3.26.83 \(\int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 (\frac {-1+x}{1+2 x})^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx\)
Optimal. Leaf size=223 \[ \frac {\left (\frac {x-1}{2 x+1}\right )^{3/4} (-2 x-1)}{2 (x+1)}+\frac {\sqrt [4]{\frac {x-1}{2 x+1}} (2 x+1)}{6 (x+1)}-\frac {1}{72} \sqrt {24420+55819 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {x-1}{2 x+1}}}{\sqrt [4]{2}}\right )+\frac {2}{9} \tan ^{-1}\left (\frac {2 \sqrt [4]{\frac {x-1}{2 x+1}}}{2 \sqrt {\frac {x-1}{2 x+1}}-1}\right )+\frac {1}{72} \sqrt {55819 \sqrt {2}-24420} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {x-1}{2 x+1}}}{\sqrt [4]{2}}\right )+\frac {10}{9} \tanh ^{-1}\left (\frac {2 \sqrt [4]{\frac {x-1}{2 x+1}}}{2 \sqrt {\frac {x-1}{2 x+1}}+1}\right ) \]
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Rubi [A] time = 1.33, antiderivative size = 373, normalized size of antiderivative = 1.67,
number of steps used = 18, number of rules used = 9, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used
= {6725, 634, 617, 204, 628, 1179, 1167, 207, 203} \begin {gather*} \frac {\sqrt [4]{-\frac {1-x}{2 x+1}} (2 x+1) \left (1-3 \sqrt {-\frac {1-x}{2 x+1}}\right )}{6 (x+1)}-\frac {5}{9} \log \left (2 \sqrt {-\frac {1-x}{2 x+1}}-2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )+\frac {5}{9} \log \left (2 \sqrt {-\frac {1-x}{2 x+1}}+2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )-\frac {8}{9} \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )+\frac {3 \left (1+\sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}+\frac {2}{9} \tan ^{-1}\left (1-2 \sqrt [4]{-\frac {1-x}{2 x+1}}\right )-\frac {2}{9} \tan ^{-1}\left (2 \sqrt [4]{-\frac {1-x}{2 x+1}}+1\right )+\frac {3 \left (1-\sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {8}{9} \sqrt [4]{2} \left (1-3 \sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{2 x+1}}}{\sqrt [4]{2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(((-1 + x)/(1 + 2*x))^(1/4) - 3*((-1 + x)/(1 + 2*x))^(3/4))/((-1 + x)*(1 + x)^2*(-1 + 2*x)),x]
[Out]
((-((1 - x)/(1 + 2*x)))^(1/4)*(1 + 2*x)*(1 - 3*Sqrt[-((1 - x)/(1 + 2*x))]))/(6*(1 + x)) + (3*(1 + Sqrt[2])*Arc
Tan[(-((1 - x)/(1 + 2*x)))^(1/4)/2^(1/4)])/(4*2^(3/4)) - (8*2^(1/4)*(1 + 3*Sqrt[2])*ArcTan[(-((1 - x)/(1 + 2*x
)))^(1/4)/2^(1/4)])/9 + (2*ArcTan[1 - 2*(-((1 - x)/(1 + 2*x)))^(1/4)])/9 - (2*ArcTan[1 + 2*(-((1 - x)/(1 + 2*x
)))^(1/4)])/9 - (8*2^(1/4)*(1 - 3*Sqrt[2])*ArcTanh[(-((1 - x)/(1 + 2*x)))^(1/4)/2^(1/4)])/9 + (3*(1 - Sqrt[2])
*ArcTanh[(-((1 - x)/(1 + 2*x)))^(1/4)/2^(1/4)])/(4*2^(3/4)) - (5*Log[1 - 2*(-((1 - x)/(1 + 2*x)))^(1/4) + 2*Sq
rt[-((1 - x)/(1 + 2*x))]])/9 + (5*Log[1 + 2*(-((1 - x)/(1 + 2*x)))^(1/4) + 2*Sqrt[-((1 - x)/(1 + 2*x))]])/9
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 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 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), 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 1167
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q)
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[-(a*c)]
Rule 1179
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)*(a + c*x^4)^(p + 1))/(
4*a*(p + 1)), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x
] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Rule 6725
