3.43.41 \(\int \frac {-64 x^3-32 x^4-4 x^7+(-192 x^2-160 x^3-8 x^4-24 x^6) \log (49)+(-192 x-288 x^2-36 x^3-60 x^5) \log ^2(49)+(-64-224 x-60 x^2-80 x^4) \log ^3(49)+(-64-44 x-60 x^3) \log ^4(49)+(-12-24 x^2) \log ^5(49)-4 x \log ^6(49)}{-4096-12288 x-15360 x^2-10240 x^3-3072 x^4+768 x^5+1088 x^6+384 x^7-48 x^9-12 x^{10}+x^{12}+(-18432-46080 x-46080 x^2-21504 x^3-384 x^4+5184 x^5+2496 x^6+192 x^7-264 x^8-84 x^9+6 x^{11}) \log (49)+(-34560-69120 x-51072 x^2-11136 x^3+8496 x^4+6432 x^5+984 x^6-576 x^7-243 x^8+15 x^{10}) \log ^2(49)+(-34560-51840 x-23616 x^2+4320 x^3+8208 x^4+2040 x^5-624 x^6-372 x^7+20 x^9) \log ^3(49)+(-19440-19440 x-2268 x^2+5184 x^3+2139 x^4-336 x^5-318 x^6+15 x^8) \log ^4(49)+(-5832-2916 x+1296 x^2+1134 x^3-72 x^4-144 x^5+6 x^7) \log ^5(49)+(-729+243 x^2-27 x^4+x^6) \log ^6(49)} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{\left (x^2-\left (3-\frac {-4+x}{x+\log (49)}\right )^2\right )^2} \]

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Rubi [B]  time = 1.98, antiderivative size = 893, normalized size of antiderivative = 38.83, number of steps used = 20, number of rules used = 6, integrand size = 412, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {2074, 638, 618, 204, 206, 614} \begin {gather*} -\frac {\left (8+\log ^3(49)+\log (2401)\right ) \tan ^{-1}\left (\frac {2 x+\log (49)+2}{\sqrt {12+8 \log (49)-\log ^2(49)}}\right )}{2 (4+3 \log (49))^3 \sqrt {12+8 \log (49)-\log ^2(49)}}+\frac {\left (96+280 \log (49)+248 \log ^2(49)+58 \log ^3(49)-10 \log ^4(49)-\log ^5(49)\right ) \tan ^{-1}\left (\frac {2 x+\log (49)+2}{\sqrt {12+8 \log (49)-\log ^2(49)}}\right )}{2 (4+3 \log (49))^3 \left (12+8 \log (49)-\log ^2(49)\right )^{3/2}}-\frac {3 \log (49) \left (8+4 \log (49)-\log ^2(49)\right ) \tan ^{-1}\left (\frac {2 x+\log (49)+2}{\sqrt {12+8 \log (49)-\log ^2(49)}}\right )}{(4+3 \log (49))^2 \left (12+8 \log (49)-\log ^2(49)\right )^{3/2}}+\frac {-\log (49) \left (8+4 \log (49)-\log ^2(49)\right ) x+\log ^4(49)-5 \log ^3(49)-11 \log ^2(49)+8 \log (49)+16}{4 (4+3 \log (49))^2 \left (x^2+(2+\log (49)) x+3 \log (49)+4\right )^2}+\frac {\log (49) \left (8+4 \log (49)+\log ^2(49)\right ) x+\log ^4(49)+5 \log ^3(49)+13 \log ^2(49)+8 \log (49)+16}{4 (4+3 \log (49))^2 \left (-x^2+(2-\log (49)) x+3 \log (49)+4\right )^2}+\frac {\left (96+280 \log (49)+248 \log ^2(49)+58 \log ^3(49)-10 \log ^4(49)-\log ^5(49)\right ) x+\log (49) (4+\log (49)) \left (36+64 \log (49)+31 \log ^2(49)+\log ^3(49)-\log ^4(49)\right )}{4 (4+3 \log (49))^3 \left (12+8 \log (49)-\log ^2(49)\right ) \left (x^2+(2+\log (49)) x+3 \log (49)+4\right )}-\frac {-\left (\left (160-88 \log (49)-216 \log ^2(49)-74 \log ^3(49)-10 \log ^4(49)+\log ^5(49)\right ) x\right )-\log ^6(49)+3 \log ^5(49)+27 \log ^4(49)+84 \log ^3(49)+4 \log ^2(49)+304 \log (49)+640}{4 (4+3 \log (49))^3 \left (-x^2+(2-\log (49)) x+3 \log (49)+4\right ) \left (20+8 \log (49)+\log ^2(49)\right )}-\frac {3 \log (49) (2 x+\log (49)+2) \left (8+4 \log (49)-\log ^2(49)\right )}{4 (4+3 \log (49))^2 \left (12+8 \log (49)-\log ^2(49)\right ) \left (x^2+(2+\log (49)) x+3 \log (49)+4\right )}+\frac {\tanh ^{-1}\left (\frac {-2 x-\log (49)+2}{\sqrt {20+8 \log (49)+\log ^2(49)}}\right ) \left (8+\log ^3(49)+\log (2401)\right )}{2 (4+3 \log (49))^3 \sqrt {20+8 \log (49)+\log ^2(49)}}-\frac {\tanh ^{-1}\left (\frac {-2 x-\log (49)+2}{\sqrt {20+8 \log (49)+\log ^2(49)}}\right ) \left (160-88 \log (49)-216 \log ^2(49)-74 \log ^3(49)-10 \log ^4(49)+\log ^5(49)\right )}{2 (4+3 \log (49))^3 \left (20+8 \log (49)+\log ^2(49)\right )^{3/2}}-\frac {3 (-2 x-\log (49)+2) \log (49) \left (8+4 \log (49)+\log ^2(49)\right )}{4 (4+3 \log (49))^2 \left (-x^2+(2-\log (49)) x+3 \log (49)+4\right ) \left (20+8 \log (49)+\log ^2(49)\right )}-\frac {3 \tanh ^{-1}\left (\frac {-2 x-\log (49)+2}{\sqrt {20+8 \log (49)+\log ^2(49)}}\right ) \log (49) \left (8+4 \log (49)+\log ^2(49)\right )}{(4+3 \log (49))^2 \left (20+8 \log (49)+\log ^2(49)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-64*x^3 - 32*x^4 - 4*x^7 + (-192*x^2 - 160*x^3 - 8*x^4 - 24*x^6)*Log[49] + (-192*x - 288*x^2 - 36*x^3 - 6
0*x^5)*Log[49]^2 + (-64 - 224*x - 60*x^2 - 80*x^4)*Log[49]^3 + (-64 - 44*x - 60*x^3)*Log[49]^4 + (-12 - 24*x^2
)*Log[49]^5 - 4*x*Log[49]^6)/(-4096 - 12288*x - 15360*x^2 - 10240*x^3 - 3072*x^4 + 768*x^5 + 1088*x^6 + 384*x^
7 - 48*x^9 - 12*x^10 + x^12 + (-18432 - 46080*x - 46080*x^2 - 21504*x^3 - 384*x^4 + 5184*x^5 + 2496*x^6 + 192*
x^7 - 264*x^8 - 84*x^9 + 6*x^11)*Log[49] + (-34560 - 69120*x - 51072*x^2 - 11136*x^3 + 8496*x^4 + 6432*x^5 + 9
84*x^6 - 576*x^7 - 243*x^8 + 15*x^10)*Log[49]^2 + (-34560 - 51840*x - 23616*x^2 + 4320*x^3 + 8208*x^4 + 2040*x
^5 - 624*x^6 - 372*x^7 + 20*x^9)*Log[49]^3 + (-19440 - 19440*x - 2268*x^2 + 5184*x^3 + 2139*x^4 - 336*x^5 - 31
8*x^6 + 15*x^8)*Log[49]^4 + (-5832 - 2916*x + 1296*x^2 + 1134*x^3 - 72*x^4 - 144*x^5 + 6*x^7)*Log[49]^5 + (-72
9 + 243*x^2 - 27*x^4 + x^6)*Log[49]^6),x]

