3.81.42 \(\int \frac {-28-88 x+13 x^2+16 x^3+2 x^4+e^x (-16+32 x-8 x^2-8 x^3-x^4) \log (16)+(16-64 x+72 x^2-8 x^3-15 x^4-2 x^5) \log (16)}{16-32 x+8 x^2+8 x^3+x^4} \, dx\) [8042]

Optimal. Leaf size=30 \[ 2 x+\frac {3}{x-\frac {4}{4+x}}-\left (e^x+(-1+x) x\right ) \log (16) \]

[Out]

2*x-4*ln(2)*((-1+x)*x+exp(x))+3/(x-4/(4+x))

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(257\) vs. \(2(30)=60\).
time = 0.93, antiderivative size = 257, normalized size of antiderivative = 8.57, number of steps used = 24, number of rules used = 13, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {6820, 6874, 628, 632, 212, 652, 736, 752, 787, 646, 31, 814, 2225} \begin {gather*} \frac {2 (2-x) x^2}{-x^2-4 x+4}-\frac {x^2}{4}+\frac {13 (2-x) x}{8 \left (-x^2-4 x+4\right )}-\frac {11 (2-x)}{-x^2-4 x+4}-\frac {7 (x+2)}{4 \left (-x^2-4 x+4\right )}+\frac {(2-x) x^3}{4 \left (-x^2-4 x+4\right )}+\frac {3 x}{2}+\left (8-5 \sqrt {2}\right ) \log \left (x+2 \left (1-\sqrt {2}\right )\right )-\frac {1}{4} \left (32-23 \sqrt {2}\right ) \log \left (x+2 \left (1-\sqrt {2}\right )\right )-\frac {1}{4} \left (32+23 \sqrt {2}\right ) \log \left (x+2 \left (1+\sqrt {2}\right )\right )+\left (8+5 \sqrt {2}\right ) \log \left (x+2 \left (1+\sqrt {2}\right )\right )-e^x \log (16)-\frac {1}{4} (1-2 x)^2 \log (16)+\frac {3 \tanh ^{-1}\left (\frac {x+2}{2 \sqrt {2}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-28 - 88*x + 13*x^2 + 16*x^3 + 2*x^4 + E^x*(-16 + 32*x - 8*x^2 - 8*x^3 - x^4)*Log[16] + (16 - 64*x + 72*x
^2 - 8*x^3 - 15*x^4 - 2*x^5)*Log[16])/(16 - 32*x + 8*x^2 + 8*x^3 + x^4),x]

[Out]

(3*x)/2 - x^2/4 - (11*(2 - x))/(4 - 4*x - x^2) + (13*(2 - x)*x)/(8*(4 - 4*x - x^2)) + (2*(2 - x)*x^2)/(4 - 4*x
- x^2) + ((2 - x)*x^3)/(4*(4 - 4*x - x^2)) - (7*(2 + x))/(4*(4 - 4*x - x^2)) + (3*ArcTanh[(2 + x)/(2*Sqrt[2])
])/Sqrt[2] - E^x*Log[16] - ((1 - 2*x)^2*Log[16])/4 - ((32 - 23*Sqrt[2])*Log[2*(1 - Sqrt[2]) + x])/4 + (8 - 5*S
qrt[2])*Log[2*(1 - Sqrt[2]) + x] + (8 + 5*Sqrt[2])*Log[2*(1 + Sqrt[2]) + x] - ((32 + 23*Sqrt[2])*Log[2*(1 + Sq
rt[2]) + x])/4

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 212

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

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 632

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 646

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 652

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)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), 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 736

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

Rule 752

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

Rule 787

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*g*(x/c
), x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2225