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{\frac {-1+x}{1+2 x}}-3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{(-1+x) (1+x)^2 (-1+2 x)} \, dx &=-\left (4 \operatorname {Subst}\left (\int \frac {\left (-1+3 x^2\right ) \left (1-2 x^4\right )^2}{\left (-2+x^4\right )^2 \left (1+4 x^4\right )} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \left (\frac {-2+5 x}{9 \left (1-2 x+2 x^2\right )}+\frac {-2-5 x}{9 \left (1+2 x+2 x^2\right )}+\frac {-1+3 x^2}{\left (-2+x^4\right )^2}+\frac {8 \left (-1+3 x^2\right )}{9 \left (-2+x^4\right )}\right ) \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\right )\\ &=-\left (\frac {4}{9} \operatorname {Subst}\left (\int \frac {-2+5 x}{1-2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\right )-\frac {4}{9} \operatorname {Subst}\left (\int \frac {-2-5 x}{1+2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {32}{9} \operatorname {Subst}\left (\int \frac {-1+3 x^2}{-2+x^4} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-4 \operatorname {Subst}\left (\int \frac {-1+3 x^2}{\left (-2+x^4\right )^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\\ &=\frac {\sqrt [4]{-\frac {1-x}{1+2 x}} (1+2 x) \left (1-3 \sqrt {-\frac {1-x}{1+2 x}}\right )}{6 (1+x)}-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{1-2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{1+2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {3-3 x^2}{-2+x^4} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {5}{9} \operatorname {Subst}\left (\int \frac {-2+4 x}{1-2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )+\frac {5}{9} \operatorname {Subst}\left (\int \frac {2+4 x}{1+2 x+2 x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {1}{9} \left (8 \left (6-\sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {2}+x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )-\frac {1}{9} \left (8 \left (6+\sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\\ &=\frac {\sqrt [4]{-\frac {1-x}{1+2 x}} (1+2 x) \left (1-3 \sqrt {-\frac {1-x}{1+2 x}}\right )}{6 (1+x)}-\frac {8}{9} \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )-\frac {8}{9} \sqrt [4]{2} \left (1-3 \sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )-\frac {5}{9} \log \left (1-2 \sqrt [4]{-\frac {1-x}{1+2 x}}+2 \sqrt {-\frac {1-x}{1+2 x}}\right )+\frac {5}{9} \log \left (1+2 \sqrt [4]{-\frac {1-x}{1+2 x}}+2 \sqrt {-\frac {1-x}{1+2 x}}\right )-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-2 \sqrt [4]{\frac {-1+x}{1+2 x}}\right )+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+2 \sqrt [4]{\frac {-1+x}{1+2 x}}\right )+\frac {1}{8} \left (3 \left (2-\sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {2}+x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )+\frac {1}{8} \left (3 \left (2+\sqrt {2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{\frac {-1+x}{1+2 x}}\right )\\ &=\frac {\sqrt [4]{-\frac {1-x}{1+2 x}} (1+2 x) \left (1-3 \sqrt {-\frac {1-x}{1+2 x}}\right )}{6 (1+x)}+\frac {3 \left (1+\sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {8}{9} \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {2}{9} \tan ^{-1}\left (1-2 \sqrt [4]{-\frac {1-x}{1+2 x}}\right )-\frac {2}{9} \tan ^{-1}\left (1+2 \sqrt [4]{-\frac {1-x}{1+2 x}}\right )-\frac {8}{9} \sqrt [4]{2} \left (1-3 \sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {3 \left (1-\sqrt {2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {1-x}{1+2 x}}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac {5}{9} \log \left (1-2 \sqrt [4]{-\frac {1-x}{1+2 x}}+2 \sqrt {-\frac {1-x}{1+2 x}}\right )+\frac {5}{9} \log \left (1+2 \sqrt [4]{-\frac {1-x}{1+2 x}}+2 \sqrt {-\frac {1-x}{1+2 x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.60, size = 398, normalized size = 1.78 \begin {gather*} \frac {1}{18} \left (-18 \left (\frac {x-1}{2 x+1}\right )^{3/4} \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};\frac {x-1}{4 x+2}\right )+\frac {3 \sqrt [4]{\frac {x-1}{2 x+1}} (2 x+1)}{x+1}+8 \sqrt [4]{2} \left (1-3 \sqrt {2}\right ) \log \left (\sqrt [4]{2}-\sqrt [4]{\frac {x-1}{2 x+1}}\right )-8 i \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \log \left (\sqrt [4]{2}-i \sqrt [4]{\frac {x-1}{2 x+1}}\right )+8 i \sqrt [4]{2} \left (1+3 \sqrt {2}\right ) \log \left (\sqrt [4]{2}+i \sqrt [4]{\frac {x-1}{2 x+1}}\right )+8 \sqrt [4]{2} \left (3 \sqrt {2}-1\right ) \log \left (\sqrt [4]{\frac {x-1}{2 x+1}}+\sqrt [4]{2}\right )-10 \log \left (2 \sqrt {\frac {x-1}{2 x+1}}-2 \sqrt [4]{\frac {x-1}{2 x+1}}+1\right )+10 \log \left (2 \sqrt {\frac {x-1}{2 x+1}}+2 \sqrt [4]{\frac {x-1}{2 x+1}}+1\right )+4 \tan ^{-1}\left (1-2 \sqrt [4]{\frac {x-1}{2 x+1}}\right )-4 \tan ^{-1}\left (2 \sqrt [4]{\frac {x-1}{2 x+1}}+1\right )+\frac {27 \left (\tan ^{-1}\left (\sqrt [4]{\frac {x-1}{4 x+2}}\right )+\tanh ^{-1}\left (\sqrt [4]{\frac {x-1}{4 x+2}}\right )\right )}{2\ 2^{3/4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(((-1 + x)/(1 + 2*x))^(1/4) - 3*((-1 + x)/(1 + 2*x))^(3/4))/((-1 + x)*(1 + x)^2*(-1 + 2*x)),x]
[Out]
((3*((-1 + x)/(1 + 2*x))^(1/4)*(1 + 2*x))/(1 + x) + 4*ArcTan[1 - 2*((-1 + x)/(1 + 2*x))^(1/4)] - 4*ArcTan[1 +
2*((-1 + x)/(1 + 2*x))^(1/4)] + (27*(ArcTan[((-1 + x)/(2 + 4*x))^(1/4)] + ArcTanh[((-1 + x)/(2 + 4*x))^(1/4)])
)/(2*2^(3/4)) - 18*((-1 + x)/(1 + 2*x))^(3/4)*Hypergeometric2F1[3/4, 2, 7/4, (-1 + x)/(2 + 4*x)] + 8*2^(1/4)*(
1 - 3*Sqrt[2])*Log[2^(1/4) - ((-1 + x)/(1 + 2*x))^(1/4)] - (8*I)*2^(1/4)*(1 + 3*Sqrt[2])*Log[2^(1/4) - I*((-1
+ x)/(1 + 2*x))^(1/4)] + (8*I)*2^(1/4)*(1 + 3*Sqrt[2])*Log[2^(1/4) + I*((-1 + x)/(1 + 2*x))^(1/4)] + 8*2^(1/4)
*(-1 + 3*Sqrt[2])*Log[2^(1/4) + ((-1 + x)/(1 + 2*x))^(1/4)] - 10*Log[1 - 2*((-1 + x)/(1 + 2*x))^(1/4) + 2*Sqrt
[(-1 + x)/(1 + 2*x)]] + 10*Log[1 + 2*((-1 + x)/(1 + 2*x))^(1/4) + 2*Sqrt[(-1 + x)/(1 + 2*x)]])/18
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IntegrateAlgebraic [A] time = 0.56, size = 216, normalized size = 0.97 \begin {gather*} \frac {-\sqrt [4]{\frac {-1+x}{1+2 x}}+3 \left (\frac {-1+x}{1+2 x}\right )^{3/4}}{2 \left (-2+\frac {-1+x}{1+2 x}\right )}-\frac {1}{72} \sqrt {24420+55819 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {-1+x}{1+2 x}}}{\sqrt [4]{2}}\right )-\frac {2}{9} \tan ^{-1}\left (\frac {-\frac {1}{2}+\sqrt {\frac {-1+x}{1+2 x}}}{\sqrt [4]{\frac {-1+x}{1+2 x}}}\right )+\frac {1}{72} \sqrt {-24420+55819 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {-1+x}{1+2 x}}}{\sqrt [4]{2}}\right )+\frac {10}{9} \tanh ^{-1}\left (\frac {2 \sqrt [4]{\frac {-1+x}{1+2 x}}}{1+2 \sqrt {\frac {-1+x}{1+2 x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(((-1 + x)/(1 + 2*x))^(1/4) - 3*((-1 + x)/(1 + 2*x))^(3/4))/((-1 + x)*(1 + x)^2*(-1 + 2*x))
,x]
[Out]
(-((-1 + x)/(1 + 2*x))^(1/4) + 3*((-1 + x)/(1 + 2*x))^(3/4))/(2*(-2 + (-1 + x)/(1 + 2*x))) - (Sqrt[24420 + 558
19*Sqrt[2]]*ArcTan[((-1 + x)/(1 + 2*x))^(1/4)/2^(1/4)])/72 - (2*ArcTan[(-1/2 + Sqrt[(-1 + x)/(1 + 2*x)])/((-1
+ x)/(1 + 2*x))^(1/4)])/9 + (Sqrt[-24420 + 55819*Sqrt[2]]*ArcTanh[((-1 + x)/(1 + 2*x))^(1/4)/2^(1/4)])/72 + (1
0*ArcTanh[(2*((-1 + x)/(1 + 2*x))^(1/4))/(1 + 2*Sqrt[(-1 + x)/(1 + 2*x)])])/9
