3.20.20 \(\int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-3375 x^3-2025 x^4-405 x^5-27 x^6+e^6 (1+3 x+3 x^2+x^3)+e^4 (-45 x-99 x^2-63 x^3-9 x^4)+e^2 (675 x^2+945 x^3+297 x^4+27 x^5)+(2025 x^3+810 x^4+81 x^5+e^4 (9 x+18 x^2+9 x^3)+e^2 (-270 x^2-324 x^3-54 x^4)) \log (5)+(-405 x^3-81 x^4+e^2 (27 x^2+27 x^3)) \log ^2(5)+27 x^3 \log ^3(5)} \, dx\)

Optimal. Leaf size=34 \[ \frac {-2+x}{x^2 \left (5+\frac {e^2 \left (-1+\frac {2-x}{3}\right )}{x}+x-\log (5)\right )^2} \]

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Rubi [B]  time = 0.77, antiderivative size = 273, normalized size of antiderivative = 8.03, number of steps used = 10, number of rules used = 6, integrand size = 213, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6, 2074, 614, 618, 206, 638} \begin {gather*} \frac {27 \left (e^4+9 \left (25+\log ^2(5)-14 \log (5)+\log (625)\right )-6 e^2 (3-\log (5))\right ) \left (6 x-e^2+15-3 \log (5)\right )}{\left (e^4-6 e^2 (3-\log (5))+9 (5-\log (5))^2\right )^2 \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )}-\frac {9 \left (-\left (x \left (e^4+9 \left (25+\log ^2(5)-14 \log (5)+\log (625)\right )-6 e^2 (3-\log (5))\right )\right )+2 e^4-3 e^2 (12-\log (625))+9 (5-\log (5)) (10-\log (25))\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )^2}-\frac {27 \left (6 x-e^2+15-3 \log (5)\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-540 + E^2*(45 - 9*x) - 81*x + 81*x^2 + (108 - 27*x)*Log[5])/(-3375*x^3 - 2025*x^4 - 405*x^5 - 27*x^6 + E
^6*(1 + 3*x + 3*x^2 + x^3) + E^4*(-45*x - 99*x^2 - 63*x^3 - 9*x^4) + E^2*(675*x^2 + 945*x^3 + 297*x^4 + 27*x^5
) + (2025*x^3 + 810*x^4 + 81*x^5 + E^4*(9*x + 18*x^2 + 9*x^3) + E^2*(-270*x^2 - 324*x^3 - 54*x^4))*Log[5] + (-
405*x^3 - 81*x^4 + E^2*(27*x^2 + 27*x^3))*Log[5]^2 + 27*x^3*Log[5]^3),x]

[Out]