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 6820

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

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-28-88 x+13 x^2+16 x^3+2 x^4-e^x \left (-4+4 x+x^2\right )^2 \log (16)-(-1+2 x) \left (-4+4 x+x^2\right )^2 \log (16)}{\left (4-4 x-x^2\right )^2} \, dx\\ &=\int \left (-\frac {28}{\left (-4+4 x+x^2\right )^2}-\frac {88 x}{\left (-4+4 x+x^2\right )^2}+\frac {13 x^2}{\left (-4+4 x+x^2\right )^2}+\frac {16 x^3}{\left (-4+4 x+x^2\right )^2}+\frac {2 x^4}{\left (-4+4 x+x^2\right )^2}-e^x \log (16)-(-1+2 x) \log (16)\right ) \, dx\\ &=-\frac {1}{4} (1-2 x)^2 \log (16)+2 \int \frac {x^4}{\left (-4+4 x+x^2\right )^2} \, dx+13 \int \frac {x^2}{\left (-4+4 x+x^2\right )^2} \, dx+16 \int \frac {x^3}{\left (-4+4 x+x^2\right )^2} \, dx-28 \int \frac {1}{\left (-4+4 x+x^2\right )^2} \, dx-88 \int \frac {x}{\left (-4+4 x+x^2\right )^2} \, dx-\log (16) \int e^x \, dx\\ &=-\frac {11 (2-x)}{4-4 x-x^2}+\frac {13 (2-x) x}{8 \left (4-4 x-x^2\right )}+\frac {2 (2-x) x^2}{4-4 x-x^2}+\frac {(2-x) x^3}{4 \left (4-4 x-x^2\right )}-\frac {7 (2+x)}{4 \left (4-4 x-x^2\right )}-e^x \log (16)-\frac {1}{4} (1-2 x)^2 \log (16)-\frac {1}{16} \int \frac {x^2 (-24+8 x)}{-4+4 x+x^2} \, dx-\frac {1}{2} \int \frac {x (-16+4 x)}{-4+4 x+x^2} \, dx+\frac {7}{4} \int \frac {1}{-4+4 x+x^2} \, dx+\frac {13}{4} \int \frac {1}{-4+4 x+x^2} \, dx-11 \int \frac {1}{-4+4 x+x^2} \, dx\\ &=-2 x-\frac {11 (2-x)}{4-4 x-x^2}+\frac {13 (2-x) x}{8 \left (4-4 x-x^2\right )}+\frac {2 (2-x) x^2}{4-4 x-x^2}+\frac {(2-x) x^3}{4 \left (4-4 x-x^2\right )}-\frac {7 (2+x)}{4 \left (4-4 x-x^2\right )}-e^x \log (16)-\frac {1}{4} (1-2 x)^2 \log (16)-\frac {1}{16} \int \left (-56+8 x-\frac {32 (7-8 x)}{-4+4 x+x^2}\right ) \, dx-\frac {1}{2} \int \frac {16-32 x}{-4+4 x+x^2} \, dx-\frac {7}{2} \text {Subst}\left (\int \frac {1}{32-x^2} \, dx,x,4+2 x\right )-\frac {13}{2} \text {Subst}\left (\int \frac {1}{32-x^2} \, dx,x,4+2 x\right )+22 \text {Subst}\left (\int \frac {1}{32-x^2} \, dx,x,4+2 x\right )\\ &=\frac {3 x}{2}-\frac {x^2}{4}-\frac {11 (2-x)}{4-4 x-x^2}+\frac {13 (2-x) x}{8 \left (4-4 x-x^2\right )}+\frac {2 (2-x) x^2}{4-4 x-x^2}+\frac {(2-x) x^3}{4 \left (4-4 x-x^2\right )}-\frac {7 (2+x)}{4 \left (4-4 x-x^2\right )}+\frac {3 \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {2}}\right )}{\sqrt {2}}-e^x \log (16)-\frac {1}{4} (1-2 x)^2 \log (16)+2 \int \frac {7-8 x}{-4+4 x+x^2} \, dx-\left (-8+5 \sqrt {2}\right ) \int \frac {1}{2-2 \sqrt {2}+x} \, dx+\left (8+5 \sqrt {2}\right ) \int \frac {1}{2+2 \sqrt {2}+x} \, dx\\ &=\frac {3 x}{2}-\frac {x^2}{4}-\frac {11 (2-x)}{4-4 x-x^2}+\frac {13 (2-x) x}{8 \left (4-4 x-x^2\right )}+\frac {2 (2-x) x^2}{4-4 x-x^2}+\frac {(2-x) x^3}{4 \left (4-4 x-x^2\right )}-\frac {7 (2+x)}{4 \left (4-4 x-x^2\right )}+\frac {3 \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {2}}\right )}{\sqrt {2}}-e^x \log (16)-\frac {1}{4} (1-2 x)^2 \log (16)+\left (8-5 \sqrt {2}\right ) \log \left (2 \left (1-\sqrt {2}\right )+x\right )+\left (8+5 \sqrt {2}\right ) \log \left (2 \left (1+\sqrt {2}\right )+x\right )+\frac {1}{4} \left (-32+23 \sqrt {2}\right ) \int \frac {1}{2-2 \sqrt {2}+x} \, dx-\frac {1}{4} \left (32+23 \sqrt {2}\right ) \int \frac {1}{2+2 \sqrt {2}+x} \, dx\\ &=\frac {3 x}{2}-\frac {x^2}{4}-\frac {11 (2-x)}{4-4 x-x^2}+\frac {13 (2-x) x}{8 \left (4-4 x-x^2\right )}+\frac {2 (2-x) x^2}{4-4 x-x^2}+\frac {(2-x) x^3}{4 \left (4-4 x-x^2\right )}-\frac {7 (2+x)}{4 \left (4-4 x-x^2\right )}+\frac {3 \tanh ^{-1}\left (\frac {2+x}{2 \sqrt {2}}\right )}{\sqrt {2}}-e^x \log (16)-\frac {1}{4} (1-2 x)^2 \log (16)-\frac {1}{4} \left (32-23 \sqrt {2}\right ) \log \left (2 \left (1-\sqrt {2}\right )+x\right )+\left (8-5 \sqrt {2}\right ) \log \left (2 \left (1-\sqrt {2}\right )+x\right )+\left (8+5 \sqrt {2}\right ) \log \left (2 \left (1+\sqrt {2}\right )+x\right )-\frac {1}{4} \left (32+23 \sqrt {2}\right ) \log \left (2 \left (1+\sqrt {2}\right )+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.41, size = 39, normalized size = 1.30 \begin {gather*} \frac {3 (4+x)}{-4+4 x+x^2}-x (-2-\log (16))-e^x \log (16)-x^2 \log (16) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-28 - 88*x + 13*x^2 + 16*x^3 + 2*x^4 + E^x*(-16 + 32*x - 8*x^2 - 8*x^3 - x^4)*Log[16] + (16 - 64*x
+ 72*x^2 - 8*x^3 - 15*x^4 - 2*x^5)*Log[16])/(16 - 32*x + 8*x^2 + 8*x^3 + x^4),x]

