3.26.86 \(\int \frac {e^{-x} (60+60 x-30 x^2-10 x^3+e^{2 x} (-45 x+15 x^2+55 x^3-15 x^4-15 x^5+5 x^6)+e^x (45 x+30 x^2-25 x^3-10 x^4+5 x^5) \log ^2(2))}{9 x^3+6 x^4-5 x^5-2 x^6+x^7} \, dx\)

Optimal. Leaf size=38 \[ \frac {5 \left (e^x-x-\frac {2 e^{-x}}{x \left (3+x-x^2\right )}-\log ^2(2)\right )}{x} \]

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Rubi [C]  time = 7.72, antiderivative size = 632, normalized size of antiderivative = 16.63, number of steps used = 95, number of rules used = 7, integrand size = 114, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6688, 12, 6742, 2197, 2177, 2178, 6728} \begin {gather*} -\frac {10}{117} \left (39+7 \sqrt {13}\right ) e^{\frac {1}{2} \left (\sqrt {13}-1\right )} \text {Ei}\left (\frac {1}{2} \left (-2 x-\sqrt {13}+1\right )\right )+\frac {20}{117} \left (13+4 \sqrt {13}\right ) e^{\frac {1}{2} \left (\sqrt {13}-1\right )} \text {Ei}\left (\frac {1}{2} \left (-2 x-\sqrt {13}+1\right )\right )+\frac {5}{39} \left (13+\sqrt {13}\right ) e^{\frac {1}{2} \left (\sqrt {13}-1\right )} \text {Ei}\left (\frac {1}{2} \left (-2 x-\sqrt {13}+1\right )\right )+\frac {35}{117} \left (1-\sqrt {13}\right ) e^{\frac {1}{2} \left (\sqrt {13}-1\right )} \text {Ei}\left (\frac {1}{2} \left (-2 x-\sqrt {13}+1\right )\right )+\frac {10 e^{\frac {1}{2} \left (\sqrt {13}-1\right )} \text {Ei}\left (\frac {1}{2} \left (-2 x-\sqrt {13}+1\right )\right )}{9 \sqrt {13}}-\frac {100}{117} e^{\frac {1}{2} \left (\sqrt {13}-1\right )} \text {Ei}\left (\frac {1}{2} \left (-2 x-\sqrt {13}+1\right )\right )+\frac {35}{117} \left (1+\sqrt {13}\right ) e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (-2 x+\sqrt {13}+1\right )\right )+\frac {5}{39} \left (13-\sqrt {13}\right ) e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (-2 x+\sqrt {13}+1\right )\right )+\frac {20}{117} \left (13-4 \sqrt {13}\right ) e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (-2 x+\sqrt {13}+1\right )\right )-\frac {10}{117} \left (39-7 \sqrt {13}\right ) e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (-2 x+\sqrt {13}+1\right )\right )-\frac {10 e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (-2 x+\sqrt {13}+1\right )\right )}{9 \sqrt {13}}-\frac {100}{117} e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (-2 x+\sqrt {13}+1\right )\right )-\frac {10 e^{-x}}{3 x^2}-\frac {70 \left (1-\sqrt {13}\right ) e^{-x}}{117 \left (-2 x-\sqrt {13}+1\right )}+\frac {200 e^{-x}}{117 \left (-2 x-\sqrt {13}+1\right )}-\frac {70 \left (1+\sqrt {13}\right ) e^{-x}}{117 \left (-2 x+\sqrt {13}+1\right )}+\frac {200 e^{-x}}{117 \left (-2 x+\sqrt {13}+1\right )}+\frac {10 e^{-x}}{9 x}+\frac {5 e^x}{x}-\frac {5 \log ^2(2)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(60 + 60*x - 30*x^2 - 10*x^3 + E^(2*x)*(-45*x + 15*x^2 + 55*x^3 - 15*x^4 - 15*x^5 + 5*x^6) + E^x*(45*x + 3
0*x^2 - 25*x^3 - 10*x^4 + 5*x^5)*Log[2]^2)/(E^x*(9*x^3 + 6*x^4 - 5*x^5 - 2*x^6 + x^7)),x]

[Out]

