3.65.59 \(\int \frac {28080-18720 e^8 x+4320 e^{16} x^2+e^x (-18720-18720 x+e^8 (12480 x+14880 x^2)+e^{16} (-2880 x^2-2880 x^3))+e^{2 x} (3120+6240 x+e^8 (-2080 x-4960 x^2)+e^{16} (480 x^2+960 x^3))}{169-156 e^8 x+36 e^{16} x^2} \, dx\)

Optimal. Leaf size=33 \[ \frac {16 \left (-3+e^x\right )^2 x^2}{\frac {6 x}{5}-\frac {x}{3-e^8 x}} \]

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Rubi [B]  time = 0.76, antiderivative size = 123, normalized size of antiderivative = 3.73, number of steps used = 21, number of rules used = 8, integrand size = 111, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {27, 6742, 683, 2199, 2194, 2176, 2177, 2178} \begin {gather*} -80 e^x x+\frac {40}{3} e^{2 x} x+120 x+80 e^x-\frac {20 e^{2 x}}{3}-\frac {2600 e^{x-8}}{3 \left (13-6 e^8 x\right )}+\frac {1300 e^{2 x-8}}{9 \left (13-6 e^8 x\right )}+\frac {1300}{e^8 \left (13-6 e^8 x\right )}-\frac {20}{9} \left (5-3 e^8\right ) e^{2 x-8}+\frac {40}{3} \left (5-6 e^8\right ) e^{x-8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(28080 - 18720*E^8*x + 4320*E^16*x^2 + E^x*(-18720 - 18720*x + E^8*(12480*x + 14880*x^2) + E^16*(-2880*x^2
- 2880*x^3)) + E^(2*x)*(3120 + 6240*x + E^8*(-2080*x - 4960*x^2) + E^16*(480*x^2 + 960*x^3)))/(169 - 156*E^8*
x + 36*E^16*x^2),x]

[Out]

80*E^x - (20*E^(2*x))/3 + (40*E^(-8 + x)*(5 - 6*E^8))/3 - (20*E^(-8 + 2*x)*(5 - 3*E^8))/9 + 120*x - 80*E^x*x +
(40*E^(2*x)*x)/3 + 1300/(E^8*(13 - 6*E^8*x)) - (2600*E^(-8 + x))/(3*(13 - 6*E^8*x)) + (1300*E^(-8 + 2*x))/(9*
(13 - 6*E^8*x))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

