\(\int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 (524288 x-6144 x^2+16 x^3)+(131072 x-132608 x^2+1540 x^3-4 x^4+e^3 (131072 x-1536 x^2+4 x^3)) \log (1+e^3-x)}{65536+65536 e^3-65536 x} \, dx\) [8908]

Optimal result

Integrand size = 98, antiderivative size = 27 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\left (1+\left (-x+\frac {x^2}{256}\right )^2\right ) \left (4+\log \left (1+e^3-x\right )\right ) \]

[Out]

(ln(exp(3)-x+1)+4)*(1+(1/256*x^2-x)^2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(303\) vs. \(2(27)=54\).

Time = 0.33 (sec) , antiderivative size = 303, normalized size of antiderivative = 11.22, number of steps used = 23, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6874, 45, 147, 2464, 2442} \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {x^4}{16384}+\frac {x^4 \log \left (-x+e^3+1\right )}{65536}+\frac {\left (1+e^3\right ) x^3}{12288}-\frac {e^3 x^3}{12288}-\frac {385 x^3}{12288}-\frac {1}{128} x^3 \log \left (-x+e^3+1\right )+\frac {\left (1+e^3\right )^2 x^2}{8192}-\frac {385 \left (1+e^3\right ) x^2}{8192}+\frac {e^3 \left (383-e^3\right ) x^2}{8192}+\frac {259 x^2}{64}+x^2 \log \left (-x+e^3+1\right )-\frac {e^3 \left (32385-382 e^3+e^6\right ) x}{4096}+\frac {\left (1+e^3\right )^3 x}{4096}-\frac {385 \left (1+e^3\right )^2 x}{4096}+\frac {259}{32} \left (1+e^3\right ) x-8 x-\frac {e^3 \left (32385+32003 e^3-381 e^6+e^9\right ) \log \left (-x+e^3+1\right )}{4096}+\frac {\left (1+e^3\right )^4 \log \left (-x+e^3+1\right )}{4096}-\frac {385 \left (1+e^3\right )^3 \log \left (-x+e^3+1\right )}{4096}+\frac {259}{32} \left (1+e^3\right )^2 \log \left (-x+e^3+1\right )-8 \left (1+e^3\right ) \log \left (-x+e^3+1\right )+\log \left (-x+e^3+1\right ) \]

[In]

Int[(-65536 + 524288*x - 595968*x^2 + 6672*x^3 - 17*x^4 + E^3*(524288*x - 6144*x^2 + 16*x^3) + (131072*x - 132
608*x^2 + 1540*x^3 - 4*x^4 + E^3*(131072*x - 1536*x^2 + 4*x^3))*Log[1 + E^3 - x])/(65536 + 65536*E^3 - 65536*x
),x]

[Out]

-8*x + (259*(1 + E^3)*x)/32 - (385*(1 + E^3)^2*x)/4096 + ((1 + E^3)^3*x)/4096 - (E^3*(32385 - 382*E^3 + E^6)*x
)/4096 + (259*x^2)/64 + (E^3*(383 - E^3)*x^2)/8192 - (385*(1 + E^3)*x^2)/8192 + ((1 + E^3)^2*x^2)/8192 - (385*
x^3)/12288 - (E^3*x^3)/12288 + ((1 + E^3)*x^3)/12288 + x^4/16384 + Log[1 + E^3 - x] - 8*(1 + E^3)*Log[1 + E^3
- x] + (259*(1 + E^3)^2*Log[1 + E^3 - x])/32 - (385*(1 + E^3)^3*Log[1 + E^3 - x])/4096 + ((1 + E^3)^4*Log[1 +
E^3 - x])/4096 - (E^3*(32385 + 32003*E^3 - 381*E^6 + E^9)*Log[1 + E^3 - x])/4096 + x^2*Log[1 + E^3 - x] - (x^3
*Log[1 + E^3 - x])/128 + (x^4*Log[1 + E^3 - x])/65536

Rule 45

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 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && (IGtQ[m, 0] || IntegersQ[m, n])