[Out]

(-3*ArcTan[(2 + 2*x + Log[49])/Sqrt[12 + 8*Log[49] - Log[49]^2]]*Log[49]*(8 + 4*Log[49] - Log[49]^2))/((4 + 3*
Log[49])^2*(12 + 8*Log[49] - Log[49]^2)^(3/2)) - (3*ArcTanh[(2 - 2*x - Log[49])/Sqrt[20 + 8*Log[49] + Log[49]^
2]]*Log[49]*(8 + 4*Log[49] + Log[49]^2))/((4 + 3*Log[49])^2*(20 + 8*Log[49] + Log[49]^2)^(3/2)) - (3*(2 - 2*x
- Log[49])*Log[49]*(8 + 4*Log[49] + Log[49]^2))/(4*(4 + 3*Log[49])^2*(4 - x^2 + x*(2 - Log[49]) + 3*Log[49])*(
20 + 8*Log[49] + Log[49]^2)) + (ArcTan[(2 + 2*x + Log[49])/Sqrt[12 + 8*Log[49] - Log[49]^2]]*(96 + 280*Log[49]
+ 248*Log[49]^2 + 58*Log[49]^3 - 10*Log[49]^4 - Log[49]^5))/(2*(4 + 3*Log[49])^3*(12 + 8*Log[49] - Log[49]^2)
^(3/2)) - (ArcTanh[(2 - 2*x - Log[49])/Sqrt[20 + 8*Log[49] + Log[49]^2]]*(160 - 88*Log[49] - 216*Log[49]^2 - 7
4*Log[49]^3 - 10*Log[49]^4 + Log[49]^5))/(2*(4 + 3*Log[49])^3*(20 + 8*Log[49] + Log[49]^2)^(3/2)) - (3*Log[49]
*(2 + 2*x + Log[49])*(8 + 4*Log[49] - Log[49]^2))/(4*(4 + 3*Log[49])^2*(12 + 8*Log[49] - Log[49]^2)*(4 + x^2 +
3*Log[49] + x*(2 + Log[49]))) + (16 + 8*Log[49] - 11*Log[49]^2 - 5*Log[49]^3 + Log[49]^4 - x*Log[49]*(8 + 4*L
og[49] - Log[49]^2))/(4*(4 + 3*Log[49])^2*(4 + x^2 + 3*Log[49] + x*(2 + Log[49]))^2) + (16 + 8*Log[49] + 13*Lo
g[49]^2 + 5*Log[49]^3 + Log[49]^4 + x*Log[49]*(8 + 4*Log[49] + Log[49]^2))/(4*(4 + 3*Log[49])^2*(4 - x^2 + x*(
2 - Log[49]) + 3*Log[49])^2) + (Log[49]*(4 + Log[49])*(36 + 64*Log[49] + 31*Log[49]^2 + Log[49]^3 - Log[49]^4)
+ x*(96 + 280*Log[49] + 248*Log[49]^2 + 58*Log[49]^3 - 10*Log[49]^4 - Log[49]^5))/(4*(4 + 3*Log[49])^3*(12 +
8*Log[49] - Log[49]^2)*(4 + x^2 + 3*Log[49] + x*(2 + Log[49]))) - (640 + 304*Log[49] + 4*Log[49]^2 + 84*Log[49
]^3 + 27*Log[49]^4 + 3*Log[49]^5 - Log[49]^6 - x*(160 - 88*Log[49] - 216*Log[49]^2 - 74*Log[49]^3 - 10*Log[49]
^4 + Log[49]^5))/(4*(4 + 3*Log[49])^3*(4 - x^2 + x*(2 - Log[49]) + 3*Log[49])*(20 + 8*Log[49] + Log[49]^2)) -
(ArcTan[(2 + 2*x + Log[49])/Sqrt[12 + 8*Log[49] - Log[49]^2]]*(8 + Log[49]^3 + Log[2401]))/(2*(4 + 3*Log[49])^
3*Sqrt[12 + 8*Log[49] - Log[49]^2]) + (ArcTanh[(2 - 2*x - Log[49])/Sqrt[20 + 8*Log[49] + Log[49]^2]]*(8 + Log[
49]^3 + Log[2401]))/(2*(4 + 3*Log[49])^3*Sqrt[20 + 8*Log[49] + Log[49]^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 206

