegral \left (d x\right )}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*log(d*x)^p),x, algorithm="fricas")

[Out]

(d*p*x*log(log(d*x)) + d*x*log(c) - p*log_integral(d*x))/d

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Sympy [A]  time = 1.31437, size = 19, normalized size = 0.86 \begin{align*} x \log{\left (c \log{\left (d x \right )}^{p} \right )} - \frac{p \operatorname{li}{\left (d x \right )}}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*ln(d*x)**p),x)

[Out]

x*log(c*log(d*x)**p) - p*li(d*x)/d

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Giac [A]  time = 1.20494, size = 35, normalized size = 1.59 \begin{align*} p x \log \left (\log \left (d\right ) + \log \left (x\right )\right ) + x \log \left (c\right ) - \frac{p{\rm Ei}\left (\log \left (d\right ) + \log \left (x\right )\right )}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*log(d*x)^p),x, algorithm="giac")

[Out]

p*x*log(log(d) + log(x)) + x*log(c) - p*Ei(log(d) + log(x))/d