3.17.8 $\int \frac{e^{8x}(48x^3+96x^4)\log (16-24e^4+9e^8+e^{16x}{x}^8+e^{8x}(8x^4-6e^4{x}^4))}{4-3e^4+e^{8x}{x}^4}dx$

Optimal. Leaf size=23 $$3{\log }^2\left({(4-3e^4+e^{8x}{x}^4)}^2\right)$$

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Rubi [A]  time = 1.30, antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 4, integrand size = 77, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.052, Rules used = {1593, 6684, 6741, 6686} $$\begin{array}{c}3{\log }^2\left({(-e^{8x}{x}^4+3e^4-4)}^2\right)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^(8*x)*(48*x^3 + 96*x^4)*Log[16 - 24*E^4 + 9*E^8 + E^(16*x)*x^8 + E^(8*x)*(8*x^4 - 6*E^4*x^4)])/(4 - 3*E
^4 + E^(8*x)*x^4),x]

[Out]

3*Log[(-4 + 3*E^4 - E^(8*x)*x^4)^2]^2

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{8x}{x}^3(48+96x)\log (16-24e^4+9e^8+e^{16x}{x}^8+e^{8x}(8x^4-6e^4{x}^4))}{4-3e^4+e^{8x}{x}^4}dx\\ & =\int \frac{e^{8x}{x}^3(48+96x)\log \left({(-4(1-\frac{3e^4}{4})-e^{8x}{x}^4)}^2\right)}{4(1-\frac{3e^4}{4})+e^{8x}{x}^4}dx\\ & =3{\log }^2\left({(-4+3e^4-e^{8x}{x}^4)}^2\right)\end{array}\end{array}$$

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Mathematica [A]  time = 0.08, size = 23, normalized size = 1.00 $$\begin{array}{c}3{\log }^2\left({(4-3e^4+e^{8x}{x}^4)}^2\right)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(8*x)*(48*x^3 + 96*x^4)*Log[16 - 24*E^4 + 9*E^8 + E^(16*x)*x^8 + E^(8*x)*(8*x^4 - 6*E^4*x^4)])/(4
- 3*E^4 + E^(8*x)*x^4),x]

[Out]

3*Log[(4 - 3*E^4 + E^(8*x)*x^4)^2]^2

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fricas [A]  time = 0.67, size = 42, normalized size = 1.83 $$\begin{array}{c}3\log {(x^8{e}^{\left(16x\right)}-2(3x^4{e}^4-4x^4)e^{\left(8x\right)}+9e^8-24e^4+16)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((96*x^4+48*x^3)*exp(2*x)^4*log(x^8*exp(2*x)^8+(-6*x^4*exp(4)+8*x^4)*exp(2*x)^4+9*exp(4)^2-24*exp(4)+
16)/(x^4*exp(2*x)^4-3*exp(4)+4),x, algorithm="fricas")

[Out]

3*log(x^8*e^(16*x) - 2*(3*x^4*e^4 - 4*x^4)*e^(8*x) + 9*e^8 - 24*e^4 + 16)^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int \frac{48(2x^4+x^3)e^{\left(8x\right)}\log (x^8{e}^{\left(16x\right)}-2(3x^4{e}^4-4x^4)e^{\left(8x\right)}+9e^8-24e^4+16)}{x^4{e}^{\left(8x\right)}-3e^4+4}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((96*x^4+48*x^3)*exp(2*x)^4*log(x^8*exp(2*x)^8+(-6*x^4*exp(4)+8*x^4)*exp(2*x)^4+9*exp(4)^2-24*exp(4)+
16)/(x^4*exp(2*x)^4-3*exp(4)+4),x, algorithm="giac")

[Out]

integrate(48*(2*x^4 + x^3)*e^(8*x)*log(x^8*e^(16*x) - 2*(3*x^4*e^4 - 4*x^4)*e^(8*x) + 9*e^8 - 24*e^4 + 16)/(x^
4*e^(8*x) - 3*e^4 + 4), x)

