3.56.76 \(\int \frac {33-48 x+18 x^2-2 x^3+e^5 (1-16 x+20 x^2-4 x^3)+e^{10} (2 x^2-2 x^3)+e^{x+x^2} (-5-7 x+2 x^2+e^5 (-1+x+2 x^2))}{16-8 x+x^2+e^{10} x^2+e^5 (-8 x+2 x^2)} \, dx\)

Optimal. Leaf size=29 \[ 2 x-x^2+\frac {-1+e^{x+x^2}}{-4+x+e^5 x} \]

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Rubi [B]  time = 0.97, antiderivative size = 400, normalized size of antiderivative = 13.79, number of steps used = 17, number of rules used = 8, integrand size = 110, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {6, 6741, 27, 6742, 43, 77, 2289, 1850} \begin {gather*} -\frac {e^{10} x^2}{\left (1+e^5\right )^2}-\frac {2 e^5 x^2}{\left (1+e^5\right )^2}-\frac {x^2}{\left (1+e^5\right )^2}-\frac {e^{x^2+x}}{4-\left (1+e^5\right ) x}+\frac {18 x}{\left (1+e^5\right )^2}-\frac {2 e^{10} \left (7-e^5\right ) x}{\left (1+e^5\right )^3}-\frac {4 e^5 \left (3-5 e^5\right ) x}{\left (1+e^5\right )^3}-\frac {16 x}{\left (1+e^5\right )^3}+\frac {e^5 \left (1+195 e^5-61 e^{10}+e^{15}\right )}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}+\frac {33}{\left (1+e^5\right ) \left (4-\left (1+e^5\right ) x\right )}-\frac {192}{\left (1+e^5\right )^2 \left (4-\left (1+e^5\right ) x\right )}+\frac {288}{\left (1+e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )}-\frac {32 e^{10} \left (3-e^5\right )}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}-\frac {128}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}-\frac {16 e^5 \left (3-8 e^5+e^{10}\right ) \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4}-\frac {48 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^2}+\frac {144 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^3}-\frac {16 e^{10} \left (5-e^5\right ) \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4}-\frac {96 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(33 - 48*x + 18*x^2 - 2*x^3 + E^5*(1 - 16*x + 20*x^2 - 4*x^3) + E^10*(2*x^2 - 2*x^3) + E^(x + x^2)*(-5 - 7
*x + 2*x^2 + E^5*(-1 + x + 2*x^2)))/(16 - 8*x + x^2 + E^10*x^2 + E^5*(-8*x + 2*x^2)),x]

[Out]

(-16*x)/(1 + E^5)^3 - (4*E^5*(3 - 5*E^5)*x)/(1 + E^5)^3 - (2*E^10*(7 - E^5)*x)/(1 + E^5)^3 + (18*x)/(1 + E^5)^
2 - x^2/(1 + E^5)^2 - (2*E^5*x^2)/(1 + E^5)^2 - (E^10*x^2)/(1 + E^5)^2 - E^(x + x^2)/(4 - (1 + E^5)*x) - 128/(
(1 + E^5)^4*(4 - (1 + E^5)*x)) - (32*E^10*(3 - E^5))/((1 + E^5)^4*(4 - (1 + E^5)*x)) + 288/((1 + E^5)^3*(4 - (
1 + E^5)*x)) - 192/((1 + E^5)^2*(4 - (1 + E^5)*x)) + 33/((1 + E^5)*(4 - (1 + E^5)*x)) + (E^5*(1 + 195*E^5 - 61
*E^10 + E^15))/((1 + E^5)^4*(4 - (1 + E^5)*x)) - (96*Log[4 - (1 + E^5)*x])/(1 + E^5)^4 - (16*E^10*(5 - E^5)*Lo
g[4 - (1 + E^5)*x])/(1 + E^5)^4 + (144*Log[4 - (1 + E^5)*x])/(1 + E^5)^3 - (48*Log[4 - (1 + E^5)*x])/(1 + E^5)
^2 - (16*E^5*(3 - 8*E^5 + E^10)*Log[4 - (1 + E^5)*x])/(1 + E^5)^4

