3.22.47 \(\int \frac {-1+x^2}{(1+x^2) \sqrt [3]{-x^2+x^4}} \, dx\)
Optimal. Leaf size=157 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^4-x^2}}\right )}{\sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} x \sqrt [3]{x^4-x^2}}{\sqrt [3]{2} \left (x^4-x^2\right )^{2/3}-2 x^2}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tanh ^{-1}\left (\frac {\frac {\sqrt [3]{2} x^2}{\sqrt {3}}+\frac {\left (x^4-x^2\right )^{2/3}}{\sqrt [3]{2} \sqrt {3}}}{x \sqrt [3]{x^4-x^2}}\right )}{2 \sqrt [3]{2}} \]
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Rubi [C] time = 0.11, antiderivative size = 46, normalized size of antiderivative = 0.29,
number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used =
{2056, 466, 430, 429} \begin {gather*} -\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};-\frac {2}{3},1;\frac {7}{6};x^2,-x^2\right )}{\sqrt [3]{x^4-x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Int[(-1 + x^2)/((1 + x^2)*(-x^2 + x^4)^(1/3)),x]
[Out]
(-3*x*(1 - x^2)^(1/3)*AppellF1[1/6, -2/3, 1, 7/6, x^2, -x^2])/(-x^2 + x^4)^(1/3)
Rule 429
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Rule 430
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
Rule 466
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]
Rule 2056
Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] && !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] && !PolyQ[P, x, 2]
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{-x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^2}\right ) \int \frac {\left (-1+x^2\right )^{2/3}}{x^{2/3} \left (1+x^2\right )} \, dx}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^6\right )^{2/3}}{1+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \left (-1+x^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^6\right )^{2/3}}{1+x^6} \, dx,x,\sqrt [3]{x}\right )}{\left (1-x^2\right )^{2/3} \sqrt [3]{-x^2+x^4}}\\ &=-\frac {3 x \sqrt [3]{1-x^2} F_1\left (\frac {1}{6};-\frac {2}{3},1;\frac {7}{6};x^2,-x^2\right )}{\sqrt [3]{-x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 48, normalized size = 0.31 \begin {gather*} \frac {3 \left (x^2 \left (x^2-1\right )\right )^{2/3} F_1\left (\frac {1}{6};-\frac {2}{3},1;\frac {7}{6};x^2,-x^2\right )}{x \left (1-x^2\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(-1 + x^2)/((1 + x^2)*(-x^2 + x^4)^(1/3)),x]
[Out]
(3*(x^2*(-1 + x^2))^(2/3)*AppellF1[1/6, -2/3, 1, 7/6, x^2, -x^2])/(x*(1 - x^2)^(2/3))
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IntegrateAlgebraic [A] time = 0.45, size = 157, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{-x^2+x^4}}\right )}{\sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} x \sqrt [3]{-x^2+x^4}}{-2 x^2+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tanh ^{-1}\left (\frac {\frac {\sqrt [3]{2} x^2}{\sqrt {3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{\sqrt [3]{2} \sqrt {3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*(-x^2 + x^4)^(1/3)),x]