200/(117*E^x*(1 - Sqrt[13] - 2*x)) - (70*(1 - Sqrt[13]))/(117*E^x*(1 - Sqrt[13] - 2*x)) + 200/(117*E^x*(1 + Sq
rt[13] - 2*x)) - (70*(1 + Sqrt[13]))/(117*E^x*(1 + Sqrt[13] - 2*x)) - 10/(3*E^x*x^2) + 10/(9*E^x*x) + (5*E^x)/
x - (100*E^((-1 + Sqrt[13])/2)*ExpIntegralEi[(1 - Sqrt[13] - 2*x)/2])/117 + (10*E^((-1 + Sqrt[13])/2)*ExpInteg
ralEi[(1 - Sqrt[13] - 2*x)/2])/(9*Sqrt[13]) + (35*(1 - Sqrt[13])*E^((-1 + Sqrt[13])/2)*ExpIntegralEi[(1 - Sqrt
[13] - 2*x)/2])/117 + (5*(13 + Sqrt[13])*E^((-1 + Sqrt[13])/2)*ExpIntegralEi[(1 - Sqrt[13] - 2*x)/2])/39 + (20
*(13 + 4*Sqrt[13])*E^((-1 + Sqrt[13])/2)*ExpIntegralEi[(1 - Sqrt[13] - 2*x)/2])/117 - (10*(39 + 7*Sqrt[13])*E^
((-1 + Sqrt[13])/2)*ExpIntegralEi[(1 - Sqrt[13] - 2*x)/2])/117 - (100*E^(-1/2 - Sqrt[13]/2)*ExpIntegralEi[(1 +
Sqrt[13] - 2*x)/2])/117 - (10*E^(-1/2 - Sqrt[13]/2)*ExpIntegralEi[(1 + Sqrt[13] - 2*x)/2])/(9*Sqrt[13]) - (10
*(39 - 7*Sqrt[13])*E^(-1/2 - Sqrt[13]/2)*ExpIntegralEi[(1 + Sqrt[13] - 2*x)/2])/117 + (20*(13 - 4*Sqrt[13])*E^
(-1/2 - Sqrt[13]/2)*ExpIntegralEi[(1 + Sqrt[13] - 2*x)/2])/117 + (5*(13 - Sqrt[13])*E^(-1/2 - Sqrt[13]/2)*ExpI
ntegralEi[(1 + Sqrt[13] - 2*x)/2])/39 + (35*(1 + Sqrt[13])*E^(-1/2 - Sqrt[13]/2)*ExpIntegralEi[(1 + Sqrt[13] -
2*x)/2])/117 - (5*Log[2]^2)/x

