3.13.75 \(\int \frac {2944+2336 x-1152 x^2+1016 x^3-264 x^4+90 x^5-11 x^6+2 x^7+e^{\frac {9}{e^{4/3}}} (8+16 x)+e^{\frac {6}{e^{4/3}}} (176+264 x-36 x^2+24 x^3)+e^{\frac {3}{e^{4/3}}} (1248+1408 x-408 x^2+312 x^3-42 x^4+12 x^5)}{1728+8 e^{\frac {9}{e^{4/3}}}-864 x+576 x^2-152 x^3+48 x^4-6 x^5+x^6+e^{\frac {6}{e^{4/3}}} (144-24 x+12 x^2)+e^{\frac {3}{e^{4/3}}} (864-288 x+168 x^2-24 x^3+6 x^4)} \, dx\)

Optimal. Leaf size=32 \[ 2+x+\left (x+\frac {2}{6+e^{\frac {3}{e^{4/3}}}-x+\frac {x^2}{2}}\right )^2 \]

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Rubi [A]  time = 0.56, antiderivative size = 54, normalized size of antiderivative = 1.69, number of steps used = 8, number of rules used = 5, integrand size = 202, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2074, 618, 204, 629, 638} \begin {gather*} x^2+\frac {8 x}{x^2-2 x+2 \left (6+e^{\frac {3}{e^{4/3}}}\right )}+\frac {16}{\left (x^2-2 x+2 \left (6+e^{\frac {3}{e^{4/3}}}\right )\right )^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2944 + 2336*x - 1152*x^2 + 1016*x^3 - 264*x^4 + 90*x^5 - 11*x^6 + 2*x^7 + E^(9/E^(4/3))*(8 + 16*x) + E^(6
/E^(4/3))*(176 + 264*x - 36*x^2 + 24*x^3) + E^(3/E^(4/3))*(1248 + 1408*x - 408*x^2 + 312*x^3 - 42*x^4 + 12*x^5
))/(1728 + 8*E^(9/E^(4/3)) - 864*x + 576*x^2 - 152*x^3 + 48*x^4 - 6*x^5 + x^6 + E^(6/E^(4/3))*(144 - 24*x + 12
*x^2) + E^(3/E^(4/3))*(864 - 288*x + 168*x^2 - 24*x^3 + 6*x^4)),x]

[Out]

x + x^2 + 16/(2*(6 + E^(3/E^(4/3))) - 2*x + x^2)^2 + (8*x)/(2*(6 + E^(3/E^(4/3))) - 2*x + x^2)

Rule 204

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

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 629

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

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 \left (1+2 x+\frac {8}{-2 \left (6+e^{\frac {3}{e^{4/3}}}\right )+2 x-x^2}+\frac {64 (1-x)}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^3}+\frac {16 \left (12+2 e^{\frac {3}{e^{4/3}}}-x\right )}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2}\right ) \, dx\\ &=x+x^2+8 \int \frac {1}{-2 \left (6+e^{\frac {3}{e^{4/3}}}\right )+2 x-x^2} \, dx+16 \int \frac {12+2 e^{\frac {3}{e^{4/3}}}-x}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2} \, dx+64 \int \frac {1-x}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^3} \, dx\\ &=x+x^2+\frac {16}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2}+\frac {8 x}{2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2}+8 \int \frac {1}{2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2} \, dx-16 \operatorname {Subst}\left (\int \frac {1}{-4 \left (11+2 e^{\frac {3}{e^{4/3}}}\right )-x^2} \, dx,x,2-2 x\right )\\ &=x+x^2+\frac {16}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2}+\frac {8 x}{2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2}+\frac {8 \tan ^{-1}\left (\frac {1-x}{\sqrt {11+2 e^{\frac {3}{e^{4/3}}}}}\right )}{\sqrt {11+2 e^{\frac {3}{e^{4/3}}}}}-16 \operatorname {Subst}\left (\int \frac {1}{-4 \left (11+2 e^{\frac {3}{e^{4/3}}}\right )-x^2} \, dx,x,-2+2 x\right )\\ &=x+x^2+\frac {16}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2}+\frac {8 x}{2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 52, normalized size = 1.62 \begin {gather*} x+x^2+\frac {16}{\left (12+2 e^{\frac {3}{e^{4/3}}}-2 x+x^2\right )^2}+\frac {8 x}{12+2 e^{\frac {3}{e^{4/3}}}-2 x+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2944 + 2336*x - 1152*x^2 + 1016*x^3 - 264*x^4 + 90*x^5 - 11*x^6 + 2*x^7 + E^(9/E^(4/3))*(8 + 16*x)
+ E^(6/E^(4/3))*(176 + 264*x - 36*x^2 + 24*x^3) + E^(3/E^(4/3))*(1248 + 1408*x - 408*x^2 + 312*x^3 - 42*x^4 +
12*x^5))/(1728 + 8*E^(9/E^(4/3)) - 864*x + 576*x^2 - 152*x^3 + 48*x^4 - 6*x^5 + x^6 + E^(6/E^(4/3))*(144 - 24*
x + 12*x^2) + E^(3/E^(4/3))*(864 - 288*x + 168*x^2 - 24*x^3 + 6*x^4)),x]