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

Rule 614

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 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 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 (\frac {64+168 \log (49)+138 \log ^2(49)+29 \log ^3(49)-7 \log ^4(49)-\log ^5(49)+x \left (16+20 \log (49)+4 \log ^2(49)-2 \log ^3(49)-\log ^4(49)\right )}{4 (4+3 \log (49))^3 \left (4+x^2+3 \log (49)+x (2+\log (49))\right )^2}+\frac {128-40 \log (49)-118 \log ^2(49)-37 \log ^3(49)-7 \log ^4(49)+\log ^5(49)-x \left (48+28 \log (49)+4 \log ^2(49)+2 \log ^3(49)-\log ^4(49)\right )}{4 (4+3 \log (49))^3 \left (4-x^2+x (2-\log (49))+3 \log (49)\right )^2}+\frac {-32-96 \log (49)-66 \log ^2(49)+5 \log ^3(49)+9 \log ^4(49)-\log ^5(49)-x \left (32+32 \log (49)-6 \log ^2(49)-8 \log ^3(49)+\log ^4(49)\right )}{2 (4+3 \log (49))^2 \left (4+x^2+3 \log (49)+x (2+\log (49))\right )^3}+\frac {-32+64 \log (49)+62 \log ^2(49)+35 \log ^3(49)+9 \log ^4(49)+\log ^5(49)+x \left (32+32 \log (49)+26 \log ^2(49)+8 \log ^3(49)+\log ^4(49)\right )}{2 (4+3 \log (49))^2 \left (4-x^2+x (2-\log (49))+3 \log (49)\right )^3}+\frac {-8-\log ^3(49)-\log (2401)}{4 (4+3 \log (49))^3 \left (4-x^2+x (2-\log (49))+3 \log (49)\right )}+\frac {-8-\log ^3(49)-\log (2401)}{4 (4+3 \log (49))^3 \left (4+x^2+3 \log (49)+x (2+\log (49))\right )}\right ) \, dx\\ &=\frac {\int \frac {64+168 \log (49)+138 \log ^2(49)+29 \log ^3(49)-7 \log ^4(49)-\log ^5(49)+x \left (16+20 \log (49)+4 \log ^2(49)-2 \log ^3(49)-\log ^4(49)\right )}{\left (4+x^2+3 \log (49)+x (2+\log (49))\right )^2} \, dx}{4 (4+3 \log (49))^3}+\frac {\int \frac {128-40 \log (49)-118 \log ^2(49)-37 \log ^3(49)-7 \log ^4(49)+\log ^5(49)-x \left (48+28 \log (49)+4 \log ^2(49)+2 \log ^3(49)-\log ^4(49)\right )}{\left (4-x^2+x (2-\log (49))+3 \log (49)\right )^2} \, dx}{4 (4+3 \log (49))^3}+\frac {\int \frac {-32-96 \log (49)-66 \log ^2(49)+5 \log ^3(49)+9 \log ^4(49)-\log ^5(49)-x \left (32+32 \log (49)-6 \log ^2(49)-8 \log ^3(49)+\log ^4(49)\right )}{\left (4+x^2+3 \log (49)+x (2+\log (49))\right )^3} \, dx}{2 (4+3 \log (49))^2}+\frac {\int \frac {-32+64 \log (49)+62 \log ^2(49)+35 \log ^3(49)+9 \log ^4(49)+\log ^5(49)+x \left (32+32 \log (49)+26 \log ^2(49)+8 \log ^3(49)+\log ^4(49)\right )}{\left (4-x^2+x (2-\log (49))+3 \log (49)\right )^3} \, dx}{2 (4+3 \log (49))^2}-\frac {\left (8+\log ^3(49)+\log (2401)\right ) \int \frac {1}{4-x^2+x (2-\log (49))+3 \log (49)} \, dx}{4 (4+3 \log (49))^3}-\frac {\left (8+\log ^3(49)+\log (2401)\right ) \int \frac {1}{4+x^2+3 \log (49)+x (2+\log (49))} \, dx}{4 (4+3 \log (49))^3}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.23, size = 535, normalized size = 23.26 \begin {gather*} \frac {\left (x^4+2 x^3 \log (49)-4 x (4+3 \log (49))-(4+3 \log (49))^2+x^2 \left (-4+\log ^2(49)\right )\right ) \left (-2640 \log ^9(49)+108 \log ^{10}(49)+27 \log ^{11}(49)-8 \log ^8(49) (3103+15 \log (2401))-12288 (-300+41 \log (2401))-4 \log ^7(49) (1870+129 \log (2401))+64 \log ^5(49) (91161+505 \log (2401))-1536 \log (49) (-11176+701 \log (2401))+4 \log ^6(49) (228864+971 \log (2401))+32 \log ^4(49) (560004+1675 \log (2401))-448 \log ^2(49) (-71392+1723 \log (2401))-64 \log ^3(49) (-492390+2521 \log (2401))\right )+(4+3 \log (49))^2 \left (-240-256 \log (49)-56 \log ^2(49)+\log ^4(49)\right ) \left (2 x^3 \left (-164 \log ^4(49)+4 \log ^5(49)+3 \log ^6(49)+8 \log (49) (-124+\log (2401))+16 \log (2401)-4 \log ^2(49) (440+\log (2401))-2 \log ^3(49) (492+\log (2401))\right )+x^2 \left (-818 \log ^5(49)+20 \log ^6(49)+15 \log ^7(49)-128 (30+\log (2401))-80 \log (49) (84+\log (2401))-12 \log ^3(49) (782+\log (2401))-5 \log ^4(49) (984+\log (2401))-4 \log ^2(49) (2152+\log (2401))\right )+(4+3 \log (49)) \left (-652 \log ^4(49)-49 \log ^6(49)+\log ^8(49)+\log ^5(49) (-232+\log (2401))-32 (120+\log (2401))-2 \log ^3(49) (1640+19 \log (2401))-8 \log ^2(49) (1100+23 \log (2401))-8 \log (49) (1224+25 \log (2401))\right )+2 x \left (-4340 \log ^4(49)-320 \log ^6(49)+8 \log ^7(49)+6 \log ^8(49)+\log ^5(49) (-1936+\log (2401))-96 (80+3 \log (2401))-24 \log ^2(49) (718+15 \log (2401))-2 \log ^3(49) (4248+31 \log (2401))-8 \log (49) (2392+73 \log (2401))\right )\right )}{(4+3 \log (49))^3 \left (240+256 \log (49)+56 \log ^2(49)-\log ^4(49)\right )^2 \left (x^4+2 x^3 \log (49)-4 x (4+3 \log (49))-(4+3 \log (49))^2+x^2 \left (-4+\log ^2(49)\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-64*x^3 - 32*x^4 - 4*x^7 + (-192*x^2 - 160*x^3 - 8*x^4 - 24*x^6)*Log[49] + (-192*x - 288*x^2 - 36*x
^3 - 60*x^5)*Log[49]^2 + (-64 - 224*x - 60*x^2 - 80*x^4)*Log[49]^3 + (-64 - 44*x - 60*x^3)*Log[49]^4 + (-12 -
24*x^2)*Log[49]^5 - 4*x*Log[49]^6)/(-4096 - 12288*x - 15360*x^2 - 10240*x^3 - 3072*x^4 + 768*x^5 + 1088*x^6 +
384*x^7 - 48*x^9 - 12*x^10 + x^12 + (-18432 - 46080*x - 46080*x^2 - 21504*x^3 - 384*x^4 + 5184*x^5 + 2496*x^6
+ 192*x^7 - 264*x^8 - 84*x^9 + 6*x^11)*Log[49] + (-34560 - 69120*x - 51072*x^2 - 11136*x^3 + 8496*x^4 + 6432*x
^5 + 984*x^6 - 576*x^7 - 243*x^8 + 15*x^10)*Log[49]^2 + (-34560 - 51840*x - 23616*x^2 + 4320*x^3 + 8208*x^4 +
2040*x^5 - 624*x^6 - 372*x^7 + 20*x^9)*Log[49]^3 + (-19440 - 19440*x - 2268*x^2 + 5184*x^3 + 2139*x^4 - 336*x^
5 - 318*x^6 + 15*x^8)*Log[49]^4 + (-5832 - 2916*x + 1296*x^2 + 1134*x^3 - 72*x^4 - 144*x^5 + 6*x^7)*Log[49]^5
+ (-729 + 243*x^2 - 27*x^4 + x^6)*Log[49]^6),x]