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fricas [A] time = 0.52, size = 350, normalized size = 1.57 \begin {gather*} -\frac {4 \, {\left (x + 1\right )} \sqrt {55819 \, \sqrt {2} + 24420} \arctan \left (\frac {1}{106162} \, \sqrt {55819 \, \sqrt {2} + 24420} {\left (37 \, \sqrt {2} - 330\right )} \sqrt {\sqrt {2} + \sqrt {\frac {x - 1}{2 \, x + 1}}} - \frac {1}{106162} \, \sqrt {55819 \, \sqrt {2} + 24420} {\left (37 \, \sqrt {2} - 330\right )} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - {\left (x + 1\right )} \sqrt {55819 \, \sqrt {2} - 24420} \log \left (\sqrt {55819 \, \sqrt {2} - 24420} {\left (165 \, \sqrt {2} + 37\right )} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) + {\left (x + 1\right )} \sqrt {55819 \, \sqrt {2} - 24420} \log \left (-\sqrt {55819 \, \sqrt {2} - 24420} {\left (165 \, \sqrt {2} + 37\right )} + 53081 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) + 32 \, {\left (x + 1\right )} \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) + 32 \, {\left (x + 1\right )} \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) - 80 \, {\left (x + 1\right )} \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) + 80 \, {\left (x + 1\right )} \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) + 72 \, {\left (2 \, x + 1\right )} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} - 24 \, {\left (2 \, x + 1\right )} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{144 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-1+x)/(1+2*x))^(1/4)-3*((-1+x)/(1+2*x))^(3/4))/(-1+x)/(1+x)^2/(-1+2*x),x, algorithm="fricas")
[Out]
-1/144*(4*(x + 1)*sqrt(55819*sqrt(2) + 24420)*arctan(1/106162*sqrt(55819*sqrt(2) + 24420)*(37*sqrt(2) - 330)*s
qrt(sqrt(2) + sqrt((x - 1)/(2*x + 1))) - 1/106162*sqrt(55819*sqrt(2) + 24420)*(37*sqrt(2) - 330)*((x - 1)/(2*x
+ 1))^(1/4)) - (x + 1)*sqrt(55819*sqrt(2) - 24420)*log(sqrt(55819*sqrt(2) - 24420)*(165*sqrt(2) + 37) + 53081
*((x - 1)/(2*x + 1))^(1/4)) + (x + 1)*sqrt(55819*sqrt(2) - 24420)*log(-sqrt(55819*sqrt(2) - 24420)*(165*sqrt(2
) + 37) + 53081*((x - 1)/(2*x + 1))^(1/4)) + 32*(x + 1)*arctan(2*((x - 1)/(2*x + 1))^(1/4) + 1) + 32*(x + 1)*a
rctan(2*((x - 1)/(2*x + 1))^(1/4) - 1) - 80*(x + 1)*log(2*sqrt((x - 1)/(2*x + 1)) + 2*((x - 1)/(2*x + 1))^(1/4
) + 1) + 80*(x + 1)*log(2*sqrt((x - 1)/(2*x + 1)) - 2*((x - 1)/(2*x + 1))^(1/4) + 1) + 72*(2*x + 1)*((x - 1)/(
2*x + 1))^(3/4) - 24*(2*x + 1)*((x - 1)/(2*x + 1))^(1/4))/(x + 1)
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giac [A] time = 0.65, size = 250, normalized size = 1.12 \begin {gather*} -\frac {1}{72} \, \sqrt {55819 \, \sqrt {2} + 24420} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) + \frac {1}{144} \, \sqrt {55819 \, \sqrt {2} - 24420} \log \left (2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - \frac {1}{144} \, \sqrt {55819 \, \sqrt {2} - 24420} \log \left ({\left | -2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} \right |}\right ) + \frac {3 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2 \, {\left (\frac {x - 1}{2 \, x + 1} - 2\right )}} - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) + \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-1+x)/(1+2*x))^(1/4)-3*((-1+x)/(1+2*x))^(3/4))/(-1+x)/(1+x)^2/(-1+2*x),x, algorithm="giac")
[Out]
-1/72*sqrt(55819*sqrt(2) + 24420)*arctan(1/2*2^(3/4)*((x - 1)/(2*x + 1))^(1/4)) + 1/144*sqrt(55819*sqrt(2) - 2
4420)*log(2^(1/4) + ((x - 1)/(2*x + 1))^(1/4)) - 1/144*sqrt(55819*sqrt(2) - 24420)*log(abs(-2^(1/4) + ((x - 1)