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 {28080-18720 e^8 x+4320 e^{16} x^2+e^x \left (-18720-18720 x+e^8 \left (12480 x+14880 x^2\right )+e^{16} \left (-2880 x^2-2880 x^3\right )\right )+e^{2 x} \left (3120+6240 x+e^8 \left (-2080 x-4960 x^2\right )+e^{16} \left (480 x^2+960 x^3\right )\right )}{\left (-13+6 e^8 x\right )^2} \, dx\\ &=\int \left (\frac {720 \left (39-26 e^8 x+6 e^{16} x^2\right )}{\left (-13+6 e^8 x\right )^2}+\frac {480 e^x \left (-39-13 \left (3-2 e^8\right ) x+e^8 \left (31-6 e^8\right ) x^2-6 e^{16} x^3\right )}{\left (13-6 e^8 x\right )^2}+\frac {80 e^{2 x} \left (39+26 \left (3-e^8\right ) x-2 e^8 \left (31-3 e^8\right ) x^2+12 e^{16} x^3\right )}{\left (13-6 e^8 x\right )^2}\right ) \, dx\\ &=80 \int \frac {e^{2 x} \left (39+26 \left (3-e^8\right ) x-2 e^8 \left (31-3 e^8\right ) x^2+12 e^{16} x^3\right )}{\left (13-6 e^8 x\right )^2} \, dx+480 \int \frac {e^x \left (-39-13 \left (3-2 e^8\right ) x+e^8 \left (31-6 e^8\right ) x^2-6 e^{16} x^3\right )}{\left (13-6 e^8 x\right )^2} \, dx+720 \int \frac {39-26 e^8 x+6 e^{16} x^2}{\left (-13+6 e^8 x\right )^2} \, dx\\ &=80 \int \left (\frac {1}{18} e^{-8+2 x} \left (-5+3 e^8\right )+\frac {1}{3} e^{2 x} x+\frac {65 e^{2 x}}{6 \left (-13+6 e^8 x\right )^2}-\frac {65 e^{-8+2 x}}{18 \left (-13+6 e^8 x\right )}\right ) \, dx+480 \int \left (\frac {1}{36} e^{-8+x} \left (5-6 e^8\right )-\frac {e^x x}{6}-\frac {65 e^x}{6 \left (-13+6 e^8 x\right )^2}+\frac {65 e^{-8+x}}{36 \left (-13+6 e^8 x\right )}\right ) \, dx+720 \int \left (\frac {1}{6}+\frac {65}{6 \left (-13+6 e^8 x\right )^2}\right ) \, dx\\ &=120 x+\frac {1300}{e^8 \left (13-6 e^8 x\right )}+\frac {80}{3} \int e^{2 x} x \, dx-80 \int e^x x \, dx-\frac {2600}{9} \int \frac {e^{-8+2 x}}{-13+6 e^8 x} \, dx+\frac {2600}{3} \int \frac {e^{2 x}}{\left (-13+6 e^8 x\right )^2} \, dx+\frac {2600}{3} \int \frac {e^{-8+x}}{-13+6 e^8 x} \, dx-5200 \int \frac {e^x}{\left (-13+6 e^8 x\right )^2} \, dx+\frac {1}{3} \left (40 \left (5-6 e^8\right )\right ) \int e^{-8+x} \, dx-\frac {1}{9} \left (40 \left (5-3 e^8\right )\right ) \int e^{-8+2 x} \, dx\\ &=\frac {40}{3} e^{-8+x} \left (5-6 e^8\right )-\frac {20}{9} e^{-8+2 x} \left (5-3 e^8\right )+120 x-80 e^x x+\frac {40}{3} e^{2 x} x+\frac {1300}{e^8 \left (13-6 e^8 x\right )}-\frac {2600 e^{-8+x}}{3 \left (13-6 e^8 x\right )}+\frac {1300 e^{-8+2 x}}{9 \left (13-6 e^8 x\right )}-\frac {1300}{27} e^{-16+\frac {13}{3 e^8}} \text {Ei}\left (-\frac {13-6 e^8 x}{3 e^8}\right )+\frac {1300}{9} e^{-16+\frac {13}{6 e^8}} \text {Ei}\left (-\frac {13-6 e^8 x}{6 e^8}\right )-\frac {40}{3} \int e^{2 x} \, dx+80 \int e^x \, dx+\frac {2600 \int \frac {e^{2 x}}{-13+6 e^8 x} \, dx}{9 e^8}-\frac {2600 \int \frac {e^x}{-13+6 e^8 x} \, dx}{3 e^8}\\ &=80 e^x-\frac {20 e^{2 x}}{3}+\frac {40}{3} e^{-8+x} \left (5-6 e^8\right )-\frac {20}{9} e^{-8+2 x} \left (5-3 e^8\right )+120 x-80 e^x x+\frac {40}{3} e^{2 x} x+\frac {1300}{e^8 \left (13-6 e^8 x\right )}-\frac {2600 e^{-8+x}}{3 \left (13-6 e^8 x\right )}+\frac {1300 e^{-8+2 x}}{9 \left (13-6 e^8 x\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 72, normalized size = 2.18 \begin {gather*} \frac {80 \left (65+78 e^8 x-72 e^{8+x} x+12 e^{8+2 x} x-36 e^{16} x^2+24 e^{16+x} x^2-4 e^{16+2 x} x^2\right )}{52 e^8-24 e^{16} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(28080 - 18720*E^8*x + 4320*E^16*x^2 + E^x*(-18720 - 18720*x + E^8*(12480*x + 14880*x^2) + E^16*(-28
80*x^2 - 2880*x^3)) + E^(2*x)*(3120 + 6240*x + E^8*(-2080*x - 4960*x^2) + E^16*(480*x^2 + 960*x^3)))/(169 - 15
6*E^8*x + 36*E^16*x^2),x]