Rule 2442

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

Rule 2464

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Poly
x*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolynomialQ[Polyx, x]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {8 x}{1+e^3-x}+\frac {e^3 (-256+x) (-128+x) x}{4096 \left (1+e^3-x\right )}-\frac {291 x^2}{32 \left (1+e^3-x\right )}+\frac {417 x^3}{4096 \left (1+e^3-x\right )}-\frac {17 x^4}{65536 \left (1+e^3-x\right )}+\frac {1}{-1-e^3+x}+\frac {(-256+x) (-128+x) x \log \left (1+e^3-x\right )}{16384}\right ) \, dx \\ & = \log \left (1+e^3-x\right )+\frac {\int (-256+x) (-128+x) x \log \left (1+e^3-x\right ) \, dx}{16384}-\frac {17 \int \frac {x^4}{1+e^3-x} \, dx}{65536}+\frac {417 \int \frac {x^3}{1+e^3-x} \, dx}{4096}+8 \int \frac {x}{1+e^3-x} \, dx-\frac {291}{32} \int \frac {x^2}{1+e^3-x} \, dx+\frac {e^3 \int \frac {(-256+x) (-128+x) x}{1+e^3-x} \, dx}{4096} \\ & = \log \left (1+e^3-x\right )+\frac {\int \left (32768 x \log \left (1+e^3-x\right )-384 x^2 \log \left (1+e^3-x\right )+x^3 \log \left (1+e^3-x\right )\right ) \, dx}{16384}-\frac {17 \int \left (-\left (1+e^3\right )^3+\frac {\left (1+e^3\right )^4}{1+e^3-x}-\left (1+e^3\right )^2 x-\left (1+e^3\right ) x^2-x^3\right ) \, dx}{65536}+\frac {417 \int \left (-\left (1+e^3\right )^2+\frac {\left (1+e^3\right )^3}{1+e^3-x}-\left (1+e^3\right ) x-x^2\right ) \, dx}{4096}+8 \int \left (-1+\frac {1+e^3}{1+e^3-x}\right ) \, dx-\frac {291}{32} \int \left (-1-e^3+\frac {\left (1+e^3\right )^2}{1+e^3-x}-x\right ) \, dx+\frac {e^3 \int \left (-32385 \left (1+\frac {e^3 \left (-382+e^3\right )}{32385}\right )+\frac {32385+32003 e^3-381 e^6+e^9}{1+e^3-x}-\left (-383+e^3\right ) x-x^2\right ) \, dx}{4096} \\ & = -8 x+\frac {291}{32} \left (1+e^3\right ) x-\frac {417 \left (1+e^3\right )^2 x}{4096}+\frac {17 \left (1+e^3\right )^3 x}{65536}-\frac {e^3 \left (32385-382 e^3+e^6\right ) x}{4096}+\frac {291 x^2}{64}+\frac {e^3 \left (383-e^3\right ) x^2}{8192}-\frac {417 \left (1+e^3\right ) x^2}{8192}+\frac {17 \left (1+e^3\right )^2 x^2}{131072}-\frac {139 x^3}{4096}-\frac {e^3 x^3}{12288}+\frac {17 \left (1+e^3\right ) x^3}{196608}+\frac {17 x^4}{262144}+\log \left (1+e^3-x\right )-8 \left (1+e^3\right ) \log \left (1+e^3-x\right )+\frac {291}{32} \left (1+e^3\right )^2 \log \left (1+e^3-x\right )-\frac {417 \left (1+e^3\right )^3 \log \left (1+e^3-x\right )}{4096}+\frac {17 \left (1+e^3\right )^4 \log \left (1+e^3-x\right )}{65536}-\frac {e^3 \left (32385+32003 e^3-381 e^6+e^9\right ) \log \left (1+e^3-x\right )}{4096}+\frac {\int x^3 \log \left (1+e^3-x\right ) \, dx}{16384}-\frac {3}{128} \int x^2 \log \left (1+e^3-x\right ) \, dx+2 \int x \log \left (1+e^3-x\right ) \, dx \\ & = -8 x+\frac {291}{32} \left (1+e^3\right ) x-\frac {417 \left (1+e^3\right )^2 x}{4096}+\frac {17 \left (1+e^3\right )^3 x}{65536}-\frac {e^3 \left (32385-382 e^3+e^6\right ) x}{4096}+\frac {291 x^2}{64}+\frac {e^3 \left (383-e^3\right ) x^2}{8192}-\frac {417 \left (1+e^3\right ) x^2}{8192}+\frac {17 \left (1+e^3\right )^2 