[Out]

x + x^2 + 16/(12 + 2*E^(3/E^(4/3)) - 2*x + x^2)^2 + (8*x)/(12 + 2*E^(3/E^(4/3)) - 2*x + x^2)

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fricas [B]  time = 0.56, size = 107, normalized size = 3.34 \begin {gather*} \frac {x^{6} - 3 \, x^{5} + 24 \, x^{4} - 12 \, x^{3} + 80 \, x^{2} + 4 \, {\left (x^{2} + x\right )} e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 4 \, {\left (x^{4} - x^{3} + 10 \, x^{2} + 16 \, x\right )} e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 240 \, x + 16}{x^{4} - 4 \, x^{3} + 28 \, x^{2} + 4 \, {\left (x^{2} - 2 \, x + 12\right )} e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} - 48 \, x + 4 \, e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x+8)*exp(3/exp(4/3))^3+(24*x^3-36*x^2+264*x+176)*exp(3/exp(4/3))^2+(12*x^5-42*x^4+312*x^3-408*x
^2+1408*x+1248)*exp(3/exp(4/3))+2*x^7-11*x^6+90*x^5-264*x^4+1016*x^3-1152*x^2+2336*x+2944)/(8*exp(3/exp(4/3))^
3+(12*x^2-24*x+144)*exp(3/exp(4/3))^2+(6*x^4-24*x^3+168*x^2-288*x+864)*exp(3/exp(4/3))+x^6-6*x^5+48*x^4-152*x^
3+576*x^2-864*x+1728),x, algorithm="fricas")

[Out]

(x^6 - 3*x^5 + 24*x^4 - 12*x^3 + 80*x^2 + 4*(x^2 + x)*e^(6*e^(-4/3)) + 4*(x^4 - x^3 + 10*x^2 + 16*x)*e^(3*e^(-
4/3)) + 240*x + 16)/(x^4 - 4*x^3 + 28*x^2 + 4*(x^2 - 2*x + 12)*e^(3*e^(-4/3)) - 48*x + 4*e^(6*e^(-4/3)) + 144)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{7} - 11 \, x^{6} + 90 \, x^{5} - 264 \, x^{4} + 1016 \, x^{3} - 1152 \, x^{2} + 8 \, {\left (2 \, x + 1\right )} e^{\left (9 \, e^{\left (-\frac {4}{3}\right )}\right )} + 4 \, {\left (6 \, x^{3} - 9 \, x^{2} + 66 \, x + 44\right )} e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 2 \, {\left (6 \, x^{5} - 21 \, x^{4} + 156 \, x^{3} - 204 \, x^{2} + 704 \, x + 624\right )} e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 2336 \, x + 2944}{x^{6} - 6 \, x^{5} + 48 \, x^{4} - 152 \, x^{3} + 576 \, x^{2} + 12 \, {\left (x^{2} - 2 \, x + 12\right )} e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 6 \, {\left (x^{4} - 4 \, x^{3} + 28 \, x^{2} - 48 \, x + 144\right )} e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} - 864 \, x + 8 \, e^{\left (9 \, e^{\left (-\frac {4}{3}\right )}\right )} + 1728}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x+8)*exp(3/exp(4/3))^3+(24*x^3-36*x^2+264*x+176)*exp(3/exp(4/3))^2+(12*x^5-42*x^4+312*x^3-408*x
^2+1408*x+1248)*exp(3/exp(4/3))+2*x^7-11*x^6+90*x^5-264*x^4+1016*x^3-1152*x^2+2336*x+2944)/(8*exp(3/exp(4/3))^
3+(12*x^2-24*x+144)*exp(3/exp(4/3))^2+(6*x^4-24*x^3+168*x^2-288*x+864)*exp(3/exp(4/3))+x^6-6*x^5+48*x^4-152*x^
3+576*x^2-864*x+1728),x, algorithm="giac")