[Out]

(80*(65 + 78*E^8*x - 72*E^(8 + x)*x + 12*E^(8 + 2*x)*x - 36*E^16*x^2 + 24*E^(16 + x)*x^2 - 4*E^(16 + 2*x)*x^2)
)/(52*E^8 - 24*E^16*x)

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fricas [B]  time = 0.69, size = 62, normalized size = 1.88 \begin {gather*} \frac {20 \, {\left (36 \, x^{2} e^{16} - 78 \, x e^{8} + 4 \, {\left (x^{2} e^{16} - 3 \, x e^{8}\right )} e^{\left (2 \, x\right )} - 24 \, {\left (x^{2} e^{16} - 3 \, x e^{8}\right )} e^{x} - 65\right )}}{6 \, x e^{16} - 13 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((960*x^3+480*x^2)*exp(4)^4+(-4960*x^2-2080*x)*exp(4)^2+6240*x+3120)*exp(x)^2+((-2880*x^3-2880*x^2)
*exp(4)^4+(14880*x^2+12480*x)*exp(4)^2-18720*x-18720)*exp(x)+4320*x^2*exp(4)^4-18720*x*exp(4)^2+28080)/(36*x^2
*exp(4)^4-156*x*exp(4)^2+169),x, algorithm="fricas")

[Out]

20*(36*x^2*e^16 - 78*x*e^8 + 4*(x^2*e^16 - 3*x*e^8)*e^(2*x) - 24*(x^2*e^16 - 3*x*e^8)*e^x - 65)/(6*x*e^16 - 13
*e^8)

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giac [B]  time = 0.18, size = 67, normalized size = 2.03 \begin {gather*} \frac {20 \, {\left (36 \, x^{2} e^{40} + 4 \, x^{2} e^{\left (2 \, x + 40\right )} - 24 \, x^{2} e^{\left (x + 40\right )} - 78 \, x e^{32} - 12 \, x e^{\left (2 \, x + 32\right )} + 72 \, x e^{\left (x + 32\right )} - 65 \, e^{24}\right )}}{6 \, x e^{40} - 13 \, e^{32}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((960*x^3+480*x^2)*exp(4)^4+(-4960*x^2-2080*x)*exp(4)^2+6240*x+3120)*exp(x)^2+((-2880*x^3-2880*x^2)
*exp(4)^4+(14880*x^2+12480*x)*exp(4)^2-18720*x-18720)*exp(x)+4320*x^2*exp(4)^4-18720*x*exp(4)^2+28080)/(36*x^2
*exp(4)^4-156*x*exp(4)^2+169),x, algorithm="giac")

[Out]

20*(36*x^2*e^40 + 4*x^2*e^(2*x + 40) - 24*x^2*e^(x + 40) - 78*x*e^32 - 12*x*e^(2*x + 32) + 72*x*e^(x + 32) - 6
5*e^24)/(6*x*e^40 - 13*e^32)

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maple [A]  time = 0.58, size = 59, normalized size = 1.79