[Out]

((x^4 + 2*x^3*Log[49] - 4*x*(4 + 3*Log[49]) - (4 + 3*Log[49])^2 + x^2*(-4 + Log[49]^2))*(-2640*Log[49]^9 + 108
*Log[49]^10 + 27*Log[49]^11 - 8*Log[49]^8*(3103 + 15*Log[2401]) - 12288*(-300 + 41*Log[2401]) - 4*Log[49]^7*(1
870 + 129*Log[2401]) + 64*Log[49]^5*(91161 + 505*Log[2401]) - 1536*Log[49]*(-11176 + 701*Log[2401]) + 4*Log[49
]^6*(228864 + 971*Log[2401]) + 32*Log[49]^4*(560004 + 1675*Log[2401]) - 448*Log[49]^2*(-71392 + 1723*Log[2401]
) - 64*Log[49]^3*(-492390 + 2521*Log[2401])) + (4 + 3*Log[49])^2*(-240 - 256*Log[49] - 56*Log[49]^2 + Log[49]^
4)*(2*x^3*(-164*Log[49]^4 + 4*Log[49]^5 + 3*Log[49]^6 + 8*Log[49]*(-124 + Log[2401]) + 16*Log[2401] - 4*Log[49
]^2*(440 + Log[2401]) - 2*Log[49]^3*(492 + Log[2401])) + x^2*(-818*Log[49]^5 + 20*Log[49]^6 + 15*Log[49]^7 - 1
28*(30 + Log[2401]) - 80*Log[49]*(84 + Log[2401]) - 12*Log[49]^3*(782 + Log[2401]) - 5*Log[49]^4*(984 + Log[24
01]) - 4*Log[49]^2*(2152 + Log[2401])) + (4 + 3*Log[49])*(-652*Log[49]^4 - 49*Log[49]^6 + Log[49]^8 + Log[49]^
5*(-232 + Log[2401]) - 32*(120 + Log[2401]) - 2*Log[49]^3*(1640 + 19*Log[2401]) - 8*Log[49]^2*(1100 + 23*Log[2
401]) - 8*Log[49]*(1224 + 25*Log[2401])) + 2*x*(-4340*Log[49]^4 - 320*Log[49]^6 + 8*Log[49]^7 + 6*Log[49]^8 +
Log[49]^5*(-1936 + Log[2401]) - 96*(80 + 3*Log[2401]) - 24*Log[49]^2*(718 + 15*Log[2401]) - 2*Log[49]^3*(4248
+ 31*Log[2401]) - 8*Log[49]*(2392 + 73*Log[2401]))))/((4 + 3*Log[49])^3*(240 + 256*Log[49] + 56*Log[49]^2 - Lo
g[49]^4)^2*(x^4 + 2*x^3*Log[49] - 4*x*(4 + 3*Log[49]) - (4 + 3*Log[49])^2 + x^2*(-4 + Log[49]^2))^2)