x^2}{131072}-\frac {139 x^3}{4096}-\frac {e^3 x^3}{12288}+\frac {17 \left (1+e^3\right ) x^3}{196608}+\frac {17 x^4}{262144}+\log \left (1+e^3-x\right )-8 \left (1+e^3\right ) \log \left (1+e^3-x\right )+\frac {291}{32} \left (1+e^3\right )^2 \log \left (1+e^3-x\right )-\frac {417 \left (1+e^3\right )^3 \log \left (1+e^3-x\right )}{4096}+\frac {17 \left (1+e^3\right )^4 \log \left (1+e^3-x\right )}{65536}-\frac {e^3 \left (32385+32003 e^3-381 e^6+e^9\right ) \log \left (1+e^3-x\right )}{4096}+x^2 \log \left (1+e^3-x\right )-\frac {1}{128} x^3 \log \left (1+e^3-x\right )+\frac {x^4 \log \left (1+e^3-x\right )}{65536}+\frac {\int \frac {x^4}{1+e^3-x} \, dx}{65536}-\frac {1}{128} \int \frac {x^3}{1+e^3-x} \, dx+\int \frac {x^2}{1+e^3-x} \, dx \\ & = -8 x+\frac {291}{32} \left (1+e^3\right ) x-\frac {417 \left (1+e^3\right )^2 x}{4096}+\frac {17 \left (1+e^3\right )^3 x}{65536}-\frac {e^3 \left (32385-382 e^3+e^6\right ) x}{4096}+\frac {291 x^2}{64}+\frac {e^3 \left (383-e^3\right ) x^2}{8192}-\frac {417 \left (1+e^3\right ) x^2}{8192}+\frac {17 \left (1+e^3\right )^2 x^2}{131072}-\frac {139 x^3}{4096}-\frac {e^3 x^3}{12288}+\frac {17 \left (1+e^3\right ) x^3}{196608}+\frac {17 x^4}{262144}+\log \left (1+e^3-x\right )-8 \left (1+e^3\right ) \log \left (1+e^3-x\right )+\frac {291}{32} \left (1+e^3\right )^2 \log \left (1+e^3-x\right )-\frac {417 \left (1+e^3\right )^3 \log \left (1+e^3-x\right )}{4096}+\frac {17 \left (1+e^3\right )^4 \log \left (1+e^3-x\right )}{65536}-\frac {e^3 \left (32385+32003 e^3-381 e^6+e^9\right ) \log \left (1+e^3-x\right )}{4096}+x^2 \log \left (1+e^3-x\right )-\frac {1}{128} x^3 \log \left (1+e^3-x\right )+\frac {x^4 \log \left (1+e^3-x\right )}{65536}+\frac {\int \left (-\left (1+e^3\right )^3+\frac {\left (1+e^3\right )^4}{1+e^3-x}-\left (1+e^3\right )^2 x-\left (1+e^3\right ) x^2-x^3\right ) \, dx}{65536}-\frac {1}{128} \int \left (-\left (1+e^3\right )^2+\frac {\left (1+e^3\right )^3}{1+e^3-x}-\left (1+e^3\right ) x-x^2\right ) \, dx+\int \left (-1-e^3+\frac {\left (1+e^3\right )^2}{1+e^3-x}-x\right ) \, dx \\ & = -8 x+\frac {259}{32} \left (1+e^3\right ) x-\frac {385 \left (1+e^3\right )^2 x}{4096}+\frac {\left (1+e^3\right )^3 x}{4096}-\frac {e^3 \left (32385-382 e^3+e^6\right ) x}{4096}+\frac {259 x^2}{64}+\frac {e^3 \left (383-e^3\right ) x^2}{8192}-\frac {385 \left (1+e^3\right ) x^2}{8192}+\frac {\left (1+e^3\right )^2 x^2}{8192}-\frac {385 x^3}{12288}-\frac {e^3 x^3}{12288}+\frac {\left (1+e^3\right ) x^3}{12288}+\frac {x^4}{16384}+\log \left (1+e^3-x\right )-8 \left (1+e^3\right ) \log \left (1+e^3-x\right )+\frac {259}{32} \left (1+e^3\right )^2 \log \left (1+e^3-x\right )-\frac {385 \left (1+e^3\right )^3 \log \left (1+e^3-x\right )}{4096}+\frac {\left (1+e^3\right )^4 \log \left (1+e^3-x\right )}{4096}-\frac {e^3 \left (32385+32003 e^3-381 e^6+e^9\right ) \log \left (1+e^3-x\right )}{4096}+x^2 \log \left (1+e^3-x\right )-\frac {1}{128} x^3 \log \left (1+e^3-x\right )+\frac {x^4 \log \left (1+e^3-x\right )}{65536} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(27)=54\).

Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {262144 x^2-2048 x^3+4 x^4+65536 \log \left (1+e^3-x\right )+65536 x^2 \log \left (1+e^3-x\right )-512 x^3 \log \left (1+e^3-x\right )+x^4 \log \left (1+e^3-x\right )}{65536} \]

[In]

Integrate[(-65536 + 524288*x - 595968*x^2 + 6672*x^3 - 17*x^4 + E^3*(524288*x - 6144*x^2 + 16*x^3) + (131072*x
- 132608*x^2 + 1540*x^3 - 4*x^4 + E^3*(131072*x - 1536*x^2 + 4*x^3))*Log[1 + E^3 - x])/(65536 + 65536*E^3 - 6
5536*x),x]

[Out]

(262144*x^2 - 2048*x^3 + 4*x^4 + 65536*Log[1 + E^3 - x] + 65536*x^2*Log[1 + E^3 - x] - 512*x^3*Log[1 + E^3 - x
] + x^4*Log[1 + E^3 - x])/65536

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78

method	result	size
risch	\(\left (\frac {1}{65536} x^{4}-\frac {1}{128} x^{3}+x^{2}\right ) \ln \left ({\mathrm e}^{3}-x +1\right )+\frac {x^{4}}{16384}-\frac {x^{3}}{32}+4 x^{2}+\ln \left (-{\mathrm e}^{3}+x -1\right )\)	\(48\)
norman	\(\ln \left ({\mathrm e}^{3}-x +1\right )+\ln \left ({\mathrm e}^{3}-x +1\right ) x^{2}+4 x^{2}-\frac {x^{3}}{32}+\frac {x^{4}}{16384}-\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) x^{3}}{128}+\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) x^{4}}{65536}\)	\(63\)
parallelrisch	\(\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) x^{4}}{65536}+\frac {x^{4}}{16384}-\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) x^{3}}{128}-\frac {x^{3}}{32}+\ln \left ({\mathrm e}^{3}-x +1\right ) x^{2}-4-4 \,{\mathrm e}^{6}+4 x^{2}-8 \,{\mathrm e}^{3}+\ln \left ({\mathrm e}^{3}-x +1\right )\)	\(74\)
parts	\(-\frac {64515 x}{65536}+\frac {x \,{\mathrm e}^{9}}{65536}-\frac {509 x \,{\mathrm e}^{6}}{65536}+\frac {64515 x \,{\mathrm e}^{3}}{65536}+\frac {x^{3} {\mathrm e}^{3}}{196608}-\frac {255 x^{2} {\mathrm e}^{3}}{65536}-\frac {32003 \left ({\mathrm e}^{3}-x +1\right )^{2}}{65536}+\frac {32385 \,{\mathrm e}^{3}}{16384}+\frac {17 x^{4}}{262144}-\frac {6655 x^{3}}{196608}+\frac {589313 x^{2}}{131072}+\frac {x^{2} {\mathrm e}^{6}}{131072}-\frac {{\mathrm e}^{9} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {381 \,{\mathrm e}^{6} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {3 \,{\mathrm e}^{6} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{16384}+\frac {32385}{16384}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{4}}{262144}-\frac {127 \left ({\mathrm e}^{3}-x +1\right )^{3}}{49152}-\frac {381 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{8192}-\frac {3 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{3}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{3}}{9}\right )}{16384}-\frac {32003 \,{\mathrm e}^{3} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{4}}{65536}+\frac {127 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{16384}+\frac {32003 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}-\frac {32385 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {\left ({\mathrm e}^{12}-508 \,{\mathrm e}^{9}+64006 \,{\mathrm e}^{6}+129540 \,{\mathrm e}^{3}+130561\right ) \ln \left (-{\mathrm e}^{3}+x -1\right )}{65536}\)	\(392\)