method	result	size
risch	\(120 x -\frac {650 \,{\mathrm e}^{-8}}{3 \left (x \,{\mathrm e}^{8}-\frac {13}{6}\right )}+\frac {80 x \left (x \,{\mathrm e}^{8}-3\right ) {\mathrm e}^{2 x}}{6 x \,{\mathrm e}^{8}-13}-\frac {480 x \left (x \,{\mathrm e}^{8}-3\right ) {\mathrm e}^{x}}{6 x \,{\mathrm e}^{8}-13}\)	\(59\)
norman	\(\frac {-2160 x -240 x \,{\mathrm e}^{2 x}+720 x^{2} {\mathrm e}^{8}+1440 \,{\mathrm e}^{x} x -480 \,{\mathrm e}^{8} {\mathrm e}^{x} x^{2}+80 \,{\mathrm e}^{8} {\mathrm e}^{2 x} x^{2}}{6 x \,{\mathrm e}^{8}-13}\)	\(62\)
default	\(\frac {960 x}{6 x \,{\mathrm e}^{8}-13}+\frac {3120 \,{\mathrm e}^{x} {\mathrm e}^{-8}}{6 x \,{\mathrm e}^{8}-13}+520 \left ({\mathrm e}^{-8}\right )^{2} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{6}} \expIntegralEi \left (1, -x +\frac {13 \,{\mathrm e}^{-8}}{6}\right )-\frac {520 \,{\mathrm e}^{2 x} {\mathrm e}^{-8}}{6 x \,{\mathrm e}^{8}-13}-\frac {520 \left ({\mathrm e}^{-8}\right )^{2} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{3}} \expIntegralEi \left (1, -2 x +\frac {13 \,{\mathrm e}^{-8}}{3}\right )}{3}-\frac {6760 \,{\mathrm e}^{2 x} {\mathrm e}^{-16}}{3 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {520 \left (3 \,{\mathrm e}^{8}+13\right ) {\mathrm e}^{-16} {\mathrm e}^{-8} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{3}} \expIntegralEi \left (1, -2 x +\frac {13 \,{\mathrm e}^{-8}}{3}\right )}{9}+\frac {-3120 x +720 x^{2} {\mathrm e}^{8}}{6 x \,{\mathrm e}^{8}-13}+\frac {6760 \,{\mathrm e}^{x} {\mathrm e}^{-16}}{6 x \,{\mathrm e}^{8}-13}+\frac {260 \left (6 \,{\mathrm e}^{8}+13\right ) {\mathrm e}^{-16} {\mathrm e}^{-8} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{6}} \expIntegralEi \left (1, -x +\frac {13 \,{\mathrm e}^{-8}}{6}\right )}{3}-2080 \,{\mathrm e}^{8} \left (-\frac {13 \,{\mathrm e}^{2 x} {\mathrm e}^{-16}}{36 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {\left (3 \,{\mathrm e}^{8}+13\right ) {\mathrm e}^{-16} {\mathrm e}^{-8} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{3}} \expIntegralEi \left (1, -2 x +\frac {13 \,{\mathrm e}^{-8}}{3}\right )}{108}\right )+12480 \,{\mathrm e}^{8} \left (-\frac {13 \,{\mathrm e}^{x} {\mathrm e}^{-16}}{36 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {\left (6 \,{\mathrm e}^{8}+13\right ) {\mathrm e}^{-16} {\mathrm e}^{-8} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{6}} \expIntegralEi \left (1, -x +\frac {13 \,{\mathrm e}^{-8}}{6}\right )}{216}\right )+14880 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{x} {\mathrm e}^{-16}}{36}-\frac {169 \,{\mathrm e}^{x} {\mathrm e}^{-24}}{216 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {13 \left (12 \,{\mathrm e}^{8}+13\right ) \left ({\mathrm e}^{-8}\right )^{2} {\mathrm e}^{-16} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{6}} \expIntegralEi \left (1, -x +\frac {13 \,{\mathrm e}^{-8}}{6}\right )}{1296}\right )-4960 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{2 x} {\mathrm e}^{-16}}{72}-\frac {169 \,{\mathrm e}^{2 x} {\mathrm e}^{-24}}{216 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {13 \left (6 \,{\mathrm e}^{8}+13\right ) \left ({\mathrm e}^{-8}\right )^{2} {\mathrm e}^{-16} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{3}} \expIntegralEi \left (1, -2 x +\frac {13 \,{\mathrm e}^{-8}}{3}\right )}{648}\right )-2880 \,{\mathrm e}^{16} \left (\frac {{\mathrm e}^{x} {\mathrm e}^{-16}}{36}-\frac {169 \,{\mathrm e}^{x} {\mathrm e}^{-24}}{216 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {13 \left (12 \,{\mathrm e}^{8}+13\right ) \left ({\mathrm e}^{-8}\right )^{2} {\mathrm e}^{-16} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{6}} \expIntegralEi \left (1, -x +\frac {13 \,{\mathrm e}^{-8}}{6}\right )}{1296}\right )-2880 \,{\mathrm e}^{16} \left (\frac {\left (3 x \,{\mathrm e}^{8}+13-3 \,{\mathrm e}^{8}\right ) {\mathrm e}^{x} {\mathrm e}^{-24}}{108}-\frac {2197 \,{\mathrm e}^{x} {\mathrm e}^{-32}}{1296 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {169 \left (18 \,{\mathrm e}^{8}+13\right ) \left ({\mathrm e}^{-8}\right )^{2} {\mathrm e}^{-24} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{6}} \expIntegralEi \left (1, -x +\frac {13 \,{\mathrm e}^{-8}}{6}\right )}{7776}\right )+480 \,{\mathrm e}^{16} \left (\frac {{\mathrm e}^{2 x} {\mathrm e}^{-16}}{72}-\frac {169 \,{\mathrm e}^{2 x} {\mathrm e}^{-24}}{216 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {13 \left (6 \,{\mathrm e}^{8}+13\right ) \left ({\mathrm e}^{-8}\right )^{2} {\mathrm e}^{-16} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{3}} \expIntegralEi \left (1, -2 x +\frac {13 \,{\mathrm e}^{-8}}{3}\right )}{648}\right )+960 \,{\mathrm e}^{16} \left (\frac {\left (6 x \,{\mathrm e}^{8}-3 \,{\mathrm e}^{8}+26\right ) {\mathrm e}^{2 x} {\mathrm e}^{-24}}{432}-\frac {2197 \,{\mathrm e}^{2 x} {\mathrm e}^{-32}}{1296 \left (6 x \,{\mathrm e}^{8}-13\right )}-\frac {169 \left (9 \,{\mathrm e}^{8}+13\right ) \left ({\mathrm e}^{-8}\right )^{2} {\mathrm e}^{-24} {\mathrm e}^{\frac {13 \,{\mathrm e}^{-8}}{3}} \expIntegralEi \left (1, -2 x +\frac {13 \,{\mathrm e}^{-8}}{3}\right )}{3888}\right )\)	\(829\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((960*x^3+480*x^2)*exp(4)^4+(-4960*x^2-2080*x)*exp(4)^2+6240*x+3120)*exp(x)^2+((-2880*x^3-2880*x^2)*exp(4
)^4+(14880*x^2+12480*x)*exp(4)^2-18720*x-18720)*exp(x)+4320*x^2*exp(4)^4-18720*x*exp(4)^2+28080)/(36*x^2*exp(4
)^4-156*x*exp(4)^2+169),x,method=_RETURNVERBOSE)