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fricas [B]  time = 0.57, size = 172, normalized size = 7.48 \begin {gather*} \frac {x^{4} + 8 \, x^{3} \log \relax (7) + 24 \, x^{2} \log \relax (7)^{2} + 32 \, x \log \relax (7)^{3} + 16 \, \log \relax (7)^{4}}{x^{8} - 8 \, x^{6} - 32 \, x^{5} + 16 \, {\left (x^{4} - 18 \, x^{2} + 81\right )} \log \relax (7)^{4} - 16 \, x^{4} + 32 \, {\left (x^{5} - 15 \, x^{3} - 12 \, x^{2} + 54 \, x + 108\right )} \log \relax (7)^{3} + 128 \, x^{3} + 8 \, {\left (3 \, x^{6} - 37 \, x^{4} - 64 \, x^{3} + 92 \, x^{2} + 432 \, x + 432\right )} \log \relax (7)^{2} + 384 \, x^{2} + 8 \, {\left (x^{7} - 10 \, x^{5} - 28 \, x^{4} + 8 \, x^{3} + 144 \, x^{2} + 288 \, x + 192\right )} \log \relax (7) + 512 \, x + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-256*x*log(7)^6+32*(-24*x^2-12)*log(7)^5+16*(-60*x^3-44*x-64)*log(7)^4+8*(-80*x^4-60*x^2-224*x-64)*
log(7)^3+4*(-60*x^5-36*x^3-288*x^2-192*x)*log(7)^2+2*(-24*x^6-8*x^4-160*x^3-192*x^2)*log(7)-4*x^7-32*x^4-64*x^
3)/(64*(x^6-27*x^4+243*x^2-729)*log(7)^6+32*(6*x^7-144*x^5-72*x^4+1134*x^3+1296*x^2-2916*x-5832)*log(7)^5+16*(
15*x^8-318*x^6-336*x^5+2139*x^4+5184*x^3-2268*x^2-19440*x-19440)*log(7)^4+8*(20*x^9-372*x^7-624*x^6+2040*x^5+8
208*x^4+4320*x^3-23616*x^2-51840*x-34560)*log(7)^3+4*(15*x^10-243*x^8-576*x^7+984*x^6+6432*x^5+8496*x^4-11136*
x^3-51072*x^2-69120*x-34560)*log(7)^2+2*(6*x^11-84*x^9-264*x^8+192*x^7+2496*x^6+5184*x^5-384*x^4-21504*x^3-460
80*x^2-46080*x-18432)*log(7)+x^12-12*x^10-48*x^9+384*x^7+1088*x^6+768*x^5-3072*x^4-10240*x^3-15360*x^2-12288*x
-4096),x, algorithm="fricas")

[Out]

(x^4 + 8*x^3*log(7) + 24*x^2*log(7)^2 + 32*x*log(7)^3 + 16*log(7)^4)/(x^8 - 8*x^6 - 32*x^5 + 16*(x^4 - 18*x^2
+ 81)*log(7)^4 - 16*x^4 + 32*(x^5 - 15*x^3 - 12*x^2 + 54*x + 108)*log(7)^3 + 128*x^3 + 8*(3*x^6 - 37*x^4 - 64*
x^3 + 92*x^2 + 432*x + 432)*log(7)^2 + 384*x^2 + 8*(x^7 - 10*x^5 - 28*x^4 + 8*x^3 + 144*x^2 + 288*x + 192)*log
(7) + 512*x + 256)

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giac [B]  time = 0.37, size = 80, normalized size = 3.48 \begin {gather*} \frac {x^{4} + 8 \, x^{3} \log \relax (7) + 24 \, x^{2} \log \relax (7)^{2} + 32 \, x \log \relax (7)^{3} + 16 \, \log \relax (7)^{4}}{{\left (x^{4} + 4 \, x^{3} \log \relax (7) + 4 \, x^{2} \log \relax (7)^{2} - 4 \, x^{2} - 24 \, x \log \relax (7) - 36 \, \log \relax (7)^{2} - 16 \, x - 48 \, \log \relax (7) - 16\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-256*x*log(7)^6+32*(-24*x^2-12)*log(7)^5+16*(-60*x^3-44*x-64)*log(7)^4+8*(-80*x^4-60*x^2-224*x-64)*
log(7)^3+4*(-60*x^5-36*x^3-288*x^2-192*x)*log(7)^2+2*(-24*x^6-8*x^4-160*x^3-192*x^2)*log(7)-4*x^7-32*x^4-64*x^
3)/(64*(x^6-27*x^4+243*x^2-729)*log(7)^6+32*(6*x^7-144*x^5-72*x^4+1134*x^3+1296*x^2-2916*x-5832)*log(7)^5+16*(
15*x^8-318*x^6-336*x^5+2139*x^4+5184*x^3-2268*x^2-19440*x-19440)*log(7)^4+8*(20*x^9-372*x^7-624*x^6+2040*x^5+8
208*x^4+4320*x^3-23616*x^2-51840*x-34560)*log(7)^3+4*(15*x^10-243*x^8-576*x^7+984*x^6+6432*x^5+8496*x^4-11136*
x^3-51072*x^2-69120*x-34560)*log(7)^2+2*(6*x^11-84*x^9-264*x^8+192*x^7+2496*x^6+5184*x^5-384*x^4-21504*x^3-460
80*x^2-46080*x-18432)*log(7)+x^12-12*x^10-48*x^9+384*x^7+1088*x^6+768*x^5-3072*x^4-10240*x^3-15360*x^2-12288*x
-4096),x, algorithm="giac")

