3.58.90 \(\int \frac {x^2+e^{10} x^2-2 x^3+2 x^4+x^6+e^{-147+x} (-1+e^5 (-1+x)+x-3 x^2+x^3)+e^5 (2 x^2+2 x^4)}{x^2+e^{10} x^2+2 x^4+x^6+e^5 (2 x^2+2 x^4)} \, dx\)

Optimal. Leaf size=23 \[ x+\frac {e^{-147+x}+x}{x \left (1+e^5+x^2\right )} \]

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Rubi [B]  time = 0.91, antiderivative size = 233, normalized size of antiderivative = 10.13, number of steps used = 16, number of rules used = 10, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 6741, 6742, 199, 203, 261, 288, 321, 385, 2288} \begin {gather*} \frac {\left (1+e^{10}\right ) x}{2 \left (1+e^5\right ) \left (x^2+e^5+1\right )}-\frac {e^{10} x}{\left (1+e^5\right ) \left (x^2+e^5+1\right )}-\frac {x}{x^2+e^5+1}+\frac {1}{x^2+e^5+1}-\frac {x^3}{2 \left (x^2+e^5+1\right )}+\frac {e^{x-147} \left (x^3+\left (1+e^5\right ) x\right )}{\left (x^2+e^5+1\right )^2 x^2}+\frac {3 x}{2}+\frac {\left (1+e^{10}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{2 \left (1+e^5\right )^{3/2}}+\frac {e^5 \left (2+e^5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\left (1+e^5\right )^{3/2}}-\frac {3}{2} \sqrt {1+e^5} \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\sqrt {1+e^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2 + E^10*x^2 - 2*x^3 + 2*x^4 + x^6 + E^(-147 + x)*(-1 + E^5*(-1 + x) + x - 3*x^2 + x^3) + E^5*(2*x^2 +
2*x^4))/(x^2 + E^10*x^2 + 2*x^4 + x^6 + E^5*(2*x^2 + 2*x^4)),x]

[Out]

(3*x)/2 + (1 + E^5 + x^2)^(-1) - x/(1 + E^5 + x^2) - (E^10*x)/((1 + E^5)*(1 + E^5 + x^2)) + ((1 + E^10)*x)/(2*
(1 + E^5)*(1 + E^5 + x^2)) - x^3/(2*(1 + E^5 + x^2)) + (E^(-147 + x)*((1 + E^5)*x + x^3))/(x^2*(1 + E^5 + x^2)
^2) + ArcTan[x/Sqrt[1 + E^5]]/Sqrt[1 + E^5] - (3*Sqrt[1 + E^5]*ArcTan[x/Sqrt[1 + E^5]])/2 + (E^5*(2 + E^5)*Arc
Tan[x/Sqrt[1 + E^5]])/(1 + E^5)^(3/2) + ((1 + E^10)*ArcTan[x/Sqrt[1 + E^5]])/(2*(1 + E^5)^(3/2))