[Out]

(3*(4 + x))/(-4 + 4*x + x^2) - x*(-2 - Log[16]) - E^x*Log[16] - x^2*Log[16]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(107\) vs. \(2(29)=58\).
time = 0.62, size = 108, normalized size = 3.60

method	result	size
risch	\(-4 x^{2} \ln \left (2\right )+4 x \ln \left (2\right )+2 x +\frac {3 x +12}{x^{2}+4 x -4}-4 \,{\mathrm e}^{x} \ln \left (2\right )\)	\(39\)
norman	\(\frac {\left (2-12 \ln \left (2\right )\right ) x^{3}+\left (-37-144 \ln \left (2\right )\right ) x -4 x^{4} \ln \left (2\right )+16 \,{\mathrm e}^{x} \ln \left (2\right )-16 x \ln \left (2\right ) {\mathrm e}^{x}-4 x^{2} \ln \left (2\right ) {\mathrm e}^{x}+44+128 \ln \left (2\right )}{x^{2}+4 x -4}\)	\(65\)
default	\(\frac {\frac {7 x}{4}+\frac {7}{2}}{x^{2}+4 x -4}+\frac {-11 x +22}{x^{2}+4 x -4}+\frac {-\frac {39 x}{4}+\frac {13}{2}}{x^{2}+4 x -4}+\frac {56 x -48}{x^{2}+4 x -4}+2 x -\frac {16 \left (\frac {17 x}{8}-\frac {7}{4}\right )}{x^{2}+4 x -4}+4 x \ln \left (2\right )-4 x^{2} \ln \left (2\right )-4 \,{\mathrm e}^{x} \ln \left (2\right )\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(-x^4-8*x^3-8*x^2+32*x-16)*ln(2)*exp(x)+4*(-2*x^5-15*x^4-8*x^3+72*x^2-64*x+16)*ln(2)+2*x^4+16*x^3+13*x^
2-88*x-28)/(x^4+8*x^3+8*x^2-32*x+16),x,method=_RETURNVERBOSE)

[Out]

