3.64.58 \(\int \frac {-9 e^{10}+6 x-13 x^2+3 x^4+e^5 (-6+12 x)+(6+18 e^5-12 x) \log (x)-9 \log ^2(x)}{x^4} \, dx\)

Optimal. Leaf size=33 \[ \frac {4+3 \left (2+x+x (5+x)+\frac {\left (e^5-x-\log (x)\right )^2}{x^2}\right )}{x} \]

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Rubi [B]  time = 0.14, antiderivative size = 98, normalized size of antiderivative = 2.97, number of steps used = 10, number of rules used = 6, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {14, 43, 2334, 12, 2305, 2304} \begin {gather*} \frac {e^5 \left (2+3 e^5\right )}{x^3}-\frac {2 \left (1+3 e^5\right )}{3 x^3}+\frac {2}{3 x^3}+\frac {3 \log ^2(x)}{x^3}+\frac {2 \log (x)}{x^3}-\frac {3 \left (1+2 e^5\right )}{x^2}+\frac {3}{x^2}-2 \left (\frac {1+3 e^5}{x^3}-\frac {3}{x^2}\right ) \log (x)+3 x+\frac {13}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9*E^10 + 6*x - 13*x^2 + 3*x^4 + E^5*(-6 + 12*x) + (6 + 18*E^5 - 12*x)*Log[x] - 9*Log[x]^2)/x^4,x]

[Out]

2/(3*x^3) - (2*(1 + 3*E^5))/(3*x^3) + (E^5*(2 + 3*E^5))/x^3 + 3/x^2 - (3*(1 + 2*E^5))/x^2 + 13/x + 3*x - 2*((1
+ 3*E^5)/x^3 - 3/x^2)*Log[x] + (2*Log[x])/x^3 + (3*Log[x]^2)/x^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-3 e^5 \left (2+3 e^5\right )+6 \left (1+2 e^5\right ) x-13 x^2+3 x^4}{x^4}+\frac {6 \left (1+3 e^5-2 x\right ) \log (x)}{x^4}-\frac {9 \log ^2(x)}{x^4}\right ) \, dx\\ &=6 \int \frac {\left (1+3 e^5-2 x\right ) \log (x)}{x^4} \, dx-9 \int \frac {\log ^2(x)}{x^4} \, dx+\int \frac {-3 e^5 \left (2+3 e^5\right )+6 \left (1+2 e^5\right ) x-13 x^2+3 x^4}{x^4} \, dx\\ &=-2 \left (\frac {1+3 e^5}{x^3}-\frac {3}{x^2}\right ) \log (x)+\frac {3 \log ^2(x)}{x^3}-6 \int \frac {-1-3 e^5+3 x}{3 x^4} \, dx-6 \int \frac {\log (x)}{x^4} \, dx+\int \left (3-\frac {3 e^5 \left (2+3 e^5\right )}{x^4}+\frac {6 \left (1+2 e^5\right )}{x^3}-\frac {13}{x^2}\right ) \, dx\\ &=\frac {2}{3 x^3}+\frac {e^5 \left (2+3 e^5\right )}{x^3}-\frac {3 \left (1+2 e^5\right )}{x^2}+\frac {13}{x}+3 x-2 \left (\frac {1+3 e^5}{x^3}-\frac {3}{x^2}\right ) \log (x)+\frac {2 \log (x)}{x^3}+\frac {3 \log ^2(x)}{x^3}-2 \int \frac {-1-3 e^5+3 x}{x^4} \, dx\\ &=\frac {2}{3 x^3}+\frac {e^5 \left (2+3 e^5\right )}{x^3}-\frac {3 \left (1+2 e^5\right )}{x^2}+\frac {13}{x}+3 x-2 \left (\frac {1+3 e^5}{x^3}-\frac {3}{x^2}\right ) \log (x)+\frac {2 \log (x)}{x^3}+\frac {3 \log ^2(x)}{x^3}-2 \int \left (\frac {-1-3 e^5}{x^4}+\frac {3}{x^3}\right ) \, dx\\ &=\frac {2}{3 x^3}-\frac {2 \left (1+3 e^5\right )}{3 x^3}+\frac {e^5 \left (2+3 e^5\right )}{x^3}+\frac {3}{x^2}-\frac {3 \left (1+2 e^5\right )}{x^2}+\frac {13}{x}+3 x-2 \left (\frac {1+3 e^5}{x^3}-\frac {3}{x^2}\right ) \log (x)+\frac {2 \log (x)}{x^3}+\frac {3 \log ^2(x)}{x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 51, normalized size = 1.55 \begin {gather*} \frac {3 e^{10}}{x^3}-\frac {6 e^5}{x^2}+\frac {13}{x}+3 x-\frac {6 e^5 \log (x)}{x^3}+\frac {6 \log (x)}{x^2}+\frac {3 \log ^2(x)}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*E^10 + 6*x - 13*x^2 + 3*x^4 + E^5*(-6 + 12*x) + (6 + 18*E^5 - 12*x)*Log[x] - 9*Log[x]^2)/x^4,x]

[Out]

(3*E^10)/x^3 - (6*E^5)/x^2 + 13/x + 3*x - (6*E^5*Log[x])/x^3 + (6*Log[x])/x^2 + (3*Log[x]^2)/x^3

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fricas [A]  time = 0.55, size = 40, normalized size = 1.21 \begin {gather*} \frac {3 \, x^{4} + 13 \, x^{2} - 6 \, x e^{5} + 6 \, {\left (x - e^{5}\right )} \log \relax (x) + 3 \, \log \relax (x)^{2} + 3 \, e^{10}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*log(x)^2+(18*exp(5)-12*x+6)*log(x)-9*exp(5)^2+(12*x-6)*exp(5)+3*x^4-13*x^2+6*x)/x^4,x, algorithm
="fricas")

