### 3.49 $$\int x (a+b \log (c \log ^p(d x^n))) \, dx$$

Optimal. Leaf size=55 $\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )-\frac{1}{2} b p x^2 \left (d x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \log \left (d x^n\right )}{n}\right )$

[Out]

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

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Rubi [A]  time = 0.0349517, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {2522, 2310, 2178} $\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )-\frac{1}{2} b p x^2 \left (d x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \log \left (d x^n\right )}{n}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[x*(a + b*Log[c*Log[d*x^n]^p]),x]

[Out]

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

Rule 2522

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

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !\$UseGamma === True

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )-\frac{1}{2} (b n p) \int \frac{x}{\log \left (d x^n\right )} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )-\frac{1}{2} \left (b p x^2 \left (d x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{x} \, dx,x,\log \left (d x^n\right )\right )\\ &=-\frac{1}{2} b p x^2 \left (d x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \log \left (d x^n\right )}{n}\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0567191, size = 49, normalized size = 0.89 $\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )-b p \left (d x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \log \left (d x^n\right )}{n}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*(a + b*Log[c*Log[d*x^n]^p]),x]

[Out]

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

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) \right ) \, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*ln(c*ln(d*x^n)^p)),x)

[Out]

int(x*(a+b*ln(c*ln(d*x^n)^p)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a x^{2} - \frac{1}{2} \,{\left (2 \, n p \int \frac{x}{2 \,{\left (\log \left (d\right ) + \log \left (x^{n}\right )\right )}}\,{d x} - x^{2} \log \left (c\right ) - x^{2} \log \left ({\left (\log \left (d\right ) + \log \left (x^{n}\right )\right )}^{p}\right )\right )} b \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2 - 1/2*(2*n*p*integrate(1/2*x/(log(d) + log(x^n)), x) - x^2*log(c) - x^2*log((log(d) + log(x^n))^p))*
b

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Fricas [A]  time = 1.85521, size = 161, normalized size = 2.93 \begin{align*} \frac{b d^{\frac{2}{n}} p x^{2} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b p \logintegral \left (d^{\frac{2}{n}} x^{2}\right ) +{\left (b x^{2} \log \left (c\right ) + a x^{2}\right )} d^{\frac{2}{n}}}{2 \, d^{\frac{2}{n}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*d^(2/n)*p*x^2*log(n*log(x) + log(d)) - b*p*log_integral(d^(2/n)*x^2) + (b*x^2*log(c) + a*x^2)*d^(2/n))/
d^(2/n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \log{\left (c \log{\left (d x^{n} \right )}^{p} \right )}\right )\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*ln(c*ln(d*x**n)**p)),x)

[Out]

Integral(x*(a + b*log(c*log(d*x**n)**p)), x)

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Giac [A]  time = 1.31949, size = 76, normalized size = 1.38 \begin{align*} \frac{1}{2} \, b p x^{2} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) + \frac{1}{2} \, b x^{2} \log \left (c\right ) + \frac{1}{2} \, a x^{2} - \frac{b p{\rm Ei}\left (\frac{2 \, \log \left (d\right )}{n} + 2 \, \log \left (x\right )\right )}{2 \, d^{\frac{2}{n}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*p*x^2*log(n*log(x) + log(d)) + 1/2*b*x^2*log(c) + 1/2*a*x^2 - 1/2*b*p*Ei(2*log(d)/n + 2*log(x))/d^(2/n)