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 203

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 385

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, 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 {x^2+e^{10} x^2-2 x^3+2 x^4+x^6+e^{-147+x} \left (-1+e^5 (-1+x)+x-3 x^2+x^3\right )+e^5 \left (2 x^2+2 x^4\right )}{\left (1+e^{10}\right ) x^2+2 x^4+x^6+e^5 \left (2 x^2+2 x^4\right )} \, dx\\ &=\int \frac {\left (1+e^{10}\right ) x^2-2 x^3+2 x^4+x^6+e^{-147+x} \left (-1+e^5 (-1+x)+x-3 x^2+x^3\right )+e^5 \left (2 x^2+2 x^4\right )}{\left (1+e^{10}\right ) x^2+2 x^4+x^6+e^5 \left (2 x^2+2 x^4\right )} \, dx\\ &=\int \frac {\left (1+e^{10}\right ) x^2-2 x^3+2 x^4+x^6+e^{-147+x} \left (-1+e^5 (-1+x)+x-3 x^2+x^3\right )+e^5 \left (2 x^2+2 x^4\right )}{x^2 \left (1+e^5+x^2\right )^2} \, dx\\ &=\int \left (\frac {1+e^{10}}{\left (1+e^5+x^2\right )^2}-\frac {2 x}{\left (1+e^5+x^2\right )^2}+\frac {2 x^2}{\left (1+e^5+x^2\right )^2}+\frac {x^4}{\left (1+e^5+x^2\right )^2}+\frac {2 e^5 \left (1+x^2\right )}{\left (1+e^5+x^2\right )^2}+\frac {e^{-147+x} \left (-1-e^5+\left (1+e^5\right ) x-3 x^2+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\left (1+e^5+x^2\right )^2} \, dx\right )+2 \int \frac {x^2}{\left (1+e^5+x^2\right )^2} \, dx+\left (2 e^5\right ) \int \frac {1+x^2}{\left (1+e^5+x^2\right )^2} \, dx+\left (1+e^{10}\right ) \int \frac {1}{\left (1+e^5+x^2\right )^2} \, dx+\int \frac {x^4}{\left (1+e^5+x^2\right )^2} \, dx+\int \frac {e^{-147+x} \left (-1-e^5+\left (1+e^5\right ) x-3 x^2+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2} \, dx\\ &=\frac {1}{1+e^5+x^2}-\frac {x}{1+e^5+x^2}-\frac {e^{10} x}{\left (1+e^5\right ) \left (1+e^5+x^2\right )}+\frac {\left (1+e^{10}\right ) x}{2 \left (1+e^5\right ) \left (1+e^5+x^2\right )}-\frac {x^3}{2 \left (1+e^5+x^2\right )}+\frac {e^{-147+x} \left (\left (1+e^5\right ) x+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2}+\frac {3}{2} \int \frac {x^2}{1+e^5+x^2} \, dx+\frac {\left (e^5 \left (2+e^5\right )\right ) \int \frac {1}{1+e^5+x^2} \, dx}{1+e^5}+\frac {\left (1+e^{10}\right ) \int \frac {1}{1+e^5+x^2} \, dx}{2 \left (1+e^5\right )}+\int \frac {1}{1+e^5+x^2} \, dx\\ &=\frac {3 x}{2}+\frac {1}{1+e^5+x^2}-\frac {x}{1+e^5+x^2}-\frac {e^{10} x}{\left (1+e^5\right ) \left (1+e^5+x^2\right )}+\frac {\left (1+e^{10}\right ) x}{2 \left (1+e^5\right ) \left (1+e^5+x^2\right )}-\frac {x^3}{2 \left (1+e^5+x^2\right )}+\frac {e^{-147+x} \left (\left (1+e^5\right ) x+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\sqrt {1+e^5}}+\frac {e^5 \left (2+e^5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\left (1+e^5\right )^{3/2}}+\frac {\left (1+e^{10}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{2 \left (1+e^5\right )^{3/2}}-\frac {1}{2} \left (3 \left (1+e^5\right )\right ) \int \frac {1}{1+e^5+x^2} \, dx\\ &=\frac {3 x}{2}+\frac {1}{1+e^5+x^2}-\frac {x}{1+e^5+x^2}-\frac {e^{10} x}{\left (1+e^5\right ) \left (1+e^5+x^2\right )}+\frac {\left (1+e^{10}\right ) x}{2 \left (1+e^5\right ) \left (1+e^5+x^2\right )}-\frac {x^3}{2 \left (1+e^5+x^2\right )}+\frac {e^{-147+x} \left (\left (1+e^5\right ) x+x^3\right )}{x^2 \left (1+e^5+x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\sqrt {1+e^5}}-\frac {3}{2} \sqrt {1+e^5} \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )+\frac {e^5 \left (2+e^5\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{\left (1+e^5\right )^{3/2}}+\frac {\left (1+e^{10}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1+e^5}}\right )}{2 \left (1+e^5\right )^{3/2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 39, normalized size = 1.70 \begin {gather*} \frac {e^x+e^{152} x^2+e^{147} x \left (1+x+x^3\right )}{e^{147} x \left (1+e^5+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + E^10*x^2 - 2*x^3 + 2*x^4 + x^6 + E^(-147 + x)*(-1 + E^5*(-1 + x) + x - 3*x^2 + x^3) + E^5*(2*
x^2 + 2*x^4))/(x^2 + E^10*x^2 + 2*x^4 + x^6 + E^5*(2*x^2 + 2*x^4)),x]

[Out]

(E^x + E^152*x^2 + E^147*x*(1 + x + x^3))/(E^147*x*(1 + E^5 + x^2))