derivativedivides	\(\frac {32385 x}{4096}+\frac {130561 \ln \left ({\mathrm e}^{3}-x +1\right )}{65536}+\frac {32003 \left ({\mathrm e}^{3}-x +1\right )^{2}}{8192}-\frac {32385 \,{\mathrm e}^{3}}{4096}-\frac {{\mathrm e}^{9} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {381 \,{\mathrm e}^{6} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {3 \,{\mathrm e}^{6} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{16384}-\frac {32385}{4096}+\frac {\left ({\mathrm e}^{3}-x +1\right )^{4}}{16384}+\frac {127 \left ({\mathrm e}^{3}-x +1\right )^{3}}{4096}-\frac {381 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{8192}-\frac {3 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{3}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{3}}{9}\right )}{16384}-\frac {32003 \,{\mathrm e}^{3} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{4}}{65536}-\frac {13 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )^{3}}{49152}+\frac {127 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{16384}-\frac {3429 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}-\frac {160015 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32003 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}-\frac {32385 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32385 \,{\mathrm e}^{3} \ln \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {{\mathrm e}^{12} \ln \left ({\mathrm e}^{3}-x +1\right )}{65536}-\frac {127 \,{\mathrm e}^{9} \ln \left ({\mathrm e}^{3}-x +1\right )}{16384}-\frac {5 \,{\mathrm e}^{9} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {27 \,{\mathrm e}^{6} \left ({\mathrm e}^{3}-x +1\right )^{2}}{65536}+\frac {1905 \,{\mathrm e}^{6} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32003 \,{\mathrm e}^{6} \ln \left ({\mathrm e}^{3}-x +1\right )}{32768}\)	\(451\)
default	\(\frac {32385 x}{4096}+\frac {130561 \ln \left ({\mathrm e}^{3}-x +1\right )}{65536}+\frac {32003 \left ({\mathrm e}^{3}-x +1\right )^{2}}{8192}-\frac {32385 \,{\mathrm e}^{3}}{4096}-\frac {{\mathrm e}^{9} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {381 \,{\mathrm e}^{6} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {3 \,{\mathrm e}^{6} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{16384}-\frac {32385}{4096}+\frac {\left ({\mathrm e}^{3}-x +1\right )^{4}}{16384}+\frac {127 \left ({\mathrm e}^{3}-x +1\right )^{3}}{4096}-\frac {381 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{2}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{2}}{4}\right )}{8192}-\frac {3 \,{\mathrm e}^{3} \left (\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{3}-\frac {\left ({\mathrm e}^{3}-x +1\right )^{3}}{9}\right )}{16384}-\frac {32003 \,{\mathrm e}^{3} \left (\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )-{\mathrm e}^{3}+x -1\right )}{16384}+\frac {\ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{4}}{65536}-\frac {13 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )^{3}}{49152}+\frac {127 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{3}}{16384}-\frac {3429 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}-\frac {160015 \,{\mathrm e}^{3} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32003 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )^{2}}{32768}-\frac {32385 \ln \left ({\mathrm e}^{3}-x +1\right ) \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32385 \,{\mathrm e}^{3} \ln \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {{\mathrm e}^{12} \ln \left ({\mathrm e}^{3}-x +1\right )}{65536}-\frac {127 \,{\mathrm e}^{9} \ln \left ({\mathrm e}^{3}-x +1\right )}{16384}-\frac {5 \,{\mathrm e}^{9} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {27 \,{\mathrm e}^{6} \left ({\mathrm e}^{3}-x +1\right )^{2}}{65536}+\frac {1905 \,{\mathrm e}^{6} \left ({\mathrm e}^{3}-x +1\right )}{16384}+\frac {32003 \,{\mathrm e}^{6} \ln \left ({\mathrm e}^{3}-x +1\right )}{32768}\)	\(451\)