/(2*x + 1))^(1/4))) + 1/2*(3*((x - 1)/(2*x + 1))^(3/4) - ((x - 1)/(2*x + 1))^(1/4))/((x - 1)/(2*x + 1) - 2) -
2/9*arctan(2*((x - 1)/(2*x + 1))^(1/4) + 1) - 2/9*arctan(2*((x - 1)/(2*x + 1))^(1/4) - 1) + 5/9*log(2*sqrt((x
- 1)/(2*x + 1)) + 2*((x - 1)/(2*x + 1))^(1/4) + 1) - 5/9*log(2*sqrt((x - 1)/(2*x + 1)) - 2*((x - 1)/(2*x + 1))
^(1/4) + 1)
________________________________________________________________________________________
maple [C] time = 17.91, size = 2833, normalized size = 12.70
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result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(2833\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((-1+x)/(1+2*x))^(1/4)-3*((-1+x)/(1+2*x))^(3/4))/(-1+x)/(1+x)^2/(-1+2*x),x,method=_RETURNVERBOSE)
[Out]
1/6*(1+2*x)/(1+x)*(-(1-x)/(1+2*x))^(1/4)-1/2*(1+2*x)/(1+x)*(-(1-x)/(1+2*x))^(3/4)-1/72*RootOf(_Z^2+2304*RootOf
(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*ln((6082560*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720
*_Z^2-2817592561)^2+12210)*(-(1-x)/(1+2*x))^(1/2)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2*x+3041280*
RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*(-(1-x)/(1+2*x))^(1/2)*RootOf(424673
28*_Z^4+225054720*_Z^2-2817592561)^2+852480*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2*RootOf(_Z^2+2304
*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*x+7855988*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+2250
54720*_Z^2-2817592561)^2+12210)*(-(1-x)/(1+2*x))^(1/2)*x-978388992*(-(1-x)/(1+2*x))^(1/4)*RootOf(42467328*_Z^4
+225054720*_Z^2-2817592561)^2*x+5925856678*(-(1-x)/(1+2*x))^(3/4)*x+170496*RootOf(42467328*_Z^4+225054720*_Z^2
-2817592561)^2*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)+3927994*RootOf(_Z^2+2
304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*(-(1-x)/(1+2*x))^(1/2)-489194496*(-(1-x)/(1+2*x))
^(1/4)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+2962928339*(-(1-x)/(1+2*x))^(3/4)-43791825*RootOf(_Z^
2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*x-2592476040*(-(1-x)/(1+2*x))^(1/4)*x-8758365*
RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)-1296238020*(-(1-x)/(1+2*x))^(1/4))/(
1+x))+2/3*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*ln((291962880*RootOf(42467328*_Z^4+225054720*_Z^2-28
17592561)^3*(-(1-x)/(1+2*x))^(1/2)*x+145981440*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*(-(1-x)/(1+2*
x))^(1/2)+40919040*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*x+978388992*(-(1-x)/(1+2*x))^(1/4)*RootOf
(42467328*_Z^4+225054720*_Z^2-2817592561)^2*x+1170163776*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*(-(1-
x)/(1+2*x))^(1/2)*x+5925856678*(-(1-x)/(1+2*x))^(3/4)*x+8183808*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561
)^3+489194496*(-(1-x)/(1+2*x))^(1/4)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+585081888*RootOf(424673
28*_Z^4+225054720*_Z^2-2817592561)*(-(1-x)/(1+2*x))^(1/2)+2962928339*(-(1-x)/(1+2*x))^(3/4)+2318857200*RootOf(
42467328*_Z^4+225054720*_Z^2-2817592561)*x+2592476040*(-(1-x)/(1+2*x))^(1/4)*x+463771440*RootOf(42467328*_Z^4+
225054720*_Z^2-2817592561)+1296238020*(-(1-x)/(1+2*x))^(1/4))/(1+x))+5/9*ln((16*(-(1-x)/(1+2*x))^(3/4)*x+8*(-(
1-x)/(1+2*x))^(3/4)+16*(-(1-x)/(1+2*x))^(1/2)*x+8*(-(1-x)/(1+2*x))^(1/2)+8*(-(1-x)/(1+2*x))^(1/4)*x+4*(-(1-x)/
(1+2*x))^(1/4)+6*x-3)/(-1+2*x))+98304/2962928339*ln((16*(-(1-x)/(1+2*x))^(3/4)*x+8*(-(1-x)/(1+2*x))^(3/4)+16*(