[Out]
-(ArcTan[(2^(1/3)*x)/(-x^2 + x^4)^(1/3)]/2^(1/3)) - ArcTan[(2^(2/3)*x*(-x^2 + x^4)^(1/3))/(-2*x^2 + 2^(1/3)*(-
x^2 + x^4)^(2/3))]/(2*2^(1/3)) - (Sqrt[3]*ArcTanh[((2^(1/3)*x^2)/Sqrt[3] + (-x^2 + x^4)^(2/3)/(2^(1/3)*Sqrt[3]
))/(x*(-x^2 + x^4)^(1/3))])/(2*2^(1/3))
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fricas [B] time = 2.89, size = 2094, normalized size = 13.34
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-1)/(x^2+1)/(x^4-x^2)^(1/3),x, algorithm="fricas")
[Out]
-1/32*sqrt(3)*2^(2/3)*log(8500000*(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*2^(2/3)*(x^2 -
1) + 6*2^(2/3)*x) + 2^(1/3)*(x^5 + 2*x^3 + x) + 4*(x^4 - x^2)^(1/3)*(3*x^3 + 2*sqrt(3)*x^2 - 3*x))/(x^5 + 2*x
^3 + x)) - 1/32*sqrt(3)*2^(2/3)*log(2125000*(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*2^(2
/3)*(x^2 - 1) + 6*2^(2/3)*x) + 2^(1/3)*(x^5 + 2*x^3 + x) + 4*(x^4 - x^2)^(1/3)*(3*x^3 + 2*sqrt(3)*x^2 - 3*x))/
(x^5 + 2*x^3 + x)) + 1/32*sqrt(3)*2^(2/3)*log(-2125000*(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(s
qrt(3)*2^(2/3)*(x^2 - 1) - 6*2^(2/3)*x) - 2^(1/3)*(x^5 + 2*x^3 + x) - 4*(x^4 - x^2)^(1/3)*(3*x^3 - 2*sqrt(3)*x
^2 - 3*x))/(x^5 + 2*x^3 + x)) + 1/32*sqrt(3)*2^(2/3)*log(-8500000*(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^
2)^(2/3)*(sqrt(3)*2^(2/3)*(x^2 - 1) - 6*2^(2/3)*x) - 2^(1/3)*(x^5 + 2*x^3 + x) - 4*(x^4 - x^2)^(1/3)*(3*x^3 -
2*sqrt(3)*x^2 - 3*x))/(x^5 + 2*x^3 + x)) + 1/4*2^(2/3)*arctan(-(74071498415429632*x^9 + 1645279755446275808*x^
8 - 2346817955632029696*x^7 - 11516958288123930656*x^6 + 5730636889080074240*x^5 + 11516958288123930656*x^4 -
2346817955632029696*x^3 - 1645279755446275808*x^2 - 125*sqrt(34)*(4*sqrt(3)*2^(1/3)*(78465570355328*x^9 - 3301
419835659*x^8 + 1100839094578688*x^7 - 595767752585659*x^6 - 3614058455553280*x^5 + 595767752585659*x^4 + 1100
839094578688*x^3 + 3301419835659*x^2 + 78465570355328*x) + 16*(x^4 - x^2)^(2/3)*(4*sqrt(3)*2^(2/3)*(1513688563
712*x^6 + 57183135266496*x^5 - 26977277846305*x^4 - 167158338888320*x^3 + 26977277846305*x^2 + 57183135266496*
x - 1513688563712) - 2^(2/3)*(79163286177664*x^6 - 56815411732213*x^5 - 187311276664960*x^4 + 112551186315710*
x^3 + 187311276664960*x^2 - 56815411732213*x - 79163286177664)) - 2^(1/3)*(36167723835659*x^9 + 47385984376852
48*x^8 - 1343569332842636*x^7 - 16069401562314752*x^6 + 2036119636643410*x^5 + 16069401562314752*x^4 - 1343569
332842636*x^3 - 4738598437685248*x^2 + 36167723835659*x) - 4*(183204669874443*x^7 + 4116235393055744*x^6 - 222
5700627116645*x^5 - 10698715224852480*x^4 + 2225700627116645*x^3 + 4116235393055744*x^2 - 531250*sqrt(3)*(1009