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 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 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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2289

Int[(F_)^(u_)*(v_)^(n_.)*(w_), x_Symbol] :> With[{z = Log[F]*v*D[u, x] + (n + 1)*D[v, x]}, Simp[(Coefficient[w
, x, Exponent[w, x]]*F^u*v^(n + 1))/Coefficient[z, x, Exponent[z, x]], x] /; EqQ[Exponent[w, x], Exponent[z, x
]] && EqQ[w*Coefficient[z, x, Exponent[z, x]], z*Coefficient[w, x, Exponent[w, x]]]] /; FreeQ[{F, n}, x] && Po
lynomialQ[u, x] && PolynomialQ[v, x] && PolynomialQ[w, x]

Rule 6741

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

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 {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 x+\left (1+e^{10}\right ) x^2+e^5 \left (-8 x+2 x^2\right )} \, dx\\ &=\int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{16-8 \left (1+e^5\right ) x+\left (1+e^5\right )^2 x^2} \, dx\\ &=\int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{\left (-4+x+e^5 x\right )^2} \, dx\\ &=\int \frac {33-48 x+18 x^2-2 x^3+e^5 \left (1-16 x+20 x^2-4 x^3\right )+e^{10} \left (2 x^2-2 x^3\right )+e^{x+x^2} \left (-5-7 x+2 x^2+e^5 \left (-1+x+2 x^2\right )\right )}{\left (-4+\left (1+e^5\right ) x\right )^2} \, dx\\ &=\int \left (\frac {33}{\left (4-\left (1+e^5\right ) x\right )^2}-\frac {48 x}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {18 x^2}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {2 e^{10} (1-x) x^2}{\left (4-\left (1+e^5\right ) x\right )^2}-\frac {2 x^3}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {e^{x+x^2} \left (-5-e^5-\left (7-e^5\right ) x+2 \left (1+e^5\right ) x^2\right )}{\left (4-\left (1+e^5\right ) x\right )^2}+\frac {e^5 \left (1-16 x+20 x^2-4 x^3\right )}{\left (4-\left (1+e^5\right ) x\right )^2}\right ) \, dx\\ &=\frac {33}{\left (1+e^5\right ) \left (4-\left (1+e^5\right ) x\right )}-2 \int \frac {x^3}{\left (4-\left (1+e^5\right ) x\right )^2} \, dx+18 \int \frac {x^2}{\left (4-\left (1+e^5\right ) x\right )^2} \, dx-48 \int \frac {x}{\left (4-\left (1+e^5\right ) x\right )^2} \, dx+e^5 \int \frac {1-16 x+20 x^2-4 x^3}{\left (4-\left (1+e^5\right ) x\right )^2} \, dx+\left (2 e^{10}\right ) \int \frac {(1-x) x^2}{\left (4-\left (1+e^5\right ) x\right )^2} \, dx+\int \frac {e^{x+x^2} \left (-5-e^5-\left (7-e^5\right ) x+2 \left (1+e^5\right ) x^2\right )}{\left (4-\left (1+e^5\right ) x\right )^2} \, dx\\ &=-\frac {e^{x+x^2}}{4-\left (1+e^5\right ) x}+\frac {33}{\left (1+e^5\right ) \left (4-\left (1+e^5\right ) x\right )}-2 \int \left (\frac {8}{\left (1+e^5\right )^3}+\frac {x}{\left (1+e^5\right )^2}+\frac {64}{\left (1+e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )^2}+\frac {48}{\left (-1-e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )}\right ) \, dx+18 \int \left (\frac {1}{\left (1+e^5\right )^2}+\frac {16}{\left (1+e^5\right )^2 \left (4-\left (1+e^5\right ) x\right )^2}+\frac {8}{\left (1+e^5\right )^2 \left (-4+\left (1+e^5\right ) x\right )}\right ) \, dx-48 \int \left (\frac {4}{\left (1+e^5\right ) \left (4-\left (1+e^5\right ) x\right )^2}+\frac {1}{\left (-1-e^5\right ) \left (4-\left (1+e^5\right ) x\right )}\right ) \, dx+e^5 \int \left (\frac {4 \left (-3+5 e^5\right )}{\left (1+e^5\right )^3}-\frac {4 x}{\left (1+e^5\right )^2}+\frac {1+195 e^5-61 e^{10}+e^{15}}{\left (1+e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )^2}+\frac {16 \left (3-8 e^5+e^{10}\right )}{\left (1+e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )}\right ) \, dx+\left (2 e^{10}\right ) \int \left (\frac {-7+e^5}{\left (1+e^5\right )^3}-\frac {x}{\left (1+e^5\right )^2}+\frac {16 \left (-3+e^5\right )}{\left (1+e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )^2}+\frac {8 \left (5-e^5\right )}{\left (1+e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )}\right ) \, dx\\ &=-\frac {16 x}{\left (1+e^5\right )^3}-\frac {4 e^5 \left (3-5 e^5\right ) x}{\left (1+e^5\right )^3}-\frac {2 e^{10} \left (7-e^5\right ) x}{\left (1+e^5\right )^3}+\frac {18 x}{\left (1+e^5\right )^2}-\frac {x^2}{\left (1+e^5\right )^2}-\frac {2 e^5 x^2}{\left (1+e^5\right )^2}-\frac {e^{10} x^2}{\left (1+e^5\right )^2}-\frac {e^{x+x^2}}{4-\left (1+e^5\right ) x}-\frac {128}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}-\frac {32 e^{10} \left (3-e^5\right )}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}+\frac {288}{\left (1+e^5\right )^3 \left (4-\left (1+e^5\right ) x\right )}-\frac {192}{\left (1+e^5\right )^2 \left (4-\left (1+e^5\right ) x\right )}+\frac {33}{\left (1+e^5\right ) \left (4-\left (1+e^5\right ) x\right )}+\frac {e^5 \left (1+195 e^5-61 e^{10}+e^{15}\right )}{\left (1+e^5\right )^4 \left (4-\left (1+e^5\right ) x\right )}-\frac {96 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4}-\frac {16 e^{10} \left (5-e^5\right ) \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4}+\frac {144 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^3}-\frac {48 \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^2}-\frac {16 e^5 \left (3-8 e^5+e^{10}\right ) \log \left (4-\left (1+e^5\right ) x\right )}{\left (1+e^5\right )^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 29, normalized size = 1.00 \begin {gather*} 2 x-x^2+\frac {-1+e^{x+x^2}}{-4+x+e^5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(33 - 48*x + 18*x^2 - 2*x^3 + E^5*(1 - 16*x + 20*x^2 - 4*x^3) + E^10*(2*x^2 - 2*x^3) + E^(x + x^2)*(
-5 - 7*x + 2*x^2 + E^5*(-1 + x + 2*x^2)))/(16 - 8*x + x^2 + E^10*x^2 + E^5*(-8*x + 2*x^2)),x]