-(1-x)/(1+2*x))^(1/2)*x+8*(-(1-x)/(1+2*x))^(1/2)+8*(-(1-x)/(1+2*x))^(1/4)*x+4*(-(1-x)/(1+2*x))^(1/4)+6*x-3)/(-
1+2*x))*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2
-2817592561)^2+12210)+260480/2962928339*ln((16*(-(1-x)/(1+2*x))^(3/4)*x+8*(-(1-x)/(1+2*x))^(3/4)+16*(-(1-x)/(1
+2*x))^(1/2)*x+8*(-(1-x)/(1+2*x))^(1/2)+8*(-(1-x)/(1+2*x))^(1/4)*x+4*(-(1-x)/(1+2*x))^(1/4)+6*x-3)/(-1+2*x))*R
ootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561
)^2+12210)-196608/2962928339*ln((7077888*(-(1-x)/(1+2*x))^(1/2)*x*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054
720*_Z^2-2817592561)^2+12210)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3+3538944*(-(1-x)/(1+2*x))^(1/2)
*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-281759
2561)^2+12210)+7077888*(-(1-x)/(1+2*x))^(1/4)*x*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-281759256
1)^2+12210)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3+3538944*(-(1-x)/(1+2*x))^(1/4)*RootOf(42467328*_
Z^4+225054720*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)-176
9472*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-28
17592561)^2+12210)*x+18754560*(-(1-x)/(1+2*x))^(1/2)*x*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-28
17592561)^2+12210)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)+4423680*RootOf(42467328*_Z^4+225054720*_Z^2
-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)+9377280*(-(1-x)/(1+2*
x))^(1/2)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2
-2817592561)^2+12210)+18754560*(-(1-x)/(1+2*x))^(1/4)*x*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2
817592561)^2+12210)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)+47406853424*(-(1-x)/(1+2*x))^(3/4)*x+93772
80*(-(1-x)/(1+2*x))^(1/4)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304*RootOf(42467328*_Z^
4+225054720*_Z^2-2817592561)^2+12210)-4688640*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304
*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*x+23703426712*(-(1-x)/(1+2*x))^(3/4)+23703426712*(-(
1-x)/(1+2*x))^(1/2)*x+11721600*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304*RootOf(4246732
8*_Z^4+225054720*_Z^2-2817592561)^2+12210)+11851713356*(-(1-x)/(1+2*x))^(1/2)+5925856678*x-14814641695)/(-1+2*
x))*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-281
7592561)^2+12210)-520960/2962928339*ln((7077888*(-(1-x)/(1+2*x))^(1/2)*x*RootOf(_Z^2+2304*RootOf(42467328*_Z^4
+225054720*_Z^2-2817592561)^2+12210)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3+3538944*(-(1-x)/(1+2*x)
)^(1/2)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2
-2817592561)^2+12210)+7077888*(-(1-x)/(1+2*x))^(1/4)*x*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-28
17592561)^2+12210)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3+3538944*(-(1-x)/(1+2*x))^(1/4)*RootOf(424
67328*_Z^4+225054720*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+122
10)-1769472*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*
_Z^2-2817592561)^2+12210)*x+18754560*(-(1-x)/(1+2*x))^(1/2)*x*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*
_Z^2-2817592561)^2+12210)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)+4423680*RootOf(42467328*_Z^4+2250547
20*_Z^2-2817592561)^3*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)+9377280*(-(1-x
)/(1+2*x))^(1/2)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+2250547
20*_Z^2-2817592561)^2+12210)+18754560*(-(1-x)/(1+2*x))^(1/4)*x*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720