[In]

int((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131072*x)*ln(exp(3)-x+1)+(16*x^3-6144*x^2+524
288*x)*exp(3)-17*x^4+6672*x^3-595968*x^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x,method=_RETURNVERBOSE)

[Out]

(1/65536*x^4-1/128*x^3+x^2)*ln(exp(3)-x+1)+1/16384*x^4-1/32*x^3+4*x^2+ln(-exp(3)+x-1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {1}{16384} \, x^{4} - \frac {1}{32} \, x^{3} + 4 \, x^{2} + \frac {1}{65536} \, {\left (x^{4} - 512 \, x^{3} + 65536 \, x^{2} + 65536\right )} \log \left (-x + e^{3} + 1\right ) \]

[In]

integrate((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131072*x)*log(exp(3)-x+1)+(16*x^3-6144*
x^2+524288*x)*exp(3)-17*x^4+6672*x^3-595968*x^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x, algorithm="fri
cas")

[Out]

1/16384*x^4 - 1/32*x^3 + 4*x^2 + 1/65536*(x^4 - 512*x^3 + 65536*x^2 + 65536)*log(-x + e^3 + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).

Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {x^{4}}{16384} - \frac {x^{3}}{32} + 4 x^{2} + \left (\frac {x^{4}}{65536} - \frac {x^{3}}{128} + x^{2}\right ) \log {\left (- x + 1 + e^{3} \right )} + \log {\left (x - e^{3} - 1 \right )} \]

[In]

integrate((((4*x**3-1536*x**2+131072*x)*exp(3)-4*x**4+1540*x**3-132608*x**2+131072*x)*ln(exp(3)-x+1)+(16*x**3-
6144*x**2+524288*x)*exp(3)-17*x**4+6672*x**3-595968*x**2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x)

[Out]

x**4/16384 - x**3/32 + 4*x**2 + (x**4/65536 - x**3/128 + x**2)*log(-x + 1 + exp(3)) + log(x - exp(3) - 1)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (23) = 46\).

Time = 0.21 (sec) , antiderivative size = 856, normalized size of antiderivative = 31.70 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\text {Too large to display} \]

[In]

integrate((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131072*x)*log(exp(3)-x+1)+(16*x^3-6144*
x^2+524288*x)*exp(3)-17*x^4+6672*x^3-595968*x^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x, algorithm="max
ima")

[Out]

1/16384*x^4 + 11/147456*x^3*(e^3 + 1) - 4619/147456*x^3 + 19/196608*x^2*(e^6 + 2*e^3 + 1) - 8083/196608*x^2*(e
^3 + 1) - 1/32768*(e^12 + 4*e^9 + 6*e^6 + 4*e^3 + 1)*log(x - e^3 - 1)^2 + 385/32768*(e^9 + 3*e^6 + 3*e^3 + 1)*
log(x - e^3 - 1)^2 - 259/256*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1)^2 + (e^3 + 1)*log(x - e^3 - 1)^2 - 1/98304*(2*
x^3 + 3*x^2*(e^3 + 1) + 6*x*(e^6 + 2*e^3 + 1) + 6*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1))*e^3*log(-x + e^3
+ 1) + 3/256*(x^2 + 2*x*(e^3 + 1) + 2*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1))*e^3*log(-x + e^3 + 1) - 2*((e^3 + 1
)*log(x - e^3 - 1) + x)*e^3*log(-x + e^3 + 1) + 2069/512*x^2 + 13/98304*x*(e^9 + 3*e^6 + 3*e^3 + 1) - 5773/983
04*x*(e^6 + 2*e^3 + 1) + 1551/256*x*(e^3 + 1) + 1/589824*(4*x^3 + 15*x^2*(e^3 + 1) + 18*(e^9 + 3*e^6 + 3*e^3 +
1)*log(x - e^3 - 1)^2 + 66*x*(e^6 + 2*e^3 + 1) + 66*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1))*e^3 - 1/24576
*(2*x^3 + 3*x^2*(e^3 + 1) + 6*x*(e^6 + 2*e^3 + 1) + 6*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1))*e^3 - 3/512*
(2*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1)^2 + x^2 + 6*x*(e^3 + 1) + 6*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1))*e^3 + ((
e^3 + 1)*log(x - e^3 - 1)^2 + 2*(e^3 + 1)*log(x - e^3 - 1) + 2*x)*e^3 + 3/64*(x^2 + 2*x*(e^3 + 1) + 2*(e^6 + 2
*e^3 + 1)*log(x - e^3 - 1))*e^3 - 8*((e^3 + 1)*log(x - e^3 - 1) + x)*e^3 + 13/98304*(e^12 + 4*e^9 + 6*e^6 + 4*
e^3 + 1)*log(x - e^3 - 1) - 5773/98304*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1) + 1551/256*(e^6 + 2*e^3 + 1)
*log(x - e^3 - 1) - 6*(e^3 + 1)*log(x - e^3 - 1) + 1/196608*(3*x^4 + 4*x^3*(e^3 + 1) + 6*x^2*(e^6 + 2*e^3 + 1)
+ 12*x*(e^9 + 3*e^6 + 3*e^3 + 1) + 12*(e^12 + 4*e^9 + 6*e^6 + 4*e^3 + 1)*log(x - e^3 - 1))*log(-x + e^3 + 1)
- 385/98304*(2*x^3 + 3*x^2*(e^3 + 1) + 6*x*(e^6 + 2*e^3 + 1) + 6*(e^9 + 3*e^6 + 3*e^3 + 1)*log(x - e^3 - 1))*l
og(-x + e^3 + 1) + 259/256*(x^2 + 2*x*(e^3 + 1) + 2*(e^6 + 2*e^3 + 1)*log(x - e^3 - 1))*log(-x + e^3 + 1) - 2*
((e^3 + 1)*log(x - e^3 - 1) + x)*log(-x + e^3 + 1) - 6*x + log(x - e^3 - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (23) = 46\).