306368*x^7 - 511421263*x^6 - 4316628224*x^5 + 1207618962*x^4 + 4316628224*x^3 - 511421263*x^2 - 1009306368*x)
- 183204669874443*x)*(x^4 - x^2)^(1/3))*sqrt((8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*2^(
2/3)*(x^2 - 1) + 6*2^(2/3)*x) + 2^(1/3)*(x^5 + 2*x^3 + x) + 4*(x^4 - x^2)^(1/3)*(3*x^3 + 2*sqrt(3)*x^2 - 3*x))
/(x^5 + 2*x^3 + x)) - 1062500*(x^4 - x^2)^(2/3)*(2*sqrt(3)*2^(1/3)*(23651383808*x^6 + 470146644789*x^5 - 22638
6757120*x^4 - 71809982630*x^3 + 226386757120*x^2 + 470146644789*x - 23651383808) - 2^(1/3)*(618463173263*x^6 -
733160605696*x^5 - 6989546598945*x^4 + 2615047352320*x^3 + 6989546598945*x^2 - 733160605696*x - 618463173263)
) - 265625*sqrt(3)*(613012268401*x^9 - 500076281856*x^8 - 1596364015228*x^7 + 3500533972992*x^6 + 117748997880
70*x^5 - 3500533972992*x^4 - 1596364015228*x^3 + 500076281856*x^2 + 613012268401*x) - 1062500*(x^4 - x^2)^(1/3
)*(sqrt(3)*2^(2/3)*(217120826737*x^7 + 155432605696*x^6 + 1229224098945*x^5 - 689287352320*x^4 - 1229224098945
*x^3 + 155432605696*x^2 - 217120826737*x) - 2*2^(2/3)*(71795383808*x^7 + 1283539269789*x^6 - 948546757120*x^5
- 5040931232630*x^4 + 948546757120*x^3 + 1283539269789*x^2 - 71795383808*x)) + 74071498415429632*x)/(479958568
556831351*x^9 - 1202832749691437056*x^8 - 12744795130528777828*x^7 + 8419829247840059392*x^6 + 322090102208531
94570*x^5 - 8419829247840059392*x^4 - 12744795130528777828*x^3 + 1202832749691437056*x^2 + 479958568556831351*
x)) - 1/4*2^(2/3)*arctan((74071498415429632*x^9 + 1645279755446275808*x^8 - 2346817955632029696*x^7 - 11516958
288123930656*x^6 + 5730636889080074240*x^5 + 11516958288123930656*x^4 - 2346817955632029696*x^3 - 164527975544
6275808*x^2 + 125*sqrt(34)*(4*sqrt(3)*2^(1/3)*(78465570355328*x^9 - 3301419835659*x^8 + 1100839094578688*x^7 -
595767752585659*x^6 - 3614058455553280*x^5 + 595767752585659*x^4 + 1100839094578688*x^3 + 3301419835659*x^2 +
78465570355328*x) + 16*(x^4 - x^2)^(2/3)*(4*sqrt(3)*2^(2/3)*(1513688563712*x^6 + 57183135266496*x^5 - 2697727
7846305*x^4 - 167158338888320*x^3 + 26977277846305*x^2 + 57183135266496*x - 1513688563712) + 2^(2/3)*(79163286
177664*x^6 - 56815411732213*x^5 - 187311276664960*x^4 + 112551186315710*x^3 + 187311276664960*x^2 - 5681541173
2213*x - 79163286177664)) + 2^(1/3)*(36167723835659*x^9 + 4738598437685248*x^8 - 1343569332842636*x^7 - 160694
01562314752*x^6 + 2036119636643410*x^5 + 16069401562314752*x^4 - 1343569332842636*x^3 - 4738598437685248*x^2 +
36167723835659*x) + 4*(183204669874443*x^7 + 4116235393055744*x^6 - 2225700627116645*x^5 - 10698715224852480*
x^4 + 2225700627116645*x^3 + 4116235393055744*x^2 + 531250*sqrt(3)*(1009306368*x^7 - 511421263*x^6 - 431662822
4*x^5 + 1207618962*x^4 + 4316628224*x^3 - 511421263*x^2 - 1009306368*x) - 183204669874443*x)*(x^4 - x^2)^(1/3)