*_Z^2-2817592561)^2+12210)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)+47406853424*(-(1-x)/(1+2*x))^(3/4)*
x+9377280*(-(1-x)/(1+2*x))^(1/4)*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304*RootOf(42467
328*_Z^4+225054720*_Z^2-2817592561)^2+12210)-4688640*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z
^2+2304*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)*x+23703426712*(-(1-x)/(1+2*x))^(3/4)+23703426
712*(-(1-x)/(1+2*x))^(1/2)*x+11721600*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304*RootOf(
42467328*_Z^4+225054720*_Z^2-2817592561)^2+12210)+11851713356*(-(1-x)/(1+2*x))^(1/2)+5925856678*x-14814641695)
/(-1+2*x))*RootOf(42467328*_Z^4+225054720*_Z^2-2817592561)*RootOf(_Z^2+2304*RootOf(42467328*_Z^4+225054720*_Z^
2-2817592561)^2+12210)
________________________________________________________________________________________
maxima [A] time = 0.42, size = 243, normalized size = 1.09 \begin {gather*} -\frac {1}{72} \cdot 2^{\frac {1}{4}} {\left (165 \, \sqrt {2} + 37\right )} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}\right ) - \frac {1}{144} \cdot 2^{\frac {1}{4}} {\left (165 \, \sqrt {2} - 37\right )} \log \left (-\frac {2^{\frac {1}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} + \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}\right ) + \frac {3 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {3}{4}} - \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}}}{2 \, {\left (\frac {x - 1}{2 \, x + 1} - 2\right )}} - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {2}{9} \, \arctan \left (2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} - 1\right ) + \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} + 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) - \frac {5}{9} \, \log \left (2 \, \sqrt {\frac {x - 1}{2 \, x + 1}} - 2 \, \left (\frac {x - 1}{2 \, x + 1}\right )^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-1+x)/(1+2*x))^(1/4)-3*((-1+x)/(1+2*x))^(3/4))/(-1+x)/(1+x)^2/(-1+2*x),x, algorithm="maxima")
[Out]
-1/72*2^(1/4)*(165*sqrt(2) + 37)*arctan(1/2*2^(3/4)*((x - 1)/(2*x + 1))^(1/4)) - 1/144*2^(1/4)*(165*sqrt(2) -
37)*log(-(2^(1/4) - ((x - 1)/(2*x + 1))^(1/4))/(2^(1/4) + ((x - 1)/(2*x + 1))^(1/4))) + 1/2*(3*((x - 1)/(2*x +
1))^(3/4) - ((x - 1)/(2*x + 1))^(1/4))/((x - 1)/(2*x + 1) - 2) - 2/9*arctan(2*((x - 1)/(2*x + 1))^(1/4) + 1)
- 2/9*arctan(2*((x - 1)/(2*x + 1))^(1/4) - 1) + 5/9*log(2*sqrt((x - 1)/(2*x + 1)) + 2*((x - 1)/(2*x + 1))^(1/4
) + 1) - 5/9*log(2*sqrt((x - 1)/(2*x + 1)) - 2*((x - 1)/(2*x + 1))^(1/4) + 1)
________________________________________________________________________________________
mupad [B] time = 2.27, size = 1310, normalized size = 5.87
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((x - 1)/(2*x + 1))^(1/4) - 3*((x - 1)/(2*x + 1))^(3/4))/((2*x - 1)*(x - 1)*(x + 1)^2),x)
[Out]
(8/27 + 10i/81)^(1/2)*atan((8/27 + 10i/81)^(1/2)*((x - 1)/(2*x + 1))^(1/4)*(18/13 - 27i/13))*2i - (8/27 - 10i/
81)^(1/2)*atan((8/27 - 10i/81)^(1/2)*((x - 1)/(2*x + 1))^(1/4)*(18/13 + 27i/13))*2i - (((x - 1)/(2*x + 1))^(1/
4) - 3*((x - 1)/(2*x + 1))^(3/4))/((2*x - 2)/(2*x + 1) - 4) - (2^(3/4)*atan(((2^(3/4)*((13937028229*2^(3/4)*((
x - 1)/(2*x + 1))^(1/4))/4 - (2^(3/4)*(37*2^(1/2) - 330)*((25646402817*2^(3/4))/4 + (2^(3/4)*(37*2^(1/2) - 330
)*(415942128*2^(3/4)*((x - 1)/(2*x + 1))^(1/4) - (2^(3/4)*(37*2^(1/2) - 330)*((700891947*2^(3/4))/8 + (2^(3/4)