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fricas [A]  time = 0.56, size = 30, normalized size = 1.30 \begin {gather*} \frac {x^{4} + x^{2} e^{5} + x^{2} + x + e^{\left (x - 147\right )}}{x^{3} + x e^{5} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(5)+x^3-3*x^2+x-1)*exp(x-147)+x^2*exp(5)^2+(2*x^4+2*x^2)*exp(5)+x^6+2*x^4-2*x^3+x^2)/(x^2
*exp(5)^2+(2*x^4+2*x^2)*exp(5)+x^6+2*x^4+x^2),x, algorithm="fricas")

[Out]

(x^4 + x^2*e^5 + x^2 + x + e^(x - 147))/(x^3 + x*e^5 + x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(5)+x^3-3*x^2+x-1)*exp(x-147)+x^2*exp(5)^2+(2*x^4+2*x^2)*exp(5)+x^6+2*x^4-2*x^3+x^2)/(x^2
*exp(5)^2+(2*x^4+2*x^2)*exp(5)+x^6+2*x^4+x^2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.82, size = 29, normalized size = 1.26

method	result	size
risch	\(x +\frac {1}{1+x^{2}+{\mathrm e}^{5}}+\frac {{\mathrm e}^{x -147}}{x \left (1+x^{2}+{\mathrm e}^{5}\right )}\)	\(29\)
norman	\(\frac {x^{4}+x +\left ({\mathrm e}^{5}+1\right ) x^{2}+{\mathrm e}^{x -147}}{x \left (1+x^{2}+{\mathrm e}^{5}\right )}\)	\(31\)
default	\(\frac {2 x}{\left (4 \,{\mathrm e}^{5}+4\right ) \left (1+x^{2}+{\mathrm e}^{5}\right )}+\frac {2 \arctan \left (\frac {x}{\sqrt {{\mathrm e}^{5}+1}}\right )}{\left (4 \,{\mathrm e}^{5}+4\right ) \sqrt {{\mathrm e}^{5}+1}}+x +\frac {\left ({\mathrm e}^{10}+2 \,{\mathrm e}^{5}+1\right ) x}{2 \left ({\mathrm e}^{5}+1\right ) \left (1+x^{2}+{\mathrm e}^{5}\right )}-\frac {3 \arctan \left (\frac {x}{\sqrt {{\mathrm e}^{5}+1}}\right ) {\mathrm e}^{10}}{2 \left ({\mathrm e}^{5}+1\right )^{\frac {3}{2}}}-\frac {3 \arctan \left (\frac {x}{\sqrt {{\mathrm e}^{5}+1}}\right ) {\mathrm e}^{5}}{\left ({\mathrm e}^{5}+1\right )^{\frac {3}{2}}}-\frac {3 \arctan \left (\frac {x}{\sqrt {{\mathrm e}^{5}+1}}\right )}{2 \left ({\mathrm e}^{5}+1\right )^{\frac {3}{2}}}+\frac {2 \,{\mathrm e}^{10} x}{\left (4 \,{\mathrm e}^{5}+4\right ) \left (1+x^{2}+{\mathrm e}^{5}\right )}+\frac {2 \,{\mathrm e}^{10} \arctan \left (\frac {x}{\sqrt {{\mathrm e}^{5}+1}}\right )}{\left (4 \,{\mathrm e}^{5}+4\right ) \sqrt {{\mathrm e}^{5}+1}}+{\mathrm e}^{-147} \left (\frac {{\mathrm e}^{x}}{2 \left ({\mathrm e}^{5}+1\right ) \left (1+x^{2}+{\mathrm e}^{5}\right )}-\frac {i \left (2 i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )+2 i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )+{\mathrm e}^{5} {\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )-{\mathrm e}^{5} {\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )+{\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )-{\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )\right )}{4 \left ({\mathrm e}^{5}+1\right )^{\frac {5}{2}}}-\frac {\expIntegralEi \left (1, -x \right )}{\left ({\mathrm e}^{5}+1\right )^{2}}\right )+{\mathrm e}^{-147} \left (-\frac {{\mathrm e}^{x}}{2 \left (1+x^{2}+{\mathrm e}^{5}\right )}+\frac {i \left ({\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )-{\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )\right )}{4 \sqrt {{\mathrm e}^{5}+1}}\right )+{\mathrm e}^{-147} {\mathrm e}^{5} \left (\frac {{\mathrm e}^{x}}{2 \left ({\mathrm e}^{5}+1\right ) \left (1+x^{2}+{\mathrm e}^{5}\right )}-\frac {i \left (2 i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )+2 i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )+{\mathrm e}^{5} {\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )-{\mathrm e}^{5} {\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )+{\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )-{\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )\right )}{4 \left ({\mathrm e}^{5}+1\right )^{\frac {5}{2}}}-\frac {\expIntegralEi \left (1, -x \right )}{\left ({\mathrm e}^{5}+1\right )^{2}}\right )+\frac {1}{1+x^{2}+{\mathrm e}^{5}}-\frac {x}{1+x^{2}+{\mathrm e}^{5}}+\frac {\arctan \left (\frac {x}{\sqrt {{\mathrm e}^{5}+1}}\right )}{\sqrt {{\mathrm e}^{5}+1}}+\frac {4 \,{\mathrm e}^{5} x}{\left (4 \,{\mathrm e}^{5}+4\right ) \left (1+x^{2}+{\mathrm e}^{5}\right )}+\frac {4 \,{\mathrm e}^{5} \arctan \left (\frac {x}{\sqrt {{\mathrm e}^{5}+1}}\right )}{\left (4 \,{\mathrm e}^{5}+4\right ) \sqrt {{\mathrm e}^{5}+1}}-\frac {{\mathrm e}^{5} x}{1+x^{2}+{\mathrm e}^{5}}+\frac {{\mathrm e}^{5} \arctan \left (\frac {x}{\sqrt {{\mathrm e}^{5}+1}}\right )}{\sqrt {{\mathrm e}^{5}+1}}-{\mathrm e}^{-147} \left (-\frac {{\mathrm e}^{x} \left (3 x^{2}+2 \,{\mathrm e}^{5}+2\right )}{2 \left ({\mathrm e}^{5}+1\right )^{2} \left (1+x^{2}+{\mathrm e}^{5}\right ) x}+\frac {i \left (i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )+i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )-3 \,{\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )+3 \,{\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )\right )}{4 \left ({\mathrm e}^{5}+1\right )^{\frac {5}{2}}}-\frac {\expIntegralEi \left (1, -x \right )}{\left ({\mathrm e}^{5}+1\right )^{2}}\right )-3 \,{\mathrm e}^{-147} \left (\frac {{\mathrm e}^{x} x}{2 \left ({\mathrm e}^{5}+1\right ) \left (1+x^{2}+{\mathrm e}^{5}\right )}-\frac {i \left (i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )+i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )-{\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )+{\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )\right )}{4 \left ({\mathrm e}^{5}+1\right )^{\frac {3}{2}}}\right )-{\mathrm e}^{-147} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{x} \left (3 x^{2}+2 \,{\mathrm e}^{5}+2\right )}{2 \left ({\mathrm e}^{5}+1\right )^{2} \left (1+x^{2}+{\mathrm e}^{5}\right ) x}+\frac {i \left (i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )+i \sqrt {{\mathrm e}^{5}+1}\, {\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )-3 \,{\mathrm e}^{i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x +i \sqrt {{\mathrm e}^{5}+1}\right )+3 \,{\mathrm e}^{-i \sqrt {{\mathrm e}^{5}+1}} \expIntegralEi \left (1, -x -i \sqrt {{\mathrm e}^{5}+1}\right )\right )}{4 \left ({\mathrm e}^{5}+1\right )^{\frac {5}{2}}}-\frac {\expIntegralEi \left (1, -x \right )}{\left ({\mathrm e}^{5}+1\right )^{2}}\right )\)	\(1383\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-1)*exp(5)+x^3-3*x^2+x-1)*exp(x-147)+x^2*exp(5)^2+(2*x^4+2*x^2)*exp(5)+x^6+2*x^4-2*x^3+x^2)/(x^2*exp(5
)^2+(2*x^4+2*x^2)*exp(5)+x^6+2*x^4+x^2),x,method=_RETURNVERBOSE)

[Out]

x+1/(1+x^2+exp(5))+1/x/(1+x^2+exp(5))*exp(x-147)