[Out]

(3*x^4 + 13*x^2 - 6*x*e^5 + 6*(x - e^5)*log(x) + 3*log(x)^2 + 3*e^10)/x^3

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giac [A]  time = 0.14, size = 41, normalized size = 1.24 \begin {gather*} \frac {3 \, x^{4} + 13 \, x^{2} - 6 \, x e^{5} + 6 \, x \log \relax (x) - 6 \, e^{5} \log \relax (x) + 3 \, \log \relax (x)^{2} + 3 \, e^{10}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*log(x)^2+(18*exp(5)-12*x+6)*log(x)-9*exp(5)^2+(12*x-6)*exp(5)+3*x^4-13*x^2+6*x)/x^4,x, algorithm
="giac")

[Out]

(3*x^4 + 13*x^2 - 6*x*e^5 + 6*x*log(x) - 6*e^5*log(x) + 3*log(x)^2 + 3*e^10)/x^3

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maple [A]  time = 0.07, size = 44, normalized size = 1.33

method	result	size
norman	\(\frac {13 x^{2}+3 x^{4}+3 \,{\mathrm e}^{10}+3 \ln \relax (x )^{2}-6 x \,{\mathrm e}^{5}+6 x \ln \relax (x )-6 \,{\mathrm e}^{5} \ln \relax (x )}{x^{3}}\)	\(44\)
risch	\(\frac {3 \ln \relax (x )^{2}}{x^{3}}-\frac {6 \left ({\mathrm e}^{5}-x \right ) \ln \relax (x )}{x^{3}}+\frac {3 x^{4}+3 \,{\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+13 x^{2}}{x^{3}}\)	\(48\)
default	\(3 x +\frac {3 \ln \relax (x )^{2}}{x^{3}}+18 \,{\mathrm e}^{5} \left (-\frac {\ln \relax (x )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )+\frac {6 \ln \relax (x )}{x^{2}}+\frac {3 \,{\mathrm e}^{10}}{x^{3}}-\frac {6 \,{\mathrm e}^{5}}{x^{2}}+\frac {13}{x}+\frac {2 \,{\mathrm e}^{5}}{x^{3}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-9*ln(x)^2+(18*exp(5)-12*x+6)*ln(x)-9*exp(5)^2+(12*x-6)*exp(5)+3*x^4-13*x^2+6*x)/x^4,x,method=_RETURNVERB
OSE)

[Out]

(13*x^2+3*x^4+3*exp(5)^2+3*ln(x)^2-6*x*exp(5)+6*x*ln(x)-6*exp(5)*ln(x))/x^3

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maxima [B]  time = 0.37, size = 81, normalized size = 2.45 \begin {gather*} -2 \, {\left (\frac {3 \, \log \relax (x)}{x^{3}} + \frac {1}{x^{3}}\right )} e^{5} + 3 \, x + \frac {13}{x} - \frac {6 \, e^{5}}{x^{2}} + \frac {6 \, \log \relax (x)}{x^{2}} + \frac {9 \, \log \relax (x)^{2} + 6 \, \log \relax (x) + 2}{3 \, x^{3}} + \frac {3 \, e^{10}}{x^{3}} + \frac {2 \, e^{5}}{x^{3}} - \frac {2 \, \log \relax (x)}{x^{3}} - \frac {2}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*log(x)^2+(18*exp(5)-12*x+6)*log(x)-9*exp(5)^2+(12*x-6)*exp(5)+3*x^4-13*x^2+6*x)/x^4,x, algorithm
="maxima")

[Out]

-2*(3*log(x)/x^3 + 1/x^3)*e^5 + 3*x + 13/x - 6*e^5/x^2 + 6*log(x)/x^2 + 1/3*(9*log(x)^2 + 6*log(x) + 2)/x^3 +
3*e^10/x^3 + 2*e^5/x^3 - 2*log(x)/x^3 - 2/3/x^3

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mupad [B]  time = 4.27, size = 42, normalized size = 1.27 \begin {gather*} 3\,x+\frac {3\,{\mathrm {e}}^{10}-x\,\left (6\,{\mathrm {e}}^5-6\,\ln \relax (x)\right )+3\,{\ln \relax (x)}^2-6\,{\mathrm {e}}^5\,\ln \relax (x)+13\,x^2}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - 9*exp(10) + log(x)*(18*exp(5) - 12*x + 6) - 9*log(x)^2 - 13*x^2 + 3*x^4 + exp(5)*(12*x - 6))/x^4,x)

[Out]

3*x + (3*exp(10) - x*(6*exp(5) - 6*log(x)) + 3*log(x)^2 - 6*exp(5)*log(x) + 13*x^2)/x^3

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sympy [A]  time = 0.24, size = 48, normalized size = 1.45 \begin {gather*} 3 x + \frac {\left (6 x - 6 e^{5}\right ) \log {\relax (x )}}{x^{3}} + \frac {13 x^{2} - 6 x e^{5} + 3 e^{10}}{x^{3}} + \frac {3 \log {\relax (x )}^{2}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*ln(x)**2+(18*exp(5)-12*x+6)*ln(x)-9*exp(5)**2+(12*x-6)*exp(5)+3*x**4-13*x**2+6*x)/x**4,x)

[Out]

3*x + (6*x - 6*exp(5))*log(x)/x**3 + (13*x**2 - 6*x*exp(5) + 3*exp(10))/x**3 + 3*log(x)**2/x**3

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