[Out]

2*x - x^2 + (-1 + E^(x + x^2))/(-4 + x + E^5*x)

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fricas [A]  time = 0.52, size = 44, normalized size = 1.52 \begin {gather*} -\frac {x^{3} - 6 \, x^{2} + {\left (x^{3} - 2 \, x^{2}\right )} e^{5} + 8 \, x - e^{\left (x^{2} + x\right )} + 1}{x e^{5} + x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+x-1)*exp(5)+2*x^2-7*x-5)*exp(x^2+x)+(-2*x^3+2*x^2)*exp(5)^2+(-4*x^3+20*x^2-16*x+1)*exp(5)-2
*x^3+18*x^2-48*x+33)/(x^2*exp(5)^2+(2*x^2-8*x)*exp(5)+x^2-8*x+16),x, algorithm="fricas")

[Out]

-(x^3 - 6*x^2 + (x^3 - 2*x^2)*e^5 + 8*x - e^(x^2 + x) + 1)/(x*e^5 + x - 4)

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giac [A]  time = 0.59, size = 45, normalized size = 1.55 \begin {gather*} -\frac {x^{3} e^{5} + x^{3} - 2 \, x^{2} e^{5} - 6 \, x^{2} + 8 \, x - e^{\left (x^{2} + x\right )} + 1}{x e^{5} + x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+x-1)*exp(5)+2*x^2-7*x-5)*exp(x^2+x)+(-2*x^3+2*x^2)*exp(5)^2+(-4*x^3+20*x^2-16*x+1)*exp(5)-2
*x^3+18*x^2-48*x+33)/(x^2*exp(5)^2+(2*x^2-8*x)*exp(5)+x^2-8*x+16),x, algorithm="giac")

[Out]

-(x^3*e^5 + x^3 - 2*x^2*e^5 - 6*x^2 + 8*x - e^(x^2 + x) + 1)/(x*e^5 + x - 4)

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maple [A]  time = 0.47, size = 37, normalized size = 1.28

method	result	size
risch	\(-x^{2}+2 x -\frac {1}{x +x \,{\mathrm e}^{5}-4}+\frac {{\mathrm e}^{\left (x +1\right ) x}}{x +x \,{\mathrm e}^{5}-4}\)	\(37\)
norman	\(\frac {\left (-{\mathrm e}^{5}-1\right ) x^{3}+\left (2 \,{\mathrm e}^{5}+6\right ) x^{2}+\left (-\frac {33}{4}-\frac {{\mathrm e}^{5}}{4}\right ) x +{\mathrm e}^{x^{2}+x}}{x +x \,{\mathrm e}^{5}-4}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2+x-1)*exp(5)+2*x^2-7*x-5)*exp(x^2+x)+(-2*x^3+2*x^2)*exp(5)^2+(-4*x^3+20*x^2-16*x+1)*exp(5)-2*x^3+1
8*x^2-48*x+33)/(x^2*exp(5)^2+(2*x^2-8*x)*exp(5)+x^2-8*x+16),x,method=_RETURNVERBOSE)

[Out]

-x^2+2*x-1/(x+x*exp(5)-4)+1/(x+x*exp(5)-4)*exp((x+1)*x)