)*sqrt(-(8*sqrt(3)*2^(1/3)*(x^4 - x^2) + 2*(x^4 - x^2)^(2/3)*(sqrt(3)*2^(2/3)*(x^2 - 1) - 6*2^(2/3)*x) - 2^(1/
3)*(x^5 + 2*x^3 + x) - 4*(x^4 - x^2)^(1/3)*(3*x^3 - 2*sqrt(3)*x^2 - 3*x))/(x^5 + 2*x^3 + x)) + 1062500*(x^4 -
x^2)^(2/3)*(2*sqrt(3)*2^(1/3)*(23651383808*x^6 + 470146644789*x^5 - 226386757120*x^4 - 71809982630*x^3 + 22638
6757120*x^2 + 470146644789*x - 23651383808) + 2^(1/3)*(618463173263*x^6 - 733160605696*x^5 - 6989546598945*x^4
+ 2615047352320*x^3 + 6989546598945*x^2 - 733160605696*x - 618463173263)) + 265625*sqrt(3)*(613012268401*x^9
- 500076281856*x^8 - 1596364015228*x^7 + 3500533972992*x^6 + 11774899788070*x^5 - 3500533972992*x^4 - 15963640
15228*x^3 + 500076281856*x^2 + 613012268401*x) + 1062500*(x^4 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*(217120826737*x^7
+ 155432605696*x^6 + 1229224098945*x^5 - 689287352320*x^4 - 1229224098945*x^3 + 155432605696*x^2 - 21712082673
7*x) + 2*2^(2/3)*(71795383808*x^7 + 1283539269789*x^6 - 948546757120*x^5 - 5040931232630*x^4 + 948546757120*x^
3 + 1283539269789*x^2 - 71795383808*x)) + 74071498415429632*x)/(479958568556831351*x^9 - 1202832749691437056*x
^8 - 12744795130528777828*x^7 + 8419829247840059392*x^6 + 32209010220853194570*x^5 - 8419829247840059392*x^4 -
12744795130528777828*x^3 + 1202832749691437056*x^2 + 479958568556831351*x)) + 1/2*2^(2/3)*arctan(-1/2*(356454
4*x^5 + 249106968*x^4 - 21387264*x^3 + 2125000*2^(2/3)*(x^4 - x^2)^(1/3)*(512*x^3 + 59*x^2 - 512*x) + 1062500*
2^(1/3)*(x^4 - x^2)^(2/3)*(59*x^2 - 2048*x - 59) - 249106968*x^2 - 125*sqrt(34)*2^(1/6)*(4*2^(2/3)*(x^4 - x^2)
^(2/3)*(15104*x^2 + 527769*x - 15104) + 3481*2^(1/3)*(x^5 + 2*x^3 + x) + 4*(x^4 - x^2)^(1/3)*(527769*x^3 - 604
16*x^2 - 527769*x)) + 3564544*x)/(205379*x^5 - 2168870912*x^4 - 1232274*x^3 + 2168870912*x^2 + 205379*x))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-1)/(x^2+1)/(x^4-x^2)^(1/3),x, algorithm="giac")
[Out]
integrate((x^2 - 1)/((x^4 - x^2)^(1/3)*(x^2 + 1)), x)
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{2}-1}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2-1)/(x^2+1)/(x^4-x^2)^(1/3),x)
[Out]
int((x^2-1)/(x^2+1)/(x^4-x^2)^(1/3),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2-1)/(x^2+1)/(x^4-x^2)^(1/3),x, algorithm="maxima")
[Out]
integrate((x^2 - 1)/((x^4 - x^2)^(1/3)*(x^2 + 1)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2-1}{\left (x^2+1\right )\,{\left (x^4-x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2 - 1)/((x^2 + 1)*(x^4 - x^2)^(1/3)),x)
[Out]
int((x^2 - 1)/((x^2 + 1)*(x^4 - x^2)^(1/3)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2-1)/(x**2+1)/(x**4-x**2)**(1/3),x)
[Out]
Integral((x - 1)*(x + 1)/((x**2*(x - 1)*(x + 1))**(1/3)*(x**2 + 1)), x)
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