3.96.99 \(\int \frac {-4 x-20 e^{5 \log ^4(x)} \log ^3(x)}{x} \, dx\)

Optimal. Leaf size=15 \[ -9-e^{5 \log ^4(x)}-4 x \]

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Rubi [A]  time = 0.07, antiderivative size = 14, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14, 2209} \begin {gather*} -4 x-e^{5 \log ^4(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x - 20*E^(5*Log[x]^4)*Log[x]^3)/x,x]

[Out]

-E^(5*Log[x]^4) - 4*x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-4-\frac {20 e^{5 \log ^4(x)} \log ^3(x)}{x}\right ) \, dx\\ &=-4 x-20 \int \frac {e^{5 \log ^4(x)} \log ^3(x)}{x} \, dx\\ &=-4 x-20 \operatorname {Subst}\left (\int e^{5 x^4} x^3 \, dx,x,\log (x)\right )\\ &=-e^{5 \log ^4(x)}-4 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 1.07 \begin {gather*} -4 \left (\frac {1}{4} e^{5 \log ^4(x)}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x - 20*E^(5*Log[x]^4)*Log[x]^3)/x,x]

[Out]

-4*(E^(5*Log[x]^4)/4 + x)

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fricas [A]  time = 0.67, size = 13, normalized size = 0.87 \begin {gather*} -4 \, x - e^{\left (5 \, \log \relax (x)^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*log(x)^3*exp(5*log(x)^4)-4*x)/x,x, algorithm="fricas")

[Out]

-4*x - e^(5*log(x)^4)

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giac [A]  time = 0.13, size = 13, normalized size = 0.87 \begin {gather*} -4 \, x - e^{\left (5 \, \log \relax (x)^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*log(x)^3*exp(5*log(x)^4)-4*x)/x,x, algorithm="giac")

[Out]

-4*x - e^(5*log(x)^4)

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maple [A]  time = 0.02, size = 14, normalized size = 0.93

method	result	size
default	\(-4 x -{\mathrm e}^{5 \ln \relax (x )^{4}}\)	\(14\)
norman	\(-4 x -{\mathrm e}^{5 \ln \relax (x )^{4}}\)	\(14\)
risch	\(-4 x -{\mathrm e}^{5 \ln \relax (x )^{4}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-20*ln(x)^3*exp(5*ln(x)^4)-4*x)/x,x,method=_RETURNVERBOSE)

[Out]

-4*x-exp(5*ln(x)^4)

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maxima [A]  time = 0.36, size = 13, normalized size = 0.87 \begin {gather*} -4 \, x - e^{\left (5 \, \log \relax (x)^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*log(x)^3*exp(5*log(x)^4)-4*x)/x,x, algorithm="maxima")

[Out]

-4*x - e^(5*log(x)^4)

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mupad [B]  time = 7.27, size = 13, normalized size = 0.87 \begin {gather*} -4\,x-{\mathrm {e}}^{5\,{\ln \relax (x)}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + 20*exp(5*log(x)^4)*log(x)^3)/x,x)

[Out]

- 4*x - exp(5*log(x)^4)

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sympy [A]  time = 0.25, size = 12, normalized size = 0.80 \begin {gather*} - 4 x - e^{5 \log {\relax (x )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-20*ln(x)**3*exp(5*ln(x)**4)-4*x)/x,x)

[Out]

-4*x - exp(5*log(x)**4)

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