(-27*(15 - E^2 + 6*x - 3*Log[5]))/((12*E^2 + (E^2 - 3*(5 - Log[5]))^2)*(E^2 - 3*x^2 - x*(15 - E^2 - 3*Log[5]))
) + (27*(15 - E^2 + 6*x - 3*Log[5])*(E^4 - 6*E^2*(3 - Log[5]) + 9*(25 - 14*Log[5] + Log[5]^2 + Log[625])))/((E
^2 - 3*x^2 - x*(15 - E^2 - 3*Log[5]))*(E^4 - 6*E^2*(3 - Log[5]) + 9*(5 - Log[5])^2)^2) - (9*(2*E^4 + 9*(5 - Lo
g[5])*(10 - Log[25]) - 3*E^2*(12 - Log[625]) - x*(E^4 - 6*E^2*(3 - Log[5]) + 9*(25 - 14*Log[5] + Log[5]^2 + Lo
g[625]))))/((12*E^2 + (E^2 - 3*(5 - Log[5]))^2)*(E^2 - 3*x^2 - x*(15 - E^2 - 3*Log[5]))^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+x^3 \left (-3375+27 \log ^3(5)\right )} \, dx\\ &=\int \left (-\frac {27}{\left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}+\frac {18 \left (-30+4 e^2-x \left (27-e^2-3 \log (5)\right )+3 \log (25)\right )}{\left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^3}\right ) \, dx\\ &=18 \int \frac {-30+4 e^2-x \left (27-e^2-3 \log (5)\right )+3 \log (25)}{\left (e^2-3 x^2+x \left (-15+e^2+3 \log (5)\right )\right )^3} \, dx-27 \int \frac {1}{\left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2} \, dx\\ &=-\frac {27 \left (15-e^2+6 x-3 \log (5)\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )}-\frac {9 \left (2 e^4+9 (5-\log (5)) (10-\log (25))-3 e^2 (12-\log (625))-x \left (e^4-6 e^2 (3-\log (5))+9 \left (25-14 \log (5)+\log ^2(5)+\log (625)\right )\right )\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}+27 \int \frac {1}{\left (e^2-3 x^2+x \left (-15+e^2+3 \log (5)\right )\right )^2} \, dx-\frac {162 \int \frac {1}{e^2-3 x^2+x \left (-15+e^2+3 \log (5)\right )} \, dx}{12 e^2+\left (e^2-3 (5-\log (5))\right )^2}\\ &=-\frac {9 \left (2 e^4+9 (5-\log (5)) (10-\log (25))-3 e^2 (12-\log (625))-x \left (e^4-6 e^2 (3-\log (5))+9 \left (25-14 \log (5)+\log ^2(5)+\log (625)\right )\right )\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}+\frac {162 \int \frac {1}{e^2-3 x^2+x \left (-15+e^2+3 \log (5)\right )} \, dx}{12 e^2+\left (e^2-3 (5-\log (5))\right )^2}+\frac {324 \operatorname {Subst}\left (\int \frac {1}{12 e^2-x^2+\left (e^2-3 (5-\log (5))\right )^2} \, dx,x,-15+e^2-6 x+3 \log (5)\right )}{12 e^2+\left (e^2-3 (5-\log (5))\right )^2}\\ &=\frac {324 \tanh ^{-1}\left (\frac {e^2-3 (5+2 x-\log (5))}{\sqrt {e^4-6 e^2 (3-\log (5))+9 (5-\log (5))^2}}\right )}{\left (e^4-6 e^2 (3-\log (5))+9 (5-\log (5))^2\right )^{3/2}}-\frac {9 \left (2 e^4+9 (5-\log (5)) (10-\log (25))-3 e^2 (12-\log (625))-x \left (e^4-6 e^2 (3-\log (5))+9 \left (25-14 \log (5)+\log ^2(5)+\log (625)\right )\right )\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}-\frac {324 \operatorname {Subst}\left (\int \frac {1}{12 e^2-x^2+\left (e^2-3 (5-\log (5))\right )^2} \, dx,x,-15+e^2-6 x+3 \log (5)\right )}{12 e^2+\left (e^2-3 (5-\log (5))\right )^2}\\ &=-\frac {9 \left (2 e^4+9 (5-\log (5)) (10-\log (25))-3 e^2 (12-\log (625))-x \left (e^4-6 e^2 (3-\log (5))+9 \left (25-14 \log (5)+\log ^2(5)+\log (625)\right )\right )\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 27, normalized size = 0.79 \begin {gather*} -\frac {9 (2-x)}{\left (e^2 (1+x)-3 x (5+x-\log (5))\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-540 + E^2*(45 - 9*x) - 81*x + 81*x^2 + (108 - 27*x)*Log[5])/(-3375*x^3 - 2025*x^4 - 405*x^5 - 27*x
^6 + E^6*(1 + 3*x + 3*x^2 + x^3) + E^4*(-45*x - 99*x^2 - 63*x^3 - 9*x^4) + E^2*(675*x^2 + 945*x^3 + 297*x^4 +
27*x^5) + (2025*x^3 + 810*x^4 + 81*x^5 + E^4*(9*x + 18*x^2 + 9*x^3) + E^2*(-270*x^2 - 324*x^3 - 54*x^4))*Log[5
] + (-405*x^3 - 81*x^4 + E^2*(27*x^2 + 27*x^3))*Log[5]^2 + 27*x^3*Log[5]^3),x]

[Out]