[Out]

integrate((2*x^7 - 11*x^6 + 90*x^5 - 264*x^4 + 1016*x^3 - 1152*x^2 + 8*(2*x + 1)*e^(9*e^(-4/3)) + 4*(6*x^3 - 9
*x^2 + 66*x + 44)*e^(6*e^(-4/3)) + 2*(6*x^5 - 21*x^4 + 156*x^3 - 204*x^2 + 704*x + 624)*e^(3*e^(-4/3)) + 2336*
x + 2944)/(x^6 - 6*x^5 + 48*x^4 - 152*x^3 + 576*x^2 + 12*(x^2 - 2*x + 12)*e^(6*e^(-4/3)) + 6*(x^4 - 4*x^3 + 28
*x^2 - 48*x + 144)*e^(3*e^(-4/3)) - 864*x + 8*e^(9*e^(-4/3)) + 1728), x)

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maple [B]  time = 0.17, size = 81, normalized size = 2.53

method	result	size
risch	\(x^{2}+x +\frac {2 x^{3}-4 x^{2}+\left (4 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}+24\right ) x +4}{{\mathrm e}^{6 \,{\mathrm e}^{-\frac {4}{3}}}+{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}} x^{2}-2 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}} x +\frac {x^{4}}{4}+12 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}-x^{3}+7 x^{2}-12 x +36}\)	\(81\)
norman	\(\frac {x^{6}+\left (12 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}+84\right ) x^{3}+\left (-12 \,{\mathrm e}^{6 \,{\mathrm e}^{-\frac {4}{3}}}-168 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}-592\right ) x^{2}+\left (36 \,{\mathrm e}^{6 \,{\mathrm e}^{-\frac {4}{3}}}+448 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}+1392\right ) x -3 x^{5}-16 \,{\mathrm e}^{9 \,{\mathrm e}^{-\frac {4}{3}}}-288 \,{\mathrm e}^{6 \,{\mathrm e}^{-\frac {4}{3}}}-1728 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}-3440}{\left (x^{2}+2 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}-2 x +12\right )^{2}}\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x+8)*exp(3/exp(4/3))^3+(24*x^3-36*x^2+264*x+176)*exp(3/exp(4/3))^2+(12*x^5-42*x^4+312*x^3-408*x^2+140
8*x+1248)*exp(3/exp(4/3))+2*x^7-11*x^6+90*x^5-264*x^4+1016*x^3-1152*x^2+2336*x+2944)/(8*exp(3/exp(4/3))^3+(12*
x^2-24*x+144)*exp(3/exp(4/3))^2+(6*x^4-24*x^3+168*x^2-288*x+864)*exp(3/exp(4/3))+x^6-6*x^5+48*x^4-152*x^3+576*
x^2-864*x+1728),x,method=_RETURNVERBOSE)

[Out]

x^2+x+(2*x^3-4*x^2+(4*exp(3*exp(-4/3))+24)*x+4)/(exp(6*exp(-4/3))+exp(3*exp(-4/3))*x^2-2*exp(3*exp(-4/3))*x+1/
4*x^4+12*exp(3*exp(-4/3))-x^3+7*x^2-12*x+36)

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maxima [B]  time = 0.35, size = 75, normalized size = 2.34 \begin {gather*} x^{2} + x + \frac {8 \, {\left (x^{3} - 2 \, x^{2} + 2 \, x {\left (e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 6\right )} + 2\right )}}{x^{4} - 4 \, x^{3} + 4 \, x^{2} {\left (e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 7\right )} - 8 \, x {\left (e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 6\right )} + 4 \, e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 48 \, e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x+8)*exp(3/exp(4/3))^3+(24*x^3-36*x^2+264*x+176)*exp(3/exp(4/3))^2+(12*x^5-42*x^4+312*x^3-408*x
^2+1408*x+1248)*exp(3/exp(4/3))+2*x^7-11*x^6+90*x^5-264*x^4+1016*x^3-1152*x^2+2336*x+2944)/(8*exp(3/exp(4/3))^
3+(12*x^2-24*x+144)*exp(3/exp(4/3))^2+(6*x^4-24*x^3+168*x^2-288*x+864)*exp(3/exp(4/3))+x^6-6*x^5+48*x^4-152*x^
3+576*x^2-864*x+1728),x, algorithm="maxima")