7/8*(2*x+4)/(x^2+4*x-4)+11/4*(-4*x+8)/(x^2+4*x-4)+13*(-3/4*x+1/2)/(x^2+4*x-4)+16*(7/2*x-3)/(x^2+4*x-4)+2*x-16*
(17/8*x-7/4)/(x^2+4*x-4)+4*x*ln(2)-4*x^2*ln(2)-4*exp(x)*ln(2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (29) = 58\).
time = 0.51, size = 410, normalized size = 13.67 \begin {gather*} -2 \, {\left (2 \, x^{2} - 79 \, \sqrt {2} \log \left (\frac {x - 2 \, \sqrt {2} + 2}{x + 2 \, \sqrt {2} + 2}\right ) - 32 \, x + \frac {8 \, {\left (41 \, x - 34\right )}}{x^{2} + 4 \, x - 4} + 112 \, \log \left (x^{2} + 4 \, x - 4\right )\right )} \log \left (2\right ) - \frac {15}{2} \, {\left (23 \, \sqrt {2} \log \left (\frac {x - 2 \, \sqrt {2} + 2}{x + 2 \, \sqrt {2} + 2}\right ) + 8 \, x - \frac {8 \, {\left (17 \, x - 14\right )}}{x^{2} + 4 \, x - 4} - 32 \, \log \left (x^{2} + 4 \, x - 4\right )\right )} \log \left (2\right ) + 2 \, {\left (5 \, \sqrt {2} \log \left (\frac {x - 2 \, \sqrt {2} + 2}{x + 2 \, \sqrt {2} + 2}\right ) - \frac {8 \, {\left (7 \, x - 6\right )}}{x^{2} + 4 \, x - 4} - 8 \, \log \left (x^{2} + 4 \, x - 4\right )\right )} \log \left (2\right ) + 9 \, {\left (\sqrt {2} \log \left (\frac {x - 2 \, \sqrt {2} + 2}{x + 2 \, \sqrt {2} + 2}\right ) - \frac {8 \, {\left (3 \, x - 2\right )}}{x^{2} + 4 \, x - 4}\right )} \log \left (2\right ) - \frac {1}{2} \, {\left (\sqrt {2} \log \left (\frac {x - 2 \, \sqrt {2} + 2}{x + 2 \, \sqrt {2} + 2}\right ) + \frac {8 \, {\left (x + 2\right )}}{x^{2} + 4 \, x - 4}\right )} \log \left (2\right ) - 4 \, {\left (\sqrt {2} \log \left (\frac {x - 2 \, \sqrt {2} + 2}{x + 2 \, \sqrt {2} + 2}\right ) + \frac {8 \, {\left (x - 2\right )}}{x^{2} + 4 \, x - 4}\right )} \log \left (2\right ) - 4 \, e^{x} \log \left (2\right ) + 2 \, x - \frac {2 \, {\left (17 \, x - 14\right )}}{x^{2} + 4 \, x - 4} + \frac {8 \, {\left (7 \, x - 6\right )}}{x^{2} + 4 \, x - 4} - \frac {13 \, {\left (3 \, x - 2\right )}}{4 \, {\left (x^{2} + 4 \, x - 4\right )}} + \frac {7 \, {\left (x + 2\right )}}{4 \, {\left (x^{2} + 4 \, x - 4\right )}} - \frac {11 \, {\left (x - 2\right )}}{x^{2} + 4 \, x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-x^4-8*x^3-8*x^2+32*x-16)*log(2)*exp(x)+4*(-2*x^5-15*x^4-8*x^3+72*x^2-64*x+16)*log(2)+2*x^4+16*x
^3+13*x^2-88*x-28)/(x^4+8*x^3+8*x^2-32*x+16),x, algorithm="maxima")

[Out]

-2*(2*x^2 - 79*sqrt(2)*log((x - 2*sqrt(2) + 2)/(x + 2*sqrt(2) + 2)) - 32*x + 8*(41*x - 34)/(x^2 + 4*x - 4) + 1
12*log(x^2 + 4*x - 4))*log(2) - 15/2*(23*sqrt(2)*log((x - 2*sqrt(2) + 2)/(x + 2*sqrt(2) + 2)) + 8*x - 8*(17*x
- 14)/(x^2 + 4*x - 4) - 32*log(x^2 + 4*x - 4))*log(2) + 2*(5*sqrt(2)*log((x - 2*sqrt(2) + 2)/(x + 2*sqrt(2) +
2)) - 8*(7*x - 6)/(x^2 + 4*x - 4) - 8*log(x^2 + 4*x - 4))*log(2) + 9*(sqrt(2)*log((x - 2*sqrt(2) + 2)/(x + 2*s
qrt(2) + 2)) - 8*(3*x - 2)/(x^2 + 4*x - 4))*log(2) - 1/2*(sqrt(2)*log((x - 2*sqrt(2) + 2)/(x + 2*sqrt(2) + 2))
+ 8*(x + 2)/(x^2 + 4*x - 4))*log(2) - 4*(sqrt(2)*log((x - 2*sqrt(2) + 2)/(x + 2*sqrt(2) + 2)) + 8*(x - 2)/(x^
2 + 4*x - 4))*log(2) - 4*e^x*log(2) + 2*x - 2*(17*x - 14)/(x^2 + 4*x - 4) + 8*(7*x - 6)/(x^2 + 4*x - 4) - 13/4
*(3*x - 2)/(x^2 + 4*x - 4) + 7/4*(x + 2)/(x^2 + 4*x - 4) - 11*(x - 2)/(x^2 + 4*x - 4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (29) = 58\).
time = 0.39, size = 61, normalized size = 2.03 \begin {gather*} \frac {2 \, x^{3} - 4 \, {\left (x^{2} + 4 \, x - 4\right )} e^{x} \log \left (2\right ) + 8 \, x^{2} - 4 \, {\left (x^{4} + 3 \, x^{3} - 8 \, x^{2} + 4 \, x\right )} \log \left (2\right ) - 5 \, x + 12}{x^{2} + 4 \, x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-x^4-8*x^3-8*x^2+32*x-16)*log(2)*exp(x)+4*(-2*x^5-15*x^4-8*x^3+72*x^2-64*x+16)*log(2)+2*x^4+16*x
^3+13*x^2-88*x-28)/(x^4+8*x^3+8*x^2-32*x+16),x, algorithm="fricas")