(-9*(2 - x))/(E^2*(1 + x) - 3*x*(5 + x - Log[5]))^2

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fricas [B]  time = 0.88, size = 83, normalized size = 2.44 \begin {gather*} \frac {9 \, {\left (x - 2\right )}}{9 \, x^{4} + 9 \, x^{2} \log \relax (5)^{2} + 90 \, x^{3} + 225 \, x^{2} + {\left (x^{2} + 2 \, x + 1\right )} e^{4} - 6 \, {\left (x^{3} + 6 \, x^{2} + 5 \, x\right )} e^{2} - 6 \, {\left (3 \, x^{3} + 15 \, x^{2} - {\left (x^{2} + x\right )} e^{2}\right )} \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-27*x+108)*log(5)+(-9*x+45)*exp(2)+81*x^2-81*x-540)/(27*x^3*log(5)^3+((27*x^3+27*x^2)*exp(2)-81*x^
4-405*x^3)*log(5)^2+((9*x^3+18*x^2+9*x)*exp(2)^2+(-54*x^4-324*x^3-270*x^2)*exp(2)+81*x^5+810*x^4+2025*x^3)*log
(5)+(x^3+3*x^2+3*x+1)*exp(2)^3+(-9*x^4-63*x^3-99*x^2-45*x)*exp(2)^2+(27*x^5+297*x^4+945*x^3+675*x^2)*exp(2)-27
*x^6-405*x^5-2025*x^4-3375*x^3),x, algorithm="fricas")

[Out]

9*(x - 2)/(9*x^4 + 9*x^2*log(5)^2 + 90*x^3 + 225*x^2 + (x^2 + 2*x + 1)*e^4 - 6*(x^3 + 6*x^2 + 5*x)*e^2 - 6*(3*
x^3 + 15*x^2 - (x^2 + x)*e^2)*log(5))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-27*x+108)*log(5)+(-9*x+45)*exp(2)+81*x^2-81*x-540)/(27*x^3*log(5)^3+((27*x^3+27*x^2)*exp(2)-81*x^
4-405*x^3)*log(5)^2+((9*x^3+18*x^2+9*x)*exp(2)^2+(-54*x^4-324*x^3-270*x^2)*exp(2)+81*x^5+810*x^4+2025*x^3)*log
(5)+(x^3+3*x^2+3*x+1)*exp(2)^3+(-9*x^4-63*x^3-99*x^2-45*x)*exp(2)^2+(27*x^5+297*x^4+945*x^3+675*x^2)*exp(2)-27
*x^6-405*x^5-2025*x^4-3375*x^3),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.79, size = 29, normalized size = 0.85