Rule 12

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

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 6688

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{-x} \left (12+12 x-6 x^2-2 x^3+e^{2 x} (-1+x) x \left (3+x-x^2\right )^2+e^x x \left (3+x-x^2\right )^2 \log ^2(2)\right )}{x^3 \left (3+x-x^2\right )^2} \, dx\\ &=5 \int \frac {e^{-x} \left (12+12 x-6 x^2-2 x^3+e^{2 x} (-1+x) x \left (3+x-x^2\right )^2+e^x x \left (3+x-x^2\right )^2 \log ^2(2)\right )}{x^3 \left (3+x-x^2\right )^2} \, dx\\ &=5 \int \left (\frac {e^x (-1+x)}{x^2}-\frac {2 e^{-x}}{\left (-3-x+x^2\right )^2}+\frac {12 e^{-x}}{x^3 \left (-3-x+x^2\right )^2}+\frac {12 e^{-x}}{x^2 \left (-3-x+x^2\right )^2}-\frac {6 e^{-x}}{x \left (-3-x+x^2\right )^2}+\frac {\log ^2(2)}{x^2}\right ) \, dx\\ &=-\frac {5 \log ^2(2)}{x}+5 \int \frac {e^x (-1+x)}{x^2} \, dx-10 \int \frac {e^{-x}}{\left (-3-x+x^2\right )^2} \, dx-30 \int \frac {e^{-x}}{x \left (-3-x+x^2\right )^2} \, dx+60 \int \frac {e^{-x}}{x^3 \left (-3-x+x^2\right )^2} \, dx+60 \int \frac {e^{-x}}{x^2 \left (-3-x+x^2\right )^2} \, dx\\ &=\frac {5 e^x}{x}-\frac {5 \log ^2(2)}{x}-10 \int \left (\frac {4 e^{-x}}{13 \left (1+\sqrt {13}-2 x\right )^2}+\frac {4 e^{-x}}{13 \sqrt {13} \left (1+\sqrt {13}-2 x\right )}+\frac {4 e^{-x}}{13 \left (-1+\sqrt {13}+2 x\right )^2}+\frac {4 e^{-x}}{13 \sqrt {13} \left (-1+\sqrt {13}+2 x\right )}\right ) \, dx-30 \int \left (\frac {e^{-x}}{9 x}+\frac {e^{-x} (-1+x)}{3 \left (-3-x+x^2\right )^2}+\frac {e^{-x} (1-x)}{9 \left (-3-x+x^2\right )}\right ) \, dx+60 \int \left (\frac {e^{-x}}{9 x^3}-\frac {2 e^{-x}}{27 x^2}+\frac {e^{-x}}{9 x}+\frac {e^{-x} (-7+4 x)}{27 \left (-3-x+x^2\right )^2}+\frac {e^{-x} (5-3 x)}{27 \left (-3-x+x^2\right )}\right ) \, dx+60 \int \left (\frac {e^{-x}}{9 x^2}-\frac {2 e^{-x}}{27 x}+\frac {e^{-x} (4-x)}{9 \left (-3-x+x^2\right )^2}+\frac {e^{-x} (-5+2 x)}{27 \left (-3-x+x^2\right )}\right ) \, dx\\ &=\frac {5 e^x}{x}-\frac {5 \log ^2(2)}{x}+\frac {20}{9} \int \frac {e^{-x} (-7+4 x)}{\left (-3-x+x^2\right )^2} \, dx+\frac {20}{9} \int \frac {e^{-x} (5-3 x)}{-3-x+x^2} \, dx+\frac {20}{9} \int \frac {e^{-x} (-5+2 x)}{-3-x+x^2} \, dx-\frac {40}{13} \int \frac {e^{-x}}{\left (1+\sqrt {13}-2 x\right )^2} \, dx-\frac {40}{13} \int \frac {e^{-x}}{\left (-1+\sqrt {13}+2 x\right )^2} \, dx-\frac {10}{3} \int \frac {e^{-x}}{x} \, dx-\frac {10}{3} \int \frac {e^{-x} (1-x)}{-3-x+x^2} \, dx-\frac {40}{9} \int \frac {e^{-x}}{x^2} \, dx-\frac {40}{9} \int \frac {e^{-x}}{x} \, dx+\frac {20}{3} \int \frac {e^{-x}}{x^3} \, dx+\frac {20}{3} \int \frac {e^{-x}}{x^2} \, dx+\frac {20}{3} \int \frac {e^{-x}}{x} \, dx+\frac {20}{3} \int \frac {e^{-x} (4-x)}{\left (-3-x+x^2\right )^2} \, dx-10 \int \frac {e^{-x} (-1+x)}{\left (-3-x+x^2\right )^2} \, dx-\frac {40 \int \frac {e^{-x}}{1+\sqrt {13}-2 x} \, dx}{13 \sqrt {13}}-\frac {40 \int \frac {e^{-x}}{-1+\sqrt {13}+2 x} \, dx}{13 \sqrt {13}}\\ &=-\frac {20 e^{-x}}{13 \left (1-\sqrt {13}-2 x\right )}-\frac {20 e^{-x}}{13 \left (1+\sqrt {13}-2 x\right )}-\frac {10 e^{-x}}{3 x^2}-\frac {20 e^{-x}}{9 x}+\frac {5 e^x}{x}-\frac {20 e^{\frac {1}{2} \left (-1+\sqrt {13}\right )} \text {Ei}\left (\frac {1}{2} \left (1-\sqrt {13}-2 x\right )\right )}{13 \sqrt {13}}+\frac {20 e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (1+\sqrt {13}-2 x\right )\right )}{13 \sqrt {13}}-\frac {10 \text {Ei}(-x)}{9}-\frac {5 \log ^2(2)}{x}-\frac {20}{13} \int \frac {e^{-x}}{1+\sqrt {13}-2 x} \, dx+\frac {20}{13} \int \frac {e^{-x}}{-1+\sqrt {13}+2 x} \, dx+\frac {20}{9} \int \left (\frac {\left (-3+\frac {7}{\sqrt {13}}\right ) e^{-x}}{-1-\sqrt {13}+2 x}+\frac {\left (-3-\frac {7}{\sqrt {13}}\right ) e^{-x}}{-1+\sqrt {13}+2 x}\right ) \, dx+\frac {20}{9} \int \left (\frac {\left (2-\frac {8}{\sqrt {13}}\right ) e^{-x}}{-1-\sqrt {13}+2 x}+\frac {\left (2+\frac {8}{\sqrt {13}}\right ) e^{-x}}{-1+\sqrt {13}+2 x}\right ) \, dx+\frac {20}{9} \int \left (-\frac {7 e^{-x}}{\left (-3-x+x^2\right )^2}+\frac {4 e^{-x} x}{\left (-3-x+x^2\right )^2}\right ) \, dx-\frac {10}{3} \int \frac {e^{-x}}{x^2} \, dx-\frac {10}{3} \int \left (\frac {\left (-1+\frac {1}{\sqrt {13}}\right ) e^{-x}}{-1-\sqrt {13}+2 x}+\frac {\left (-1-\frac {1}{\sqrt {13}}\right ) e^{-x}}{-1+\sqrt {13}+2 x}\right ) \, dx+\frac {40}{9} \int \frac {e^{-x}}{x} \, dx-\frac {20}{3} \int \frac {e^{-x}}{x} \, dx+\frac {20}{3} \int \left (\frac {4 e^{-x}}{\left (-3-x+x^2\right )^2}-\frac {e^{-x} x}{\left (-3-x+x^2\right )^2}\right ) \, dx-10 \int \left (-\frac {e^{-x}}{\left (-3-x+x^2\right )^2}+\frac {e^{-x} x}{\left (-3-x+x^2\right )^2}\right ) \, dx\\ &=-\frac {20 e^{-x}}{13 \left (1-\sqrt {13}-2 x\right )}-\frac {20 e^{-x}}{13 \left (1+\sqrt {13}-2 x\right )}-\frac {10 e^{-x}}{3 x^2}+\frac {10 e^{-x}}{9 x}+\frac {5 e^x}{x}+\frac {10}{13} e^{\frac {1}{2} \left (-1+\sqrt {13}\right )} \text {Ei}\left (\frac {1}{2} \left (1-\sqrt {13}-2 x\right )\right )-\frac {20 e^{\frac {1}{2} \left (-1+\sqrt {13}\right )} \text {Ei}\left (\frac {1}{2} \left (1-\sqrt {13}-2 x\right )\right )}{13 \sqrt {13}}+\frac {10}{13} e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (1+\sqrt {13}-2 x\right )\right )+\frac {20 e^{-\frac {1}{2}-\frac {\sqrt {13}}{2}} \text {Ei}\left (\frac {1}{2} \left (1+\sqrt {13}-2 x\right )\right )}{13 \sqrt {13}}-\frac {10 \text {Ei}(-x)}{3}-\frac {5 \log ^2(2)}{x}+\frac {10}{3} \int \frac {e^{-x}}{x} \, dx-\frac {20}{3} \int \frac {e^{-x} x}{\left (-3-x+x^2\right )^2} \, dx+\frac {80}{9} \int \frac {e^{-x} x}{\left (-3-x+x^2\right )^2} \, dx+10 \int \frac {e^{-x}}{\left (-3-x+x^2\right )^2} \, dx-10 \int \frac {e^{-x} x}{\left (-3-x+x^2\right )^2} \, dx-\frac {140}{9} \int \frac {e^{-x}}{\left (-3-x+x^2\right )^2} \, dx+\frac {80}{3} \int \frac {e^{-x}}{\left (-3-x+x^2\right )^2} \, dx-\frac {1}{117} \left (20 \left (39-7 \sqrt {13}\right )\right ) \int \frac {e^{-x}}{-1-\sqrt {13}+2 x} \, dx+\frac {1}{117} \left (40 \left (13-4 \sqrt {13}\right )\right ) \int \frac {e^{-x}}{-1-\sqrt {13}+2 x} \, dx+\frac {1}{39} \left (10 \left (13-\sqrt {13}\right )\right ) \int \frac {e^{-x}}{-1-\sqrt {13}+2 x} \, dx+\frac {1}{39} \left (10 \left (13+\sqrt {13}\right )\right ) \int \frac {e^{-x}}{-1+\sqrt {13}+2 x} \, dx+\frac {1}{117} \left (40 \left (13+4 \sqrt {13}\right )\right ) \int \frac {e^{-x}}{-1+\sqrt {13}+2 x} \, dx-\frac {1}{117} \left (20 \left (39+7 \sqrt {13}\right )\right ) \int \frac {e^{-x}}{-1+\sqrt {13}+2 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.48, size = 35, normalized size = 0.92 \begin {gather*} \frac {5 \left (e^x x+\frac {2 e^{-x}}{-3-x+x^2}-x \log ^2(2)\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(60 + 60*x - 30*x^2 - 10*x^3 + E^(2*x)*(-45*x + 15*x^2 + 55*x^3 - 15*x^4 - 15*x^5 + 5*x^6) + E^x*(45
*x + 30*x^2 - 25*x^3 - 10*x^4 + 5*x^5)*Log[2]^2)/(E^x*(9*x^3 + 6*x^4 - 5*x^5 - 2*x^6 + x^7)),x]