[Out]

(2*x^3 - 4*(x^2 + 4*x - 4)*e^x*log(2) + 8*x^2 - 4*(x^4 + 3*x^3 - 8*x^2 + 4*x)*log(2) - 5*x + 12)/(x^2 + 4*x -
4)

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Sympy [A]
time = 0.15, size = 42, normalized size = 1.40 \begin {gather*} - 4 x^{2} \log {\left (2 \right )} - x \left (- 4 \log {\left (2 \right )} - 2\right ) - \frac {- 3 x - 12}{x^{2} + 4 x - 4} - 4 e^{x} \log {\left (2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-x**4-8*x**3-8*x**2+32*x-16)*ln(2)*exp(x)+4*(-2*x**5-15*x**4-8*x**3+72*x**2-64*x+16)*ln(2)+2*x**
4+16*x**3+13*x**2-88*x-28)/(x**4+8*x**3+8*x**2-32*x+16),x)

[Out]

-4*x**2*log(2) - x*(-4*log(2) - 2) - (-3*x - 12)/(x**2 + 4*x - 4) - 4*exp(x)*log(2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (29) = 58\).
time = 0.43, size = 75, normalized size = 2.50 \begin {gather*} -\frac {4 \, x^{4} \log \left (2\right ) + 12 \, x^{3} \log \left (2\right ) + 4 \, x^{2} e^{x} \log \left (2\right ) - 2 \, x^{3} - 32 \, x^{2} \log \left (2\right ) + 16 \, x e^{x} \log \left (2\right ) - 8 \, x^{2} + 16 \, x \log \left (2\right ) - 16 \, e^{x} \log \left (2\right ) + 5 \, x - 12}{x^{2} + 4 \, x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-x^4-8*x^3-8*x^2+32*x-16)*log(2)*exp(x)+4*(-2*x^5-15*x^4-8*x^3+72*x^2-64*x+16)*log(2)+2*x^4+16*x
^3+13*x^2-88*x-28)/(x^4+8*x^3+8*x^2-32*x+16),x, algorithm="giac")

[Out]

-(4*x^4*log(2) + 12*x^3*log(2) + 4*x^2*e^x*log(2) - 2*x^3 - 32*x^2*log(2) + 16*x*e^x*log(2) - 8*x^2 + 16*x*log
(2) - 16*e^x*log(2) + 5*x - 12)/(x^2 + 4*x - 4)

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Mupad [B]
time = 0.73, size = 38, normalized size = 1.27 \begin {gather*} x\,\left (4\,\ln \left (2\right )+2\right )+\frac {3\,x+12}{x^2+4\,x-4}-4\,x^2\,\ln \left (2\right )-4\,{\mathrm {e}}^x\,\ln \left (2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(88*x - 13*x^2 - 16*x^3 - 2*x^4 + 4*log(2)*(64*x - 72*x^2 + 8*x^3 + 15*x^4 + 2*x^5 - 16) + 4*exp(x)*log(2
)*(8*x^2 - 32*x + 8*x^3 + x^4 + 16) + 28)/(8*x^2 - 32*x + 8*x^3 + x^4 + 16),x)

[Out]

x*(4*log(2) + 2) + (3*x + 12)/(4*x + x^2 - 4) - 4*x^2*log(2) - 4*exp(x)*log(2)

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