method	result	size
norman	\(\frac {9 x -18}{\left ({\mathrm e}^{2} x +3 x \ln \relax (5)-3 x^{2}+{\mathrm e}^{2}-15 x \right )^{2}}\)	\(29\)
risch	\(\frac {9 x -18}{x^{2} {\mathrm e}^{4}+6 \,{\mathrm e}^{2} \ln \relax (5) x^{2}-6 x^{3} {\mathrm e}^{2}+9 x^{2} \ln \relax (5)^{2}-18 x^{3} \ln \relax (5)+9 x^{4}+2 x \,{\mathrm e}^{4}+6 x \,{\mathrm e}^{2} \ln \relax (5)-36 x^{2} {\mathrm e}^{2}-90 x^{2} \ln \relax (5)+90 x^{3}+{\mathrm e}^{4}-30 \,{\mathrm e}^{2} x +225 x^{2}}\)	\(96\)
gosper	\(\frac {9 x -18}{x^{2} {\mathrm e}^{4}+6 \,{\mathrm e}^{2} \ln \relax (5) x^{2}-6 x^{3} {\mathrm e}^{2}+9 x^{2} \ln \relax (5)^{2}-18 x^{3} \ln \relax (5)+9 x^{4}+2 x \,{\mathrm e}^{4}+6 x \,{\mathrm e}^{2} \ln \relax (5)-36 x^{2} {\mathrm e}^{2}-90 x^{2} \ln \relax (5)+90 x^{3}+{\mathrm e}^{4}-30 \,{\mathrm e}^{2} x +225 x^{2}}\)	\(101\)
default	\(-3 \left (\munderset {\textit {\_R} =\RootOf \left (27 \textit {\_Z}^{6}+\left (-27 \,{\mathrm e}^{2}-81 \ln \relax (5)+405\right ) \textit {\_Z}^{5}+\left (54 \,{\mathrm e}^{2} \ln \relax (5)+81 \ln \relax (5)^{2}-297 \,{\mathrm e}^{2}-810 \ln \relax (5)+9 \,{\mathrm e}^{4}+2025\right ) \textit {\_Z}^{4}+\left (-27 \,{\mathrm e}^{2} \ln \relax (5)^{2}-27 \ln \relax (5)^{3}+324 \,{\mathrm e}^{2} \ln \relax (5)+405 \ln \relax (5)^{2}-9 \,{\mathrm e}^{4} \ln \relax (5)-945 \,{\mathrm e}^{2}-2025 \ln \relax (5)+63 \,{\mathrm e}^{4}-{\mathrm e}^{6}+3375\right ) \textit {\_Z}^{3}+\left (-27 \,{\mathrm e}^{2} \ln \relax (5)^{2}+270 \,{\mathrm e}^{2} \ln \relax (5)-18 \,{\mathrm e}^{4} \ln \relax (5)-675 \,{\mathrm e}^{2}+99 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{6}\right ) \textit {\_Z}^{2}+\left (-9 \,{\mathrm e}^{4} \ln \relax (5)+45 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{6}\right ) \textit {\_Z} -{\mathrm e}^{6}\right )}{\sum }\frac {\left (-9 \textit {\_R}^{2}+\left ({\mathrm e}^{2}+3 \ln \relax (5)+9\right ) \textit {\_R} -5 \,{\mathrm e}^{2}-12 \ln \relax (5)+60\right ) \ln \left (x -\textit {\_R} \right )}{9 \,{\mathrm e}^{4} \ln \relax (5) \textit {\_R}^{2}-324 \,{\mathrm e}^{2} \ln \relax (5) \textit {\_R}^{2}+27 \,{\mathrm e}^{2} \ln \relax (5)^{2} \textit {\_R}^{2}-72 \,{\mathrm e}^{2} \ln \relax (5) \textit {\_R}^{3}+18 \,{\mathrm e}^{2} \ln \relax (5)^{2} \textit {\_R} -180 \textit {\_R} \,{\mathrm e}^{2} \ln \relax (5)+12 \textit {\_R} \,{\mathrm e}^{4} \ln \relax (5)+{\mathrm e}^{6}-15 \,{\mathrm e}^{4}+3 \,{\mathrm e}^{4} \ln \relax (5)-63 \textit {\_R}^{2} {\mathrm e}^{4}+2 \textit {\_R} \,{\mathrm e}^{6}-66 \textit {\_R} \,{\mathrm e}^{4}+450 \,{\mathrm e}^{2} \textit {\_R} -12 \textit {\_R}^{3} {\mathrm e}^{4}+945 \textit {\_R}^{2} {\mathrm e}^{2}-675 \textit {\_R}^{4}-2700 \textit {\_R}^{3}-3375 \textit {\_R}^{2}-108 \textit {\_R}^{3} \ln \relax (5)^{2}+\textit {\_R}^{2} {\mathrm e}^{6}+2025 \textit {\_R}^{2} \ln \relax (5)-405 \textit {\_R}^{2} \ln \relax (5)^{2}+396 \textit {\_R}^{3} {\mathrm e}^{2}+45 \textit {\_R}^{4} {\mathrm e}^{2}+1080 \textit {\_R}^{3} \ln \relax (5)+135 \textit {\_R}^{4} \ln \relax (5)+27 \textit {\_R}^{2} \ln \relax (5)^{3}-54 \textit {\_R}^{5}}\right )\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-27*x+108)*ln(5)+(-9*x+45)*exp(2)+81*x^2-81*x-540)/(27*x^3*ln(5)^3+((27*x^3+27*x^2)*exp(2)-81*x^4-405*x^
3)*ln(5)^2+((9*x^3+18*x^2+9*x)*exp(2)^2+(-54*x^4-324*x^3-270*x^2)*exp(2)+81*x^5+810*x^4+2025*x^3)*ln(5)+(x^3+3
*x^2+3*x+1)*exp(2)^3+(-9*x^4-63*x^3-99*x^2-45*x)*exp(2)^2+(27*x^5+297*x^4+945*x^3+675*x^2)*exp(2)-27*x^6-405*x
^5-2025*x^4-3375*x^3),x,method=_RETURNVERBOSE)