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maxima [B]  time = 0.43, size = 703, normalized size = 24.24 \begin {gather*} -{\left (\frac {x^{2} {\left (e^{5} + 1\right )} + 16 \, x}{e^{15} + 3 \, e^{10} + 3 \, e^{5} + 1} + \frac {96 \, \log \left (x {\left (e^{5} + 1\right )} - 4\right )}{e^{20} + 4 \, e^{15} + 6 \, e^{10} + 4 \, e^{5} + 1} - \frac {128}{x {\left (e^{25} + 5 \, e^{20} + 10 \, e^{15} + 10 \, e^{10} + 5 \, e^{5} + 1\right )} - 4 \, e^{20} - 16 \, e^{15} - 24 \, e^{10} - 16 \, e^{5} - 4}\right )} e^{10} + 2 \, {\left (\frac {x}{e^{10} + 2 \, e^{5} + 1} + \frac {8 \, \log \left (x {\left (e^{5} + 1\right )} - 4\right )}{e^{15} + 3 \, e^{10} + 3 \, e^{5} + 1} - \frac {16}{x {\left (e^{20} + 4 \, e^{15} + 6 \, e^{10} + 4 \, e^{5} + 1\right )} - 4 \, e^{15} - 12 \, e^{10} - 12 \, e^{5} - 4}\right )} e^{10} - 2 \, {\left (\frac {x^{2} {\left (e^{5} + 1\right )} + 16 \, x}{e^{15} + 3 \, e^{10} + 3 \, e^{5} + 1} + \frac {96 \, \log \left (x {\left (e^{5} + 1\right )} - 4\right )}{e^{20} + 4 \, e^{15} + 6 \, e^{10} + 4 \, e^{5} + 1} - \frac {128}{x {\left (e^{25} + 5 \, e^{20} + 10 \, e^{15} + 10 \, e^{10} + 5 \, e^{5} + 1\right )} - 4 \, e^{20} - 16 \, e^{15} - 24 \, e^{10} - 16 \, e^{5} - 4}\right )} e^{5} + 20 \, {\left (\frac {x}{e^{10} + 2 \, e^{5} + 1} + \frac {8 \, \log \left (x {\left (e^{5} + 1\right )} - 4\right )}{e^{15} + 3 \, e^{10} + 3 \, e^{5} + 1} - \frac {16}{x {\left (e^{20} + 4 \, e^{15} + 6 \, e^{10} + 4 \, e^{5} + 1\right )} - 4 \, e^{15} - 12 \, e^{10} - 12 \, e^{5} - 4}\right )} e^{5} - 16 \, {\left (\frac {\log \left (x {\left (e^{5} + 1\right )} - 4\right )}{e^{10} + 2 \, e^{5} + 1} - \frac {4}{x {\left (e^{15} + 3 \, e^{10} + 3 \, e^{5} + 1\right )} - 4 \, e^{10} - 8 \, e^{5} - 4}\right )} e^{5} - \frac {x^{2} {\left (e^{5} + 1\right )} + 16 \, x}{e^{15} + 3 \, e^{10} + 3 \, e^{5} + 1} + \frac {18 \, x}{e^{10} + 2 \, e^{5} + 1} - \frac {e^{5}}{x {\left (e^{10} + 2 \, e^{5} + 1\right )} - 4 \, e^{5} - 4} + \frac {e^{\left (x^{2} + x\right )}}{x {\left (e^{5} + 1\right )} - 4} - \frac {96 \, \log \left (x {\left (e^{5} + 1\right )} - 4\right )}{e^{20} + 4 \, e^{15} + 6 \, e^{10} + 4 \, e^{5} + 1} + \frac {144 \, \log \left (x {\left (e^{5} + 1\right )} - 4\right )}{e^{15} + 3 \, e^{10} + 3 \, e^{5} + 1} - \frac {48 \, \log \left (x {\left (e^{5} + 1\right )} - 4\right )}{e^{10} + 2 \, e^{5} + 1} + \frac {128}{x {\left (e^{25} + 5 \, e^{20} + 10 \, e^{15} + 10 \, e^{10} + 5 \, e^{5} + 1\right )} - 4 \, e^{20} - 16 \, e^{15} - 24 \, e^{10} - 16 \, e^{5} - 4} - \frac {288}{x {\left (e^{20} + 4 \, e^{15} + 6 \, e^{10} + 4 \, e^{5} + 1\right )} - 4 \, e^{15} - 12 \, e^{10} - 12 \, e^{5} - 4} + \frac {192}{x {\left (e^{15} + 3 \, e^{10} + 3 \, e^{5} + 1\right )} - 4 \, e^{10} - 8 \, e^{5} - 4} - \frac {33}{x {\left (e^{10} + 2 \, e^{5} + 1\right )} - 4 \, e^{5} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2+x-1)*exp(5)+2*x^2-7*x-5)*exp(x^2+x)+(-2*x^3+2*x^2)*exp(5)^2+(-4*x^3+20*x^2-16*x+1)*exp(5)-2
*x^3+18*x^2-48*x+33)/(x^2*exp(5)^2+(2*x^2-8*x)*exp(5)+x^2-8*x+16),x, algorithm="maxima")

[Out]