[Out]

(x^4 + 8*x^3*log(7) + 24*x^2*log(7)^2 + 32*x*log(7)^3 + 16*log(7)^4)/(x^4 + 4*x^3*log(7) + 4*x^2*log(7)^2 - 4*
x^2 - 24*x*log(7) - 36*log(7)^2 - 16*x - 48*log(7) - 16)^2

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maple [B]  time = 0.74, size = 81, normalized size = 3.52

method	result	size
norman	\(\frac {16 \ln \relax (7)^{4}+32 \ln \relax (7)^{3} x +24 x^{2} \ln \relax (7)^{2}+8 \ln \relax (7) x^{3}+x^{4}}{\left (4 x^{2} \ln \relax (7)^{2}+4 \ln \relax (7) x^{3}+x^{4}-36 \ln \relax (7)^{2}-24 x \ln \relax (7)-4 x^{2}-48 \ln \relax (7)-16 x -16\right )^{2}}\)	\(81\)
gosper	\(\frac {16 \ln \relax (7)^{4}+32 \ln \relax (7)^{3} x +24 x^{2} \ln \relax (7)^{2}+8 \ln \relax (7) x^{3}+x^{4}}{256+512 x +1536 \ln \relax (7)+x^{8}-8 x^{6}-32 x^{5}-16 x^{4}+128 x^{3}+384 x^{2}+2304 x \ln \relax (7)+1296 \ln \relax (7)^{4}+736 x^{2} \ln \relax (7)^{2}+3456 \ln \relax (7)^{2}+1152 x^{2} \ln \relax (7)+16 \ln \relax (7)^{4} x^{4}+32 \ln \relax (7)^{3} x^{5}+24 \ln \relax (7)^{2} x^{6}+8 \ln \relax (7) x^{7}-288 \ln \relax (7)^{4} x^{2}-480 \ln \relax (7)^{3} x^{3}-296 \ln \relax (7)^{2} x^{4}-80 \ln \relax (7) x^{5}-384 \ln \relax (7)^{3} x^{2}-512 \ln \relax (7)^{2} x^{3}-224 \ln \relax (7) x^{4}+1728 \ln \relax (7)^{3} x +64 \ln \relax (7) x^{3}+3456 \ln \relax (7)^{2} x +3456 \ln \relax (7)^{3}}\)	\(227\)
risch	\(\frac {\ln \relax (7)^{4}+2 \ln \relax (7)^{3} x +\frac {3 x^{2} \ln \relax (7)^{2}}{2}+\frac {\ln \relax (7) x^{3}}{2}+\frac {x^{4}}{16}}{16+32 x +96 \ln \relax (7)+\frac {x^{8}}{16}-\frac {x^{6}}{2}-2 x^{5}-x^{4}+8 x^{3}+24 x^{2}+144 x \ln \relax (7)+81 \ln \relax (7)^{4}+46 x^{2} \ln \relax (7)^{2}+216 \ln \relax (7)^{2}+72 x^{2} \ln \relax (7)+\ln \relax (7)^{4} x^{4}+2 \ln \relax (7)^{3} x^{5}+\frac {3 \ln \relax (7)^{2} x^{6}}{2}+\frac {\ln \relax (7) x^{7}}{2}-18 \ln \relax (7)^{4} x^{2}-30 \ln \relax (7)^{3} x^{3}-\frac {37 \ln \relax (7)^{2} x^{4}}{2}-5 \ln \relax (7) x^{5}-24 \ln \relax (7)^{3} x^{2}-32 \ln \relax (7)^{2} x^{3}-14 \ln \relax (7) x^{4}+108 \ln \relax (7)^{3} x +4 \ln \relax (7) x^{3}+216 \ln \relax (7)^{2} x +216 \ln \relax (7)^{3}}\)	\(228\)
default	\(\frac {\left (-\frac {\ln \relax (7)^{3}}{2}-\frac {\ln \relax (7)}{4}-\frac {1}{2}\right ) x^{3}+\left (-2 \ln \relax (7)^{4}+2 \ln \relax (7)^{3}-\frac {\ln \relax (7)^{2}}{4}+\ln \relax (7)+3\right ) x^{2}+\left (-2 \ln \relax (7)^{5}+10 \ln \relax (7)^{4}+\frac {17 \ln \relax (7)^{3}}{2}+14 \ln \relax (7)^{2}+9 \ln \relax (7)-2\right ) x +12 \ln \relax (7)^{5}+17 \ln \relax (7)^{4}+24 \ln \relax (7)^{3}+9 \ln \relax (7)^{2}-8 \ln \relax (7)-4}{2 \left (3 \ln \relax (7)+2\right ) \left (9 \ln \relax (7)^{2}+12 \ln \relax (7)+4\right ) \left (2 x \ln \relax (7)+x^{2}-6 \ln \relax (7)-2 x -4\right )^{2}}-\frac {\left (-\frac {\ln \relax (7)^{3}}{2}-\frac {\ln \relax (7)}{4}-\frac {1}{2}\right ) x^{3}+\left (-2 \ln \relax (7)^{4}-2 \ln \relax (7)^{3}-\frac {\ln \relax (7)^{2}}{4}-\ln \relax (7)-1\right ) x^{2}+\left (-2 \ln \relax (7)^{5}-10 \ln \relax (7)^{4}+\frac {\ln \relax (7)^{3}}{2}+10 \ln \relax (7)^{2}+\ln \relax (7)-2\right ) x -12 \ln \relax (7)^{5}+\ln \relax (7)^{4}+24 \ln \relax (7)^{3}+9 \ln \relax (7)^{2}-8 \ln \relax (7)-4}{2 \left (3 \ln \relax (7)+2\right ) \left (9 \ln \relax (7)^{2}+12 \ln \relax (7)+4\right ) \left (2 x \ln \relax (7)+x^{2}+6 \ln \relax (7)+2 x +4\right )^{2}}\)	\(294\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-256*x*ln(7)^6+32*(-24*x^2-12)*ln(7)^5+16*(-60*x^3-44*x-64)*ln(7)^4+8*(-80*x^4-60*x^2-224*x-64)*ln(7)^3+4
*(-60*x^5-36*x^3-288*x^2-192*x)*ln(7)^2+2*(-24*x^6-8*x^4-160*x^3-192*x^2)*ln(7)-4*x^7-32*x^4-64*x^3)/(64*(x^6-
27*x^4+243*x^2-729)*ln(7)^6+32*(6*x^7-144*x^5-72*x^4+1134*x^3+1296*x^2-2916*x-5832)*ln(7)^5+16*(15*x^8-318*x^6
-336*x^5+2139*x^4+5184*x^3-2268*x^2-19440*x-19440)*ln(7)^4+8*(20*x^9-372*x^7-624*x^6+2040*x^5+8208*x^4+4320*x^
3-23616*x^2-51840*x-34560)*ln(7)^3+4*(15*x^10-243*x^8-576*x^7+984*x^6+6432*x^5+8496*x^4-11136*x^3-51072*x^2-69
120*x-34560)*ln(7)^2+2*(6*x^11-84*x^9-264*x^8+192*x^7+2496*x^6+5184*x^5-384*x^4-21504*x^3-46080*x^2-46080*x-18
432)*ln(7)+x^12-12*x^10-48*x^9+384*x^7+1088*x^6+768*x^5-3072*x^4-10240*x^3-15360*x^2-12288*x-4096),x,method=_R
ETURNVERBOSE)