[Out]

(9*x-18)/(exp(2)*x+3*x*ln(5)-3*x^2+exp(2)-15*x)^2

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maxima [B]  time = 0.49, size = 70, normalized size = 2.06 \begin {gather*} \frac {9 \, {\left (x - 2\right )}}{9 \, x^{4} - 6 \, x^{3} {\left (e^{2} + 3 \, \log \relax (5) - 15\right )} + {\left (6 \, {\left (e^{2} - 15\right )} \log \relax (5) + 9 \, \log \relax (5)^{2} + e^{4} - 36 \, e^{2} + 225\right )} x^{2} + 2 \, {\left (3 \, e^{2} \log \relax (5) + e^{4} - 15 \, e^{2}\right )} x + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-27*x+108)*log(5)+(-9*x+45)*exp(2)+81*x^2-81*x-540)/(27*x^3*log(5)^3+((27*x^3+27*x^2)*exp(2)-81*x^
4-405*x^3)*log(5)^2+((9*x^3+18*x^2+9*x)*exp(2)^2+(-54*x^4-324*x^3-270*x^2)*exp(2)+81*x^5+810*x^4+2025*x^3)*log
(5)+(x^3+3*x^2+3*x+1)*exp(2)^3+(-9*x^4-63*x^3-99*x^2-45*x)*exp(2)^2+(27*x^5+297*x^4+945*x^3+675*x^2)*exp(2)-27
*x^6-405*x^5-2025*x^4-3375*x^3),x, algorithm="maxima")

[Out]

9*(x - 2)/(9*x^4 - 6*x^3*(e^2 + 3*log(5) - 15) + (6*(e^2 - 15)*log(5) + 9*log(5)^2 + e^4 - 36*e^2 + 225)*x^2 +
2*(3*e^2*log(5) + e^4 - 15*e^2)*x + e^4)