[Out]

120*x-650/3*exp(-8)/(x*exp(8)-13/6)+80*x*(x*exp(8)-3)/(6*x*exp(8)-13)*exp(2*x)-480*x*(x*exp(8)-3)/(6*x*exp(8)-
13)*exp(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 20 \, {\left (6 \, x e^{\left (-16\right )} + 26 \, e^{\left (-24\right )} \log \left (6 \, x e^{8} - 13\right ) - \frac {169}{6 \, x e^{32} - 13 \, e^{24}}\right )} e^{16} - 520 \, {\left (e^{\left (-16\right )} \log \left (6 \, x e^{8} - 13\right ) - \frac {13}{6 \, x e^{24} - 13 \, e^{16}}\right )} e^{8} + \frac {3120 \, e^{\left (\frac {13}{6} \, e^{\left (-8\right )} - 8\right )} E_{2}\left (-\frac {1}{6} \, {\left (6 \, x e^{8} - 13\right )} e^{\left (-8\right )}\right )}{6 \, x e^{8} - 13} + \frac {80 \, {\left ({\left (x^{2} e^{8} - 3 \, x\right )} e^{\left (2 \, x\right )} - 6 \, {\left (x^{2} e^{8} - 3 \, x\right )} e^{x}\right )}}{6 \, x e^{8} - 13} - \frac {4680}{6 \, x e^{16} - 13 \, e^{8}} + 18720 \, \int \frac {e^{x}}{36 \, x^{2} e^{16} - 156 \, x e^{8} + 169}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((960*x^3+480*x^2)*exp(4)^4+(-4960*x^2-2080*x)*exp(4)^2+6240*x+3120)*exp(x)^2+((-2880*x^3-2880*x^2)
*exp(4)^4+(14880*x^2+12480*x)*exp(4)^2-18720*x-18720)*exp(x)+4320*x^2*exp(4)^4-18720*x*exp(4)^2+28080)/(36*x^2
*exp(4)^4-156*x*exp(4)^2+169),x, algorithm="maxima")

