3.102.96 \(\int \frac {4 e^4 x^3}{(-450+x^4) \log ^2(\frac {1}{225} (-450+x^4))} \, dx\)

Optimal. Leaf size=17 \[ -\frac {e^4}{\log \left (-2+\frac {x^4}{225}\right )} \]

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Rubi [A]  time = 0.10, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {12, 2475, 2390, 2302, 30} \begin {gather*} -\frac {e^4}{\log \left (\frac {x^4}{225}-2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^4*x^3)/((-450 + x^4)*Log[(-450 + x^4)/225]^2),x]

[Out]

-(E^4/Log[-2 + x^4/225])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2302

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

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (4 e^4\right ) \int \frac {x^3}{\left (-450+x^4\right ) \log ^2\left (\frac {1}{225} \left (-450+x^4\right )\right )} \, dx\\ &=e^4 \operatorname {Subst}\left (\int \frac {1}{(-450+x) \log ^2\left (\frac {1}{225} (-450+x)\right )} \, dx,x,x^4\right )\\ &=e^4 \operatorname {Subst}\left (\int \frac {1}{x \log ^2\left (\frac {x}{225}\right )} \, dx,x,-450+x^4\right )\\ &=e^4 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (-2+\frac {x^4}{225}\right )\right )\\ &=-\frac {e^4}{\log \left (-2+\frac {x^4}{225}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {e^4}{\log \left (-2+\frac {x^4}{225}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^4*x^3)/((-450 + x^4)*Log[(-450 + x^4)/225]^2),x]

[Out]

-(E^4/Log[-2 + x^4/225])

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fricas [A]  time = 0.58, size = 14, normalized size = 0.82 \begin {gather*} -\frac {e^{4}}{\log \left (\frac {1}{225} \, x^{4} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x^3*exp(4)/(x^4-450)/log(1/225*x^4-2)^2,x, algorithm="fricas")

[Out]

-e^4/log(1/225*x^4 - 2)

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giac [A]  time = 0.16, size = 14, normalized size = 0.82 \begin {gather*} -\frac {e^{4}}{\log \left (\frac {1}{225} \, x^{4} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x^3*exp(4)/(x^4-450)/log(1/225*x^4-2)^2,x, algorithm="giac")

[Out]

-e^4/log(1/225*x^4 - 2)

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maple [A]  time = 0.08, size = 15, normalized size = 0.88

method	result	size
norman	\(-\frac {{\mathrm e}^{4}}{\ln \left (\frac {x^{4}}{225}-2\right )}\)	\(15\)
risch	\(-\frac {{\mathrm e}^{4}}{\ln \left (\frac {x^{4}}{225}-2\right )}\)	\(15\)
default	\(-\frac {{\mathrm e}^{4}}{-2 \ln \left (15\right )+\ln \left (x^{4}-450\right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x^3*exp(4)/(x^4-450)/ln(1/225*x^4-2)^2,x,method=_RETURNVERBOSE)

[Out]

-exp(4)/ln(1/225*x^4-2)

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maxima [A]  time = 0.44, size = 22, normalized size = 1.29 \begin {gather*} \frac {e^{4}}{2 \, \log \relax (5) + 2 \, \log \relax (3) - \log \left (x^{4} - 450\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x^3*exp(4)/(x^4-450)/log(1/225*x^4-2)^2,x, algorithm="maxima")

[Out]

e^4/(2*log(5) + 2*log(3) - log(x^4 - 450))

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mupad [B]  time = 0.28, size = 14, normalized size = 0.82 \begin {gather*} -\frac {{\mathrm {e}}^4}{\ln \left (\frac {x^4}{225}-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*exp(4))/(log(x^4/225 - 2)^2*(x^4 - 450)),x)

[Out]

-exp(4)/log(x^4/225 - 2)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.71 \begin {gather*} - \frac {e^{4}}{\log {\left (\frac {x^{4}}{225} - 2 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x**3*exp(4)/(x**4-450)/ln(1/225*x**4-2)**2,x)

[Out]

-exp(4)/log(x**4/225 - 2)

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