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mupad [B]  time = 1.92, size = 525, normalized size = 15.44 \begin {gather*} \frac {\frac {\left (2187\,\ln \left (125\right )\,\ln \left (625\right )-26244\,{\ln \relax (5)}^2\right )\,x^2}{774\,{\mathrm {e}}^4-8100\,{\mathrm {e}}^2-36\,{\mathrm {e}}^6+{\mathrm {e}}^8-40500\,\ln \relax (5)+\ln \left (5^{12\,{\mathrm {e}}^6-396\,{\mathrm {e}}^4}\right )+5940\,{\mathrm {e}}^2\,\ln \relax (5)-1404\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+108\,{\mathrm {e}}^2\,{\ln \relax (5)}^3+54\,{\mathrm {e}}^4\,{\ln \relax (5)}^2+12150\,{\ln \relax (5)}^2-1620\,{\ln \relax (5)}^3+81\,{\ln \relax (5)}^4+50625}+\frac {9\,\left (774\,{\mathrm {e}}^4-8100\,{\mathrm {e}}^2-36\,{\mathrm {e}}^6+{\mathrm {e}}^8-24300\,\ln \relax (5)-4050\,\ln \left (625\right )+\ln \left (5^{3240\,{\mathrm {e}}^2-396\,{\mathrm {e}}^4+12\,{\mathrm {e}}^6-648\,{\ln \relax (5)}^2}\right )+2700\,{\mathrm {e}}^2\,\ln \relax (5)+1620\,\ln \relax (5)\,\ln \left (625\right )+\ln \left (625\right )\,\ln \left (\frac {1}{5^{108\,{\mathrm {e}}^2}}\right )-972\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+108\,{\mathrm {e}}^2\,{\ln \relax (5)}^3+54\,{\mathrm {e}}^4\,{\ln \relax (5)}^2+5670\,{\ln \relax (5)}^2-972\,{\ln \relax (5)}^3+81\,{\ln \relax (5)}^4+50625\right )\,x}{774\,{\mathrm {e}}^4-8100\,{\mathrm {e}}^2-36\,{\mathrm {e}}^6+{\mathrm {e}}^8-40500\,\ln \relax (5)+\ln \left (5^{12\,{\mathrm {e}}^6-396\,{\mathrm {e}}^4}\right )+5940\,{\mathrm {e}}^2\,\ln \relax (5)-1404\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+108\,{\mathrm {e}}^2\,{\ln \relax (5)}^3+54\,{\mathrm {e}}^4\,{\ln \relax (5)}^2+12150\,{\ln \relax (5)}^2-1620\,{\ln \relax (5)}^3+81\,{\ln \relax (5)}^4+50625}-\frac {\frac {9\,\ln \left (5^{2700\,{\mathrm {e}}^2-180\,{\mathrm {e}}^4+48\,{\mathrm {e}}^6+324\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+324\,{\ln \relax (5)}^3}\right )}{2}-145800\,{\mathrm {e}}^2+13932\,{\mathrm {e}}^4-648\,{\mathrm {e}}^6+18\,{\mathrm {e}}^8-546750\,\ln \relax (5)-\frac {91125\,\ln \left (625\right )}{2}+94770\,{\mathrm {e}}^2\,\ln \relax (5)-6318\,{\mathrm {e}}^4\,\ln \relax (5)+\frac {54675\,\ln \relax (5)\,\ln \left (625\right )}{2}+\frac {9\,\ln \left (625\right )\,\ln \left (5^{27\,{\mathrm {e}}^4-540\,{\mathrm {e}}^2}\right )}{2}-15552\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+486\,{\mathrm {e}}^2\,{\ln \relax (5)}^3+486\,{\mathrm {e}}^4\,{\ln \relax (5)}^2-\frac {10935\,{\ln \relax (5)}^2\,\ln \left (625\right )}{2}+109350\,{\ln \relax (5)}^2-7290\,{\ln \relax (5)}^3+911250}{{\left (\ln \left (\frac {{15625}^{{\mathrm {e}}^2}}{807793566946316088741610050849573099185363389551639556884765625}\right )-18\,{\mathrm {e}}^2+{\mathrm {e}}^4+9\,{\ln \relax (5)}^2+225\right )}^2}}{9\,x^4+\left (90-\ln \left (3814697265625\right )-6\,{\mathrm {e}}^2\right )\,x^3+\left (\ln \left (\frac {{15625}^{{\mathrm {e}}^2}}{807793566946316088741610050849573099185363389551639556884765625}\right )-36\,{\mathrm {e}}^2+{\mathrm {e}}^4+9\,{\ln \relax (5)}^2+225\right )\,x^2+\left (2\,{\mathrm {e}}^4-30\,{\mathrm {e}}^2+\ln \left (5^{6\,{\mathrm {e}}^2}\right )\right )\,x+{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((81*x + log(5)*(27*x - 108) - 81*x^2 + exp(2)*(9*x - 45) + 540)/(log(5)^2*(405*x^3 - exp(2)*(27*x^2 + 27*x
^3) + 81*x^4) - 27*x^3*log(5)^3 - exp(6)*(3*x + 3*x^2 + x^3 + 1) - log(5)*(exp(4)*(9*x + 18*x^2 + 9*x^3) - exp
(2)*(270*x^2 + 324*x^3 + 54*x^4) + 2025*x^3 + 810*x^4 + 81*x^5) + exp(4)*(45*x + 99*x^2 + 63*x^3 + 9*x^4) + 33
75*x^3 + 2025*x^4 + 405*x^5 + 27*x^6 - exp(2)*(675*x^2 + 945*x^3 + 297*x^4 + 27*x^5)),x)

[Out]