*(37*2^(1/2) - 330)*(3601989*2^(1/2)*(37*2^(1/2) - 330) - 634967019*2^(3/4)*((x - 1)/(2*x + 1))^(1/4)))/288))/
288))/288))/288)*(37*2^(1/2) - 330)*1i)/288 + (2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 + (2
^(3/4)*(37*2^(1/2) - 330)*((25646402817*2^(3/4))/4 - (2^(3/4)*(37*2^(1/2) - 330)*(415942128*2^(3/4)*((x - 1)/(
2*x + 1))^(1/4) + (2^(3/4)*(37*2^(1/2) - 330)*((700891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) - 330)*(3601989*2^
(1/2)*(37*2^(1/2) - 330) + 634967019*2^(3/4)*((x - 1)/(2*x + 1))^(1/4)))/288))/288))/288))/288)*(37*2^(1/2) -
330)*1i)/288)/((11880642501*2^(3/4))/2 + (2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 - (2^(3/4
)*(37*2^(1/2) - 330)*((25646402817*2^(3/4))/4 + (2^(3/4)*(37*2^(1/2) - 330)*(415942128*2^(3/4)*((x - 1)/(2*x +
1))^(1/4) - (2^(3/4)*(37*2^(1/2) - 330)*((700891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) - 330)*(3601989*2^(1/2)
*(37*2^(1/2) - 330) - 634967019*2^(3/4)*((x - 1)/(2*x + 1))^(1/4)))/288))/288))/288))/288)*(37*2^(1/2) - 330))
/288 - (2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 + (2^(3/4)*(37*2^(1/2) - 330)*((25646402817
*2^(3/4))/4 - (2^(3/4)*(37*2^(1/2) - 330)*(415942128*2^(3/4)*((x - 1)/(2*x + 1))^(1/4) + (2^(3/4)*(37*2^(1/2)
- 330)*((700891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) - 330)*(3601989*2^(1/2)*(37*2^(1/2) - 330) + 634967019*2^
(3/4)*((x - 1)/(2*x + 1))^(1/4)))/288))/288))/288))/288)*(37*2^(1/2) - 330))/288))*(37*2^(1/2) - 330)*1i)/144
- (2^(3/4)*atan(((2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 - (2^(3/4)*(37*2^(1/2) + 330)*((2
5646402817*2^(3/4))/4 + (2^(3/4)*(37*2^(1/2) + 330)*(415942128*2^(3/4)*((x - 1)/(2*x + 1))^(1/4) - (2^(3/4)*(3
7*2^(1/2) + 330)*((700891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) + 330)*(2^(1/2)*(37*2^(1/2) + 330)*3601989i - 6
34967019*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))*1i)/288)*1i)/288)*1i)/288)*1i)/288)*(37*2^(1/2) + 330))/288 + (2^(
3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 + (2^(3/4)*(37*2^(1/2) + 330)*((25646402817*2^(3/4))/4
- (2^(3/4)*(37*2^(1/2) + 330)*(415942128*2^(3/4)*((x - 1)/(2*x + 1))^(1/4) + (2^(3/4)*(37*2^(1/2) + 330)*((70
0891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) + 330)*(2^(1/2)*(37*2^(1/2) + 330)*3601989i + 634967019*2^(3/4)*((x
- 1)/(2*x + 1))^(1/4))*1i)/288)*1i)/288)*1i)/288)*1i)/288)*(37*2^(1/2) + 330))/288)/((11880642501*2^(3/4))/2 +
(2^(3/4)*((13937028229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 - (2^(3/4)*(37*2^(1/2) + 330)*((25646402817*2^(3/
4))/4 + (2^(3/4)*(37*2^(1/2) + 330)*(415942128*2^(3/4)*((x - 1)/(2*x + 1))^(1/4) - (2^(3/4)*(37*2^(1/2) + 330)
*((700891947*2^(3/4))/8 + (2^(3/4)*(37*2^(1/2) + 330)*(2^(1/2)*(37*2^(1/2) + 330)*3601989i - 634967019*2^(3/4)
*((x - 1)/(2*x + 1))^(1/4))*1i)/288)*1i)/288)*1i)/288)*1i)/288)*(37*2^(1/2) + 330)*1i)/288 - (2^(3/4)*((139370
28229*2^(3/4)*((x - 1)/(2*x + 1))^(1/4))/4 + (2^(3/4)*(37*2^(1/2) + 330)*((25646402817*2^(3/4))/4 - (2^(3/4)*(
37*2^(1/2) + 330)*(415942128*2^(3/4)*((x - 1)/(2*x + 1))^(1/4) + (2^(3/4)*(37*2^(1/2) + 330)*((700891947*2^(3/
4))/8 + (2^(3/4)*(37*2^(1/2) + 330)*(2^(1/2)*(37*2^(1/2) + 330)*3601989i + 634967019*2^(3/4)*((x - 1)/(2*x + 1
))^(1/4))*1i)/288)*1i)/288)*1i)/288)*1i)/288)*(37*2^(1/2) + 330)*1i)/288))*(37*2^(1/2) + 330))/144
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-1+x)/(1+2*x))**(1/4)-3*((-1+x)/(1+2*x))**(3/4))/(-1+x)/(1+x)**2/(-1+2*x),x)
[Out]
Timed out
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