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maple [B]  time = 0.43, size = 77, normalized size = 3.35

method	result	size
default	$12(\ln \left({(-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4)}^2\right)-2\ln (-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4))\ln (-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4)+12\ln {(-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4)}^2$	$77$
risch	$-24\ln (3)\ln (-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4)+12\ln {(-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4)}^2-6i\pi \ln (-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4)\mathrm{csgn}{\left(i(-\frac{x^4{\mathrm{e}}^{8x}}{3}+{\mathrm{e}}^4-\frac{4}{3})\right)}^2\mathrm{csgn}\left(i{(-\frac{x^4{\mathrm{e}}^{8x}}{3}+{\mathrm{e}}^4-\frac{4}{3})}^2\right)+12i\pi \ln (-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4)\mathrm{csgn}\left(i(-\frac{x^4{\mathrm{e}}^{8x}}{3}+{\mathrm{e}}^4-\frac{4}{3})\right)\mathrm{csgn}{\left(i{(-\frac{x^4{\mathrm{e}}^{8x}}{3}+{\mathrm{e}}^4-\frac{4}{3})}^2\right)}^2-6i\pi \ln (-x^4{\mathrm{e}}^{8x}+3{\mathrm{e}}^4-4)\mathrm{csgn}{\left(i{(-\frac{x^4{\mathrm{e}}^{8x}}{3}+{\mathrm{e}}^4-\frac{4}{3})}^2\right)}^3$	$199$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((96*x^4+48*x^3)*exp(2*x)^4*ln(x^8*exp(2*x)^8+(-6*x^4*exp(4)+8*x^4)*exp(2*x)^4+9*exp(4)^2-24*exp(4)+16)/(x^
4*exp(2*x)^4-3*exp(4)+4),x,method=_RETURNVERBOSE)

[Out]

12*(ln((-x^4*exp(8*x)+3*exp(4)-4)^2)-2*ln(-x^4*exp(8*x)+3*exp(4)-4))*ln(-x^4*exp(8*x)+3*exp(4)-4)+12*ln(-x^4*e
xp(8*x)+3*exp(4)-4)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}48\int \frac{(2x^4+x^3)e^{\left(8x\right)}\log (x^8{e}^{\left(16x\right)}-2(3x^4{e}^4-4x^4)e^{\left(8x\right)}+9e^8-24e^4+16)}{x^4{e}^{\left(8x\right)}-3e^4+4}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((96*x^4+48*x^3)*exp(2*x)^4*log(x^8*exp(2*x)^8+(-6*x^4*exp(4)+8*x^4)*exp(2*x)^4+9*exp(4)^2-24*exp(4)+
16)/(x^4*exp(2*x)^4-3*exp(4)+4),x, algorithm="maxima")

[Out]

48*integrate((2*x^4 + x^3)*e^(8*x)*log(x^8*e^(16*x) - 2*(3*x^4*e^4 - 4*x^4)*e^(8*x) + 9*e^8 - 24*e^4 + 16)/(x^
4*e^(8*x) - 3*e^4 + 4), x)

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mupad [B]  time = 1.48, size = 42, normalized size = 1.83 $$\begin{array}{c}3{\ln (9{\mathrm{e}}^8-24{\mathrm{e}}^4+x^8{\mathrm{e}}^{16x}-{\mathrm{e}}^{8x}(6x^4{\mathrm{e}}^4-8x^4)+16)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8*x)*log(9*exp(8) - 24*exp(4) + x^8*exp(16*x) - exp(8*x)*(6*x^4*exp(4) - 8*x^4) + 16)*(48*x^3 + 96*x^
4))/(x^4*exp(8*x) - 3*exp(4) + 4),x)

[Out]

3*log(9*exp(8) - 24*exp(4) + x^8*exp(16*x) - exp(8*x)*(6*x^4*exp(4) - 8*x^4) + 16)^2

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sympy [B]  time = 0.44, size = 42, normalized size = 1.83 $$\begin{array}{c}3\log {(x^8{e}^{16x}+(-6x^4{e}^4+8x^4)e^{8x}-24e^4+16+9e^8)}^2\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((96*x**4+48*x**3)*exp(2*x)**4*ln(x**8*exp(2*x)**8+(-6*x**4*exp(4)+8*x**4)*exp(2*x)**4+9*exp(4)**2-24
*exp(4)+16)/(x**4*exp(2*x)**4-3*exp(4)+4),x)

[Out]

3*log(x**8*exp(16*x) + (-6*x**4*exp(4) + 8*x**4)*exp(8*x) - 24*exp(4) + 16 + 9*exp(8))**2

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