[Out]

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

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fricas [A]  time = 0.74, size = 61, normalized size = 1.61 \begin {gather*} -\frac {5 \, {\left ({\left (x^{3} - x^{2} - 3 \, x\right )} e^{x} \log \relax (2)^{2} - {\left (x^{3} - x^{2} - 3 \, x\right )} e^{\left (2 \, x\right )} - 2\right )} e^{\left (-x\right )}}{x^{4} - x^{3} - 3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^6-15*x^5-15*x^4+55*x^3+15*x^2-45*x)*exp(x)^2+(5*x^5-10*x^4-25*x^3+30*x^2+45*x)*log(2)^2*exp(x)
-10*x^3-30*x^2+60*x+60)/(x^7-2*x^6-5*x^5+6*x^4+9*x^3)/exp(x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.33, size = 67, normalized size = 1.76 \begin {gather*} -\frac {5 \, {\left (x^{3} \log \relax (2)^{2} - x^{3} e^{x} - x^{2} \log \relax (2)^{2} + x^{2} e^{x} - 3 \, x \log \relax (2)^{2} + 3 \, x e^{x} - 2 \, e^{\left (-x\right )}\right )}}{x^{4} - x^{3} - 3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^6-15*x^5-15*x^4+55*x^3+15*x^2-45*x)*exp(x)^2+(5*x^5-10*x^4-25*x^3+30*x^2+45*x)*log(2)^2*exp(x)
-10*x^3-30*x^2+60*x+60)/(x^7-2*x^6-5*x^5+6*x^4+9*x^3)/exp(x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.20, size = 37, normalized size = 0.97