[Out]

(16*ln(7)^4+32*ln(7)^3*x+24*x^2*ln(7)^2+8*ln(7)*x^3+x^4)/(4*x^2*ln(7)^2+4*ln(7)*x^3+x^4-36*ln(7)^2-24*x*ln(7)-
4*x^2-48*ln(7)-16*x-16)^2

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maxima [B]  time = 0.36, size = 196, normalized size = 8.52 \begin {gather*} \frac {x^{4} + 8 \, x^{3} \log \relax (7) + 24 \, x^{2} \log \relax (7)^{2} + 32 \, x \log \relax (7)^{3} + 16 \, \log \relax (7)^{4}}{x^{8} + 8 \, x^{7} \log \relax (7) + 8 \, {\left (3 \, \log \relax (7)^{2} - 1\right )} x^{6} + 16 \, {\left (2 \, \log \relax (7)^{3} - 5 \, \log \relax (7) - 2\right )} x^{5} + 8 \, {\left (2 \, \log \relax (7)^{4} - 37 \, \log \relax (7)^{2} - 28 \, \log \relax (7) - 2\right )} x^{4} - 32 \, {\left (15 \, \log \relax (7)^{3} + 16 \, \log \relax (7)^{2} - 2 \, \log \relax (7) - 4\right )} x^{3} + 1296 \, \log \relax (7)^{4} - 32 \, {\left (9 \, \log \relax (7)^{4} + 12 \, \log \relax (7)^{3} - 23 \, \log \relax (7)^{2} - 36 \, \log \relax (7) - 12\right )} x^{2} + 3456 \, \log \relax (7)^{3} + 64 \, {\left (27 \, \log \relax (7)^{3} + 54 \, \log \relax (7)^{2} + 36 \, \log \relax (7) + 8\right )} x + 3456 \, \log \relax (7)^{2} + 1536 \, \log \relax (7) + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-256*x*log(7)^6+32*(-24*x^2-12)*log(7)^5+16*(-60*x^3-44*x-64)*log(7)^4+8*(-80*x^4-60*x^2-224*x-64)*
log(7)^3+4*(-60*x^5-36*x^3-288*x^2-192*x)*log(7)^2+2*(-24*x^6-8*x^4-160*x^3-192*x^2)*log(7)-4*x^7-32*x^4-64*x^
3)/(64*(x^6-27*x^4+243*x^2-729)*log(7)^6+32*(6*x^7-144*x^5-72*x^4+1134*x^3+1296*x^2-2916*x-5832)*log(7)^5+16*(
15*x^8-318*x^6-336*x^5+2139*x^4+5184*x^3-2268*x^2-19440*x-19440)*log(7)^4+8*(20*x^9-372*x^7-624*x^6+2040*x^5+8
208*x^4+4320*x^3-23616*x^2-51840*x-34560)*log(7)^3+4*(15*x^10-243*x^8-576*x^7+984*x^6+6432*x^5+8496*x^4-11136*
x^3-51072*x^2-69120*x-34560)*log(7)^2+2*(6*x^11-84*x^9-264*x^8+192*x^7+2496*x^6+5184*x^5-384*x^4-21504*x^3-460
80*x^2-46080*x-18432)*log(7)+x^12-12*x^10-48*x^9+384*x^7+1088*x^6+768*x^5-3072*x^4-10240*x^3-15360*x^2-12288*x
-4096),x, algorithm="maxima")

[Out]