[Out]

x^2 + x + 8*(x^3 - 2*x^2 + 2*x*(e^(3*e^(-4/3)) + 6) + 2)/(x^4 - 4*x^3 + 4*x^2*(e^(3*e^(-4/3)) + 7) - 8*x*(e^(3
*e^(-4/3)) + 6) + 4*e^(6*e^(-4/3)) + 48*e^(3*e^(-4/3)) + 144)

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mupad [B]  time = 0.25, size = 94, normalized size = 2.94 \begin {gather*} x+\frac {8\,x^3+\left (48\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+4\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-\frac {4}{3}}}-{\left (2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+12\right )}^2+128\right )\,x^2+\left (64\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+4\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-\frac {4}{3}}}-{\left (2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+12\right )}^2+240\right )\,x+16}{{\left (x^2-2\,x+2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+12\right )}^2}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2336*x + exp(6*exp(-4/3))*(264*x - 36*x^2 + 24*x^3 + 176) + exp(3*exp(-4/3))*(1408*x - 408*x^2 + 312*x^3
- 42*x^4 + 12*x^5 + 1248) + exp(9*exp(-4/3))*(16*x + 8) - 1152*x^2 + 1016*x^3 - 264*x^4 + 90*x^5 - 11*x^6 + 2*
x^7 + 2944)/(8*exp(9*exp(-4/3)) - 864*x + exp(3*exp(-4/3))*(168*x^2 - 288*x - 24*x^3 + 6*x^4 + 864) + 576*x^2
- 152*x^3 + 48*x^4 - 6*x^5 + x^6 + exp(6*exp(-4/3))*(12*x^2 - 24*x + 144) + 1728),x)

[Out]

x + (x^2*(48*exp(3*exp(-4/3)) + 4*exp(6*exp(-4/3)) - (2*exp(3*exp(-4/3)) + 12)^2 + 128) + 8*x^3 + x*(64*exp(3*
exp(-4/3)) + 4*exp(6*exp(-4/3)) - (2*exp(3*exp(-4/3)) + 12)^2 + 240) + 16)/(2*exp(3*exp(-4/3)) - 2*x + x^2 + 1
2)^2 + x^2

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sympy [B]  time = 1.44, size = 90, normalized size = 2.81 \begin {gather*} x^{2} + x + \frac {8 x^{3} - 16 x^{2} + x \left (16 e^{\frac {3}{e^{\frac {4}{3}}}} + 96\right ) + 16}{x^{4} - 4 x^{3} + x^{2} \left (4 e^{\frac {3}{e^{\frac {4}{3}}}} + 28\right ) + x \left (-48 - 8 e^{\frac {3}{e^{\frac {4}{3}}}}\right ) + 4 e^{\frac {6}{e^{\frac {4}{3}}}} + 48 e^{\frac {3}{e^{\frac {4}{3}}}} + 144} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x+8)*exp(3/exp(4/3))**3+(24*x**3-36*x**2+264*x+176)*exp(3/exp(4/3))**2+(12*x**5-42*x**4+312*x**
3-408*x**2+1408*x+1248)*exp(3/exp(4/3))+2*x**7-11*x**6+90*x**5-264*x**4+1016*x**3-1152*x**2+2336*x+2944)/(8*ex
p(3/exp(4/3))**3+(12*x**2-24*x+144)*exp(3/exp(4/3))**2+(6*x**4-24*x**3+168*x**2-288*x+864)*exp(3/exp(4/3))+x**
6-6*x**5+48*x**4-152*x**3+576*x**2-864*x+1728),x)

[Out]

x**2 + x + (8*x**3 - 16*x**2 + x*(16*exp(3*exp(-4/3)) + 96) + 16)/(x**4 - 4*x**3 + x**2*(4*exp(3*exp(-4/3)) +
28) + x*(-48 - 8*exp(3*exp(-4/3))) + 4*exp(6*exp(-4/3)) + 48*exp(3*exp(-4/3)) + 144)

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