((x^2*(2187*log(125)*log(625) - 26244*log(5)^2))/(774*exp(4) - 8100*exp(2) - 36*exp(6) + exp(8) - 40500*log(5)
+ log(5^(12*exp(6) - 396*exp(4))) + 5940*exp(2)*log(5) - 1404*exp(2)*log(5)^2 + 108*exp(2)*log(5)^3 + 54*exp(
4)*log(5)^2 + 12150*log(5)^2 - 1620*log(5)^3 + 81*log(5)^4 + 50625) - ((9*log(5^(2700*exp(2) - 180*exp(4) + 48
*exp(6) + 324*exp(2)*log(5)^2 + 324*log(5)^3)))/2 - 145800*exp(2) + 13932*exp(4) - 648*exp(6) + 18*exp(8) - 54
6750*log(5) - (91125*log(625))/2 + 94770*exp(2)*log(5) - 6318*exp(4)*log(5) + (54675*log(5)*log(625))/2 + (9*l
og(625)*log(5^(27*exp(4) - 540*exp(2))))/2 - 15552*exp(2)*log(5)^2 + 486*exp(2)*log(5)^3 + 486*exp(4)*log(5)^2
- (10935*log(5)^2*log(625))/2 + 109350*log(5)^2 - 7290*log(5)^3 + 911250)/(log(15625^exp(2)/80779356694631608
8741610050849573099185363389551639556884765625) - 18*exp(2) + exp(4) + 9*log(5)^2 + 225)^2 + (9*x*(774*exp(4)
- 8100*exp(2) - 36*exp(6) + exp(8) - 24300*log(5) - 4050*log(625) + log(5^(3240*exp(2) - 396*exp(4) + 12*exp(6
) - 648*log(5)^2)) + 2700*exp(2)*log(5) + 1620*log(5)*log(625) + log(625)*log(1/5^(108*exp(2))) - 972*exp(2)*l
og(5)^2 + 108*exp(2)*log(5)^3 + 54*exp(4)*log(5)^2 + 5670*log(5)^2 - 972*log(5)^3 + 81*log(5)^4 + 50625))/(774
*exp(4) - 8100*exp(2) - 36*exp(6) + exp(8) - 40500*log(5) + log(5^(12*exp(6) - 396*exp(4))) + 5940*exp(2)*log(
5) - 1404*exp(2)*log(5)^2 + 108*exp(2)*log(5)^3 + 54*exp(4)*log(5)^2 + 12150*log(5)^2 - 1620*log(5)^3 + 81*log
(5)^4 + 50625))/(exp(4) + x*(2*exp(4) - 30*exp(2) + log(5^(6*exp(2)))) + 9*x^4 + x^2*(log(15625^exp(2)/8077935
66946316088741610050849573099185363389551639556884765625) - 36*exp(2) + exp(4) + 9*log(5)^2 + 225) - x^3*(6*ex
p(2) + log(3814697265625) - 90))

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sympy [B]  time = 4.83, size = 83, normalized size = 2.44 \begin {gather*} - \frac {18 - 9 x}{9 x^{4} + x^{3} \left (- 6 e^{2} - 18 \log {\relax (5 )} + 90\right ) + x^{2} \left (- 36 e^{2} - 90 \log {\relax (5 )} + 9 \log {\relax (5 )}^{2} + e^{4} + 6 e^{2} \log {\relax (5 )} + 225\right ) + x \left (- 30 e^{2} + 6 e^{2} \log {\relax (5 )} + 2 e^{4}\right ) + e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-27*x+108)*ln(5)+(-9*x+45)*exp(2)+81*x**2-81*x-540)/(27*x**3*ln(5)**3+((27*x**3+27*x**2)*exp(2)-81
*x**4-405*x**3)*ln(5)**2+((9*x**3+18*x**2+9*x)*exp(2)**2+(-54*x**4-324*x**3-270*x**2)*exp(2)+81*x**5+810*x**4+
2025*x**3)*ln(5)+(x**3+3*x**2+3*x+1)*exp(2)**3+(-9*x**4-63*x**3-99*x**2-45*x)*exp(2)**2+(27*x**5+297*x**4+945*
x**3+675*x**2)*exp(2)-27*x**6-405*x**5-2025*x**4-3375*x**3),x)

[Out]

-(18 - 9*x)/(9*x**4 + x**3*(-6*exp(2) - 18*log(5) + 90) + x**2*(-36*exp(2) - 90*log(5) + 9*log(5)**2 + exp(4)
+ 6*exp(2)*log(5) + 225) + x*(-30*exp(2) + 6*exp(2)*log(5) + 2*exp(4)) + exp(4))

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