(x^4 + 8*x^3*log(7) + 24*x^2*log(7)^2 + 32*x*log(7)^3 + 16*log(7)^4)/(x^8 + 8*x^7*log(7) + 8*(3*log(7)^2 - 1)*
x^6 + 16*(2*log(7)^3 - 5*log(7) - 2)*x^5 + 8*(2*log(7)^4 - 37*log(7)^2 - 28*log(7) - 2)*x^4 - 32*(15*log(7)^3
+ 16*log(7)^2 - 2*log(7) - 4)*x^3 + 1296*log(7)^4 - 32*(9*log(7)^4 + 12*log(7)^3 - 23*log(7)^2 - 36*log(7) - 1
2)*x^2 + 3456*log(7)^3 + 64*(27*log(7)^3 + 54*log(7)^2 + 36*log(7) + 8)*x + 3456*log(7)^2 + 1536*log(7) + 256)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(7)*(192*x^2 + 160*x^3 + 8*x^4 + 24*x^6) + 4*log(7)^2*(192*x + 288*x^2 + 36*x^3 + 60*x^5) + 16*log(7
)^4*(44*x + 60*x^3 + 64) + 256*x*log(7)^6 + 32*log(7)^5*(24*x^2 + 12) + 8*log(7)^3*(224*x + 60*x^2 + 80*x^4 +
64) + 64*x^3 + 32*x^4 + 4*x^7)/(12288*x + 4*log(7)^2*(69120*x + 51072*x^2 + 11136*x^3 - 8496*x^4 - 6432*x^5 -
984*x^6 + 576*x^7 + 243*x^8 - 15*x^10 + 34560) + 32*log(7)^5*(2916*x - 1296*x^2 - 1134*x^3 + 72*x^4 + 144*x^5
- 6*x^7 + 5832) + 16*log(7)^4*(19440*x + 2268*x^2 - 5184*x^3 - 2139*x^4 + 336*x^5 + 318*x^6 - 15*x^8 + 19440)
- 64*log(7)^6*(243*x^2 - 27*x^4 + x^6 - 729) + 15360*x^2 + 10240*x^3 + 3072*x^4 - 768*x^5 - 1088*x^6 - 384*x^7
+ 48*x^9 + 12*x^10 - x^12 + 8*log(7)^3*(51840*x + 23616*x^2 - 4320*x^3 - 8208*x^4 - 2040*x^5 + 624*x^6 + 372*
x^7 - 20*x^9 + 34560) + 2*log(7)*(46080*x + 46080*x^2 + 21504*x^3 + 384*x^4 - 5184*x^5 - 2496*x^6 - 192*x^7 +
264*x^8 + 84*x^9 - 6*x^11 + 18432) + 4096),x)

[Out]

\text{Hanged}

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sympy [B]  time = 23.31, size = 207, normalized size = 9.00 \begin {gather*} - \frac {- x^{4} - 8 x^{3} \log {\relax (7 )} - 24 x^{2} \log {\relax (7 )}^{2} - 32 x \log {\relax (7 )}^{3} - 16 \log {\relax (7 )}^{4}}{x^{8} + 8 x^{7} \log {\relax (7 )} + x^{6} \left (-8 + 24 \log {\relax (7 )}^{2}\right ) + x^{5} \left (- 80 \log {\relax (7 )} - 32 + 32 \log {\relax (7 )}^{3}\right ) + x^{4} \left (- 296 \log {\relax (7 )}^{2} - 224 \log {\relax (7 )} - 16 + 16 \log {\relax (7 )}^{4}\right ) + x^{3} \left (- 480 \log {\relax (7 )}^{3} - 512 \log {\relax (7 )}^{2} + 64 \log {\relax (7 )} + 128\right ) + x^{2} \left (- 288 \log {\relax (7 )}^{4} - 384 \log {\relax (7 )}^{3} + 384 + 1152 \log {\relax (7 )} + 736 \log {\relax (7 )}^{2}\right ) + x \left (512 + 2304 \log {\relax (7 )} + 1728 \log {\relax (7 )}^{3} + 3456 \log {\relax (7 )}^{2}\right ) + 256 + 1536 \log {\relax (7 )} + 3456 \log {\relax (7 )}^{2} + 1296 \log {\relax (7 )}^{4} + 3456 \log {\relax (7 )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-256*x*ln(7)**6+32*(-24*x**2-12)*ln(7)**5+16*(-60*x**3-44*x-64)*ln(7)**4+8*(-80*x**4-60*x**2-224*x-
64)*ln(7)**3+4*(-60*x**5-36*x**3-288*x**2-192*x)*ln(7)**2+2*(-24*x**6-8*x**4-160*x**3-192*x**2)*ln(7)-4*x**7-3
2*x**4-64*x**3)/(64*(x**6-27*x**4+243*x**2-729)*ln(7)**6+32*(6*x**7-144*x**5-72*x**4+1134*x**3+1296*x**2-2916*
x-5832)*ln(7)**5+16*(15*x**8-318*x**6-336*x**5+2139*x**4+5184*x**3-2268*x**2-19440*x-19440)*ln(7)**4+8*(20*x**
9-372*x**7-624*x**6+2040*x**5+8208*x**4+4320*x**3-23616*x**2-51840*x-34560)*ln(7)**3+4*(15*x**10-243*x**8-576*
x**7+984*x**6+6432*x**5+8496*x**4-11136*x**3-51072*x**2-69120*x-34560)*ln(7)**2+2*(6*x**11-84*x**9-264*x**8+19
2*x**7+2496*x**6+5184*x**5-384*x**4-21504*x**3-46080*x**2-46080*x-18432)*ln(7)+x**12-12*x**10-48*x**9+384*x**7
+1088*x**6+768*x**5-3072*x**4-10240*x**3-15360*x**2-12288*x-4096),x)

[Out]

-(-x**4 - 8*x**3*log(7) - 24*x**2*log(7)**2 - 32*x*log(7)**3 - 16*log(7)**4)/(x**8 + 8*x**7*log(7) + x**6*(-8
+ 24*log(7)**2) + x**5*(-80*log(7) - 32 + 32*log(7)**3) + x**4*(-296*log(7)**2 - 224*log(7) - 16 + 16*log(7)**
4) + x**3*(-480*log(7)**3 - 512*log(7)**2 + 64*log(7) + 128) + x**2*(-288*log(7)**4 - 384*log(7)**3 + 384 + 11
52*log(7) + 736*log(7)**2) + x*(512 + 2304*log(7) + 1728*log(7)**3 + 3456*log(7)**2) + 256 + 1536*log(7) + 345
6*log(7)**2 + 1296*log(7)**4 + 3456*log(7)**3)

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