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maxima [B]  time = 0.53, size = 242, normalized size = 10.52 \begin {gather*} \frac {1}{2} \, {\left (\frac {x}{x^{2} {\left (e^{5} + 1\right )} + e^{10} + 2 \, e^{5} + 1} + \frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{{\left (e^{5} + 1\right )}^{\frac {3}{2}}}\right )} e^{10} + {\left (\frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{\sqrt {e^{5} + 1}} - \frac {x}{x^{2} + e^{5} + 1}\right )} e^{5} + {\left (\frac {x}{x^{2} {\left (e^{5} + 1\right )} + e^{10} + 2 \, e^{5} + 1} + \frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{{\left (e^{5} + 1\right )}^{\frac {3}{2}}}\right )} e^{5} - \frac {3}{2} \, \sqrt {e^{5} + 1} \arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right ) + x + \frac {x {\left (e^{5} + 1\right )}}{2 \, {\left (x^{2} + e^{5} + 1\right )}} + \frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{\sqrt {e^{5} + 1}} + \frac {x}{2 \, {\left (x^{2} {\left (e^{5} + 1\right )} + e^{10} + 2 \, e^{5} + 1\right )}} - \frac {x}{x^{2} + e^{5} + 1} + \frac {e^{x}}{x^{3} e^{147} + x {\left (e^{152} + e^{147}\right )}} + \frac {\arctan \left (\frac {x}{\sqrt {e^{5} + 1}}\right )}{2 \, {\left (e^{5} + 1\right )}^{\frac {3}{2}}} + \frac {1}{x^{2} + e^{5} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(5)+x^3-3*x^2+x-1)*exp(x-147)+x^2*exp(5)^2+(2*x^4+2*x^2)*exp(5)+x^6+2*x^4-2*x^3+x^2)/(x^2
*exp(5)^2+(2*x^4+2*x^2)*exp(5)+x^6+2*x^4+x^2),x, algorithm="maxima")

[Out]

1/2*(x/(x^2*(e^5 + 1) + e^10 + 2*e^5 + 1) + arctan(x/sqrt(e^5 + 1))/(e^5 + 1)^(3/2))*e^10 + (arctan(x/sqrt(e^5
+ 1))/sqrt(e^5 + 1) - x/(x^2 + e^5 + 1))*e^5 + (x/(x^2*(e^5 + 1) + e^10 + 2*e^5 + 1) + arctan(x/sqrt(e^5 + 1)
)/(e^5 + 1)^(3/2))*e^5 - 3/2*sqrt(e^5 + 1)*arctan(x/sqrt(e^5 + 1)) + x + 1/2*x*(e^5 + 1)/(x^2 + e^5 + 1) + arc
tan(x/sqrt(e^5 + 1))/sqrt(e^5 + 1) + 1/2*x/(x^2*(e^5 + 1) + e^10 + 2*e^5 + 1) - x/(x^2 + e^5 + 1) + e^x/(x^3*e
^147 + x*(e^152 + e^147)) + 1/2*arctan(x/sqrt(e^5 + 1))/(e^5 + 1)^(3/2) + 1/(x^2 + e^5 + 1)

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mupad [B]  time = 4.53, size = 21, normalized size = 0.91 \begin {gather*} x+\frac {x+{\mathrm {e}}^{x-147}}{x\,\left (x^2+{\mathrm {e}}^5+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - 147)*(x + exp(5)*(x - 1) - 3*x^2 + x^3 - 1) + exp(5)*(2*x^2 + 2*x^4) + x^2*exp(10) + x^2 - 2*x^3
+ 2*x^4 + x^6)/(exp(5)*(2*x^2 + 2*x^4) + x^2*exp(10) + x^2 + 2*x^4 + x^6),x)

[Out]

x + (x + exp(x - 147))/(x*(exp(5) + x^2 + 1))

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sympy [A]  time = 0.27, size = 26, normalized size = 1.13 \begin {gather*} x + \frac {e^{x - 147}}{x^{3} + x + x e^{5}} + \frac {1}{x^{2} + 1 + e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(5)+x**3-3*x**2+x-1)*exp(x-147)+x**2*exp(5)**2+(2*x**4+2*x**2)*exp(5)+x**6+2*x**4-2*x**3+
x**2)/(x**2*exp(5)**2+(2*x**4+2*x**2)*exp(5)+x**6+2*x**4+x**2),x)

[Out]

x + exp(x - 147)/(x**3 + x + x*exp(5)) + 1/(x**2 + 1 + exp(5))

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