[Out]

20*(6*x*e^(-16) + 26*e^(-24)*log(6*x*e^8 - 13) - 169/(6*x*e^32 - 13*e^24))*e^16 - 520*(e^(-16)*log(6*x*e^8 - 1
3) - 13/(6*x*e^24 - 13*e^16))*e^8 + 3120*e^(13/6*e^(-8) - 8)*exp_integral_e(2, -1/6*(6*x*e^8 - 13)*e^(-8))/(6*
x*e^8 - 13) + 80*((x^2*e^8 - 3*x)*e^(2*x) - 6*(x^2*e^8 - 3*x)*e^x)/(6*x*e^8 - 13) - 4680/(6*x*e^16 - 13*e^8) +
18720*integrate(e^x/(36*x^2*e^16 - 156*x*e^8 + 169), x)

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mupad [B]  time = 4.58, size = 64, normalized size = 1.94 \begin {gather*} 120\,x-\frac {1300\,{\mathrm {e}}^{-8}}{6\,x\,{\mathrm {e}}^8-13}+\frac {{\mathrm {e}}^x\,\left (240\,x\,{\mathrm {e}}^{-8}-80\,x^2\right )}{x-\frac {13\,{\mathrm {e}}^{-8}}{6}}-\frac {{\mathrm {e}}^{2\,x}\,\left (40\,x\,{\mathrm {e}}^{-8}-\frac {40\,x^2}{3}\right )}{x-\frac {13\,{\mathrm {e}}^{-8}}{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(6240*x - exp(8)*(2080*x + 4960*x^2) + exp(16)*(480*x^2 + 960*x^3) + 3120) - 18720*x*exp(8) - ex
p(x)*(18720*x - exp(8)*(12480*x + 14880*x^2) + exp(16)*(2880*x^2 + 2880*x^3) + 18720) + 4320*x^2*exp(16) + 280
80)/(36*x^2*exp(16) - 156*x*exp(8) + 169),x)

[Out]

120*x - (1300*exp(-8))/(6*x*exp(8) - 13) + (exp(x)*(240*x*exp(-8) - 80*x^2))/(x - (13*exp(-8))/6) - (exp(2*x)*
(40*x*exp(-8) - (40*x^2)/3))/(x - (13*exp(-8))/6)

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sympy [B]  time = 0.31, size = 82, normalized size = 2.48 \begin {gather*} 120 x + \frac {\left (- 2880 x^{3} e^{16} + 14880 x^{2} e^{8} - 18720 x\right ) e^{x} + \left (480 x^{3} e^{16} - 2480 x^{2} e^{8} + 3120 x\right ) e^{2 x}}{36 x^{2} e^{16} - 156 x e^{8} + 169} - \frac {1300}{6 x e^{16} - 13 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((960*x**3+480*x**2)*exp(4)**4+(-4960*x**2-2080*x)*exp(4)**2+6240*x+3120)*exp(x)**2+((-2880*x**3-28
80*x**2)*exp(4)**4+(14880*x**2+12480*x)*exp(4)**2-18720*x-18720)*exp(x)+4320*x**2*exp(4)**4-18720*x*exp(4)**2+
28080)/(36*x**2*exp(4)**4-156*x*exp(4)**2+169),x)

[Out]

120*x + ((-2880*x**3*exp(16) + 14880*x**2*exp(8) - 18720*x)*exp(x) + (480*x**3*exp(16) - 2480*x**2*exp(8) + 31
20*x)*exp(2*x))/(36*x**2*exp(16) - 156*x*exp(8) + 169) - 1300/(6*x*exp(16) - 13*exp(8))

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