Time = 0.28 (sec) , antiderivative size = 366, normalized size of antiderivative = 13.56 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\frac {1}{65536} \, {\left (x - e^{3} - 1\right )}^{4} \log \left (-x + e^{3} + 1\right ) + \frac {1}{16384} \, {\left (x - e^{3} - 1\right )}^{3} e^{3} \log \left (-x + e^{3} + 1\right ) + \frac {1}{16384} \, {\left (x - e^{3} - 1\right )}^{4} + \frac {1}{4096} \, {\left (x - e^{3} - 1\right )}^{3} e^{3} - \frac {127}{16384} \, {\left (x - e^{3} - 1\right )}^{3} \log \left (-x + e^{3} + 1\right ) + \frac {3}{32768} \, {\left (x - e^{3} - 1\right )}^{2} e^{6} \log \left (-x + e^{3} + 1\right ) - \frac {381}{16384} \, {\left (x - e^{3} - 1\right )}^{2} e^{3} \log \left (-x + e^{3} + 1\right ) - \frac {127}{4096} \, {\left (x - e^{3} - 1\right )}^{3} + \frac {3}{8192} \, {\left (x - e^{3} - 1\right )}^{2} e^{6} - \frac {381}{4096} \, {\left (x - e^{3} - 1\right )}^{2} e^{3} + \frac {32003}{32768} \, {\left (x - e^{3} - 1\right )}^{2} \log \left (-x + e^{3} + 1\right ) + \frac {1}{16384} \, {\left (x - e^{3} - 1\right )} e^{9} \log \left (-x + e^{3} + 1\right ) - \frac {381}{16384} \, {\left (x - e^{3} - 1\right )} e^{6} \log \left (-x + e^{3} + 1\right ) + \frac {32003}{16384} \, {\left (x - e^{3} - 1\right )} e^{3} \log \left (-x + e^{3} + 1\right ) + \frac {32003}{8192} \, {\left (x - e^{3} - 1\right )}^{2} + \frac {1}{4096} \, {\left (x - e^{3} - 1\right )} e^{9} - \frac {381}{4096} \, {\left (x - e^{3} - 1\right )} e^{6} + \frac {32003}{4096} \, {\left (x - e^{3} - 1\right )} e^{3} + \frac {32385}{16384} \, {\left (x - e^{3} - 1\right )} \log \left (-x + e^{3} + 1\right ) + \frac {1}{65536} \, e^{12} \log \left (-x + e^{3} + 1\right ) - \frac {127}{16384} \, e^{9} \log \left (-x + e^{3} + 1\right ) + \frac {32003}{32768} \, e^{6} \log \left (-x + e^{3} + 1\right ) + \frac {32385}{16384} \, e^{3} \log \left (-x + e^{3} + 1\right ) + \frac {32385}{4096} \, x - \frac {32385}{4096} \, e^{3} + \frac {130561}{65536} \, \log \left (-x + e^{3} + 1\right ) - \frac {32385}{4096} \]