-((x^2*(e^5 + 1) + 16*x)/(e^15 + 3*e^10 + 3*e^5 + 1) + 96*log(x*(e^5 + 1) - 4)/(e^20 + 4*e^15 + 6*e^10 + 4*e^5
+ 1) - 128/(x*(e^25 + 5*e^20 + 10*e^15 + 10*e^10 + 5*e^5 + 1) - 4*e^20 - 16*e^15 - 24*e^10 - 16*e^5 - 4))*e^1
0 + 2*(x/(e^10 + 2*e^5 + 1) + 8*log(x*(e^5 + 1) - 4)/(e^15 + 3*e^10 + 3*e^5 + 1) - 16/(x*(e^20 + 4*e^15 + 6*e^
10 + 4*e^5 + 1) - 4*e^15 - 12*e^10 - 12*e^5 - 4))*e^10 - 2*((x^2*(e^5 + 1) + 16*x)/(e^15 + 3*e^10 + 3*e^5 + 1)
+ 96*log(x*(e^5 + 1) - 4)/(e^20 + 4*e^15 + 6*e^10 + 4*e^5 + 1) - 128/(x*(e^25 + 5*e^20 + 10*e^15 + 10*e^10 +
5*e^5 + 1) - 4*e^20 - 16*e^15 - 24*e^10 - 16*e^5 - 4))*e^5 + 20*(x/(e^10 + 2*e^5 + 1) + 8*log(x*(e^5 + 1) - 4)
/(e^15 + 3*e^10 + 3*e^5 + 1) - 16/(x*(e^20 + 4*e^15 + 6*e^10 + 4*e^5 + 1) - 4*e^15 - 12*e^10 - 12*e^5 - 4))*e^
5 - 16*(log(x*(e^5 + 1) - 4)/(e^10 + 2*e^5 + 1) - 4/(x*(e^15 + 3*e^10 + 3*e^5 + 1) - 4*e^10 - 8*e^5 - 4))*e^5
- (x^2*(e^5 + 1) + 16*x)/(e^15 + 3*e^10 + 3*e^5 + 1) + 18*x/(e^10 + 2*e^5 + 1) - e^5/(x*(e^10 + 2*e^5 + 1) - 4
*e^5 - 4) + e^(x^2 + x)/(x*(e^5 + 1) - 4) - 96*log(x*(e^5 + 1) - 4)/(e^20 + 4*e^15 + 6*e^10 + 4*e^5 + 1) + 144
*log(x*(e^5 + 1) - 4)/(e^15 + 3*e^10 + 3*e^5 + 1) - 48*log(x*(e^5 + 1) - 4)/(e^10 + 2*e^5 + 1) + 128/(x*(e^25
+ 5*e^20 + 10*e^15 + 10*e^10 + 5*e^5 + 1) - 4*e^20 - 16*e^15 - 24*e^10 - 16*e^5 - 4) - 288/(x*(e^20 + 4*e^15 +
6*e^10 + 4*e^5 + 1) - 4*e^15 - 12*e^10 - 12*e^5 - 4) + 192/(x*(e^15 + 3*e^10 + 3*e^5 + 1) - 4*e^10 - 8*e^5 -
4) - 33/(x*(e^10 + 2*e^5 + 1) - 4*e^5 - 4)

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mupad [B]  time = 4.24, size = 28, normalized size = 0.97 \begin {gather*} 2\,x+\frac {{\mathrm {e}}^{x^2+x}-1}{x\,\left ({\mathrm {e}}^5+1\right )-4}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*x + exp(5)*(16*x - 20*x^2 + 4*x^3 - 1) - exp(10)*(2*x^2 - 2*x^3) + exp(x + x^2)*(7*x - exp(5)*(x + 2*
x^2 - 1) - 2*x^2 + 5) - 18*x^2 + 2*x^3 - 33)/(x^2*exp(10) - exp(5)*(8*x - 2*x^2) - 8*x + x^2 + 16),x)

[Out]

2*x + (exp(x + x^2) - 1)/(x*(exp(5) + 1) - 4) - x^2

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sympy [A]  time = 0.44, size = 31, normalized size = 1.07 \begin {gather*} - x^{2} + 2 x + \frac {e^{x^{2} + x}}{x + x e^{5} - 4} - \frac {1}{x \left (1 + e^{5}\right ) - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2+x-1)*exp(5)+2*x**2-7*x-5)*exp(x**2+x)+(-2*x**3+2*x**2)*exp(5)**2+(-4*x**3+20*x**2-16*x+1)*
exp(5)-2*x**3+18*x**2-48*x+33)/(x**2*exp(5)**2+(2*x**2-8*x)*exp(5)+x**2-8*x+16),x)

[Out]

-x**2 + 2*x + exp(x**2 + x)/(x + x*exp(5) - 4) - 1/(x*(1 + exp(5)) - 4)

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