method	result	size
risch	\(-\frac {5 \ln \relax (2)^{2}}{x}+\frac {5 \,{\mathrm e}^{x}}{x}+\frac {10 \,{\mathrm e}^{-x}}{x^{2} \left (x^{2}-x -3\right )}\)	\(37\)
norman	\(\frac {\left (10-5 \,{\mathrm e}^{x} \ln \relax (2)^{2} x^{3}+5 \,{\mathrm e}^{x} \ln \relax (2)^{2} x^{2}-15 x \,{\mathrm e}^{2 x}-5 \,{\mathrm e}^{2 x} x^{2}+5 \,{\mathrm e}^{2 x} x^{3}+15 x \ln \relax (2)^{2} {\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x^{2} \left (x^{2}-x -3\right )}\)	\(77\)
default	\(\frac {10 \,{\mathrm e}^{-x} \left (37 x^{3}-64 x^{2}-78 x +39\right )}{39 x^{2} \left (x^{2}-x -3\right )}-\frac {5 \ln \relax (2)^{2}}{x}-\frac {20 \,{\mathrm e}^{-x} \left (20 x^{2}-23 x -39\right )}{39 \left (x^{2}-x -3\right ) x}-\frac {10 \,{\mathrm e}^{-x} \left (x -7\right )}{13 \left (x^{2}-x -3\right )}+\frac {10 \,{\mathrm e}^{-x} \left (2 x -1\right )}{13 \left (x^{2}-x -3\right )}+\frac {5 \,{\mathrm e}^{x} \left (20 x^{2}-23 x -39\right )}{13 \left (x^{2}-x -3\right ) x}+\frac {5 \,{\mathrm e}^{x} \left (x -7\right )}{13 \left (x^{2}-x -3\right )}-\frac {55 \,{\mathrm e}^{x} \left (2 x -1\right )}{13 \left (x^{2}-x -3\right )}+\frac {15 \,{\mathrm e}^{x} \left (x +6\right )}{13 \left (x^{2}-x -3\right )}+\frac {15 \,{\mathrm e}^{x} \left (7 x +3\right )}{13 \left (x^{2}-x -3\right )}-\frac {5 \,{\mathrm e}^{x} \left (10 x +21\right )}{13 \left (x^{2}-x -3\right )}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^6-15*x^5-15*x^4+55*x^3+15*x^2-45*x)*exp(x)^2+(5*x^5-10*x^4-25*x^3+30*x^2+45*x)*ln(2)^2*exp(x)-10*x^3
-30*x^2+60*x+60)/(x^7-2*x^6-5*x^5+6*x^4+9*x^3)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-5*ln(2)^2/x+5*exp(x)/x+10/x^2/(x^2-x-3)*exp(-x)

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maxima [A]  time = 0.53, size = 65, normalized size = 1.71 \begin {gather*} -\frac {5 \, {\left (x^{3} \log \relax (2)^{2} - x^{2} \log \relax (2)^{2} - 3 \, x \log \relax (2)^{2} - {\left (x^{3} - x^{2} - 3 \, x\right )} e^{x} - 2 \, e^{\left (-x\right )}\right )}}{x^{4} - x^{3} - 3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^6-15*x^5-15*x^4+55*x^3+15*x^2-45*x)*exp(x)^2+(5*x^5-10*x^4-25*x^3+30*x^2+45*x)*log(2)^2*exp(x)
-10*x^3-30*x^2+60*x+60)/(x^7-2*x^6-5*x^5+6*x^4+9*x^3)/exp(x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.77, size = 39, normalized size = 1.03 \begin {gather*} \frac {5\,{\mathrm {e}}^x}{x}-\frac {5\,{\ln \relax (2)}^2}{x}-\frac {10\,{\mathrm {e}}^{-x}}{-x^4+x^3+3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(60*x - 30*x^2 - 10*x^3 - exp(2*x)*(45*x - 15*x^2 - 55*x^3 + 15*x^4 + 15*x^5 - 5*x^6) + exp(x)*lo
g(2)^2*(45*x + 30*x^2 - 25*x^3 - 10*x^4 + 5*x^5) + 60))/(9*x^3 + 6*x^4 - 5*x^5 - 2*x^6 + x^7),x)

[Out]

(5*exp(x))/x - (5*log(2)^2)/x - (10*exp(-x))/(3*x^2 + x^3 - x^4)

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sympy [A]  time = 0.21, size = 44, normalized size = 1.16 \begin {gather*} \frac {10 x e^{- x} + \left (5 x^{4} - 5 x^{3} - 15 x^{2}\right ) e^{x}}{x^{5} - x^{4} - 3 x^{3}} - \frac {5 \log {\relax (2 )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**6-15*x**5-15*x**4+55*x**3+15*x**2-45*x)*exp(x)**2+(5*x**5-10*x**4-25*x**3+30*x**2+45*x)*ln(2)
**2*exp(x)-10*x**3-30*x**2+60*x+60)/(x**7-2*x**6-5*x**5+6*x**4+9*x**3)/exp(x),x)

[Out]

(10*x*exp(-x) + (5*x**4 - 5*x**3 - 15*x**2)*exp(x))/(x**5 - x**4 - 3*x**3) - 5*log(2)**2/x

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