[In]

integrate((((4*x^3-1536*x^2+131072*x)*exp(3)-4*x^4+1540*x^3-132608*x^2+131072*x)*log(exp(3)-x+1)+(16*x^3-6144*
x^2+524288*x)*exp(3)-17*x^4+6672*x^3-595968*x^2+524288*x-65536)/(65536*exp(3)-65536*x+65536),x, algorithm="gia
c")

[Out]

1/65536*(x - e^3 - 1)^4*log(-x + e^3 + 1) + 1/16384*(x - e^3 - 1)^3*e^3*log(-x + e^3 + 1) + 1/16384*(x - e^3 -
1)^4 + 1/4096*(x - e^3 - 1)^3*e^3 - 127/16384*(x - e^3 - 1)^3*log(-x + e^3 + 1) + 3/32768*(x - e^3 - 1)^2*e^6
*log(-x + e^3 + 1) - 381/16384*(x - e^3 - 1)^2*e^3*log(-x + e^3 + 1) - 127/4096*(x - e^3 - 1)^3 + 3/8192*(x -
e^3 - 1)^2*e^6 - 381/4096*(x - e^3 - 1)^2*e^3 + 32003/32768*(x - e^3 - 1)^2*log(-x + e^3 + 1) + 1/16384*(x - e
^3 - 1)*e^9*log(-x + e^3 + 1) - 381/16384*(x - e^3 - 1)*e^6*log(-x + e^3 + 1) + 32003/16384*(x - e^3 - 1)*e^3*
log(-x + e^3 + 1) + 32003/8192*(x - e^3 - 1)^2 + 1/4096*(x - e^3 - 1)*e^9 - 381/4096*(x - e^3 - 1)*e^6 + 32003
/4096*(x - e^3 - 1)*e^3 + 32385/16384*(x - e^3 - 1)*log(-x + e^3 + 1) + 1/65536*e^12*log(-x + e^3 + 1) - 127/1
6384*e^9*log(-x + e^3 + 1) + 32003/32768*e^6*log(-x + e^3 + 1) + 32385/16384*e^3*log(-x + e^3 + 1) + 32385/409
6*x - 32385/4096*e^3 + 130561/65536*log(-x + e^3 + 1) - 32385/4096

Mupad [B] (verification not implemented)

Time = 13.61 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {-65536+524288 x-595968 x^2+6672 x^3-17 x^4+e^3 \left (524288 x-6144 x^2+16 x^3\right )+\left (131072 x-132608 x^2+1540 x^3-4 x^4+e^3 \left (131072 x-1536 x^2+4 x^3\right )\right ) \log \left (1+e^3-x\right )}{65536+65536 e^3-65536 x} \, dx=\ln \left (x-{\mathrm {e}}^3-1\right )+\ln \left ({\mathrm {e}}^3-x+1\right )\,\left (\frac {x^4}{65536}-\frac {x^3}{128}+x^2\right )+4\,x^2-\frac {x^3}{32}+\frac {x^4}{16384} \]

[In]

int((524288*x + exp(3)*(524288*x - 6144*x^2 + 16*x^3) + log(exp(3) - x + 1)*(131072*x + exp(3)*(131072*x - 153
6*x^2 + 4*x^3) - 132608*x^2 + 1540*x^3 - 4*x^4) - 595968*x^2 + 6672*x^3 - 17*x^4 - 65536)/(65536*exp(3) - 6553
6*x + 65536),x)

[Out]

log(x - exp(3) - 1) + log(exp(3) - x + 1)*(x^2 - x^3/128 + x^4/65536) + 4*x^2 - x^3/32 + x^4/16384