∫ 1______ (1+x) 3 √ ___

3.1.93 \(\int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx\) [93]

Optimal. Leaf size=108 \[ \frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )-\frac {1}{2} \log (1+x)+\frac {3}{4} \log \left (2+x-\sqrt [3]{2+x^3}\right )-\frac {1}{4} \log \left (-x+\sqrt [3]{2+x^3}\right ) \]

[Out]

-1/2*ln(1+x)+3/4*ln(2+x-(x^3+2)^(1/3))-1/4*ln(-x+(x^3+2)^(1/3))+1/6*arctan(1/3*(1+2*x/(x^3+2)^(1/3))*3^(1/2))*

3^(1/2)-1/2*arctan(1/3*(1+2*(2+x)/(x^3+2)^(1/3))*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2175, 245, 2176} \begin {gather*} \frac {3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac {1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log (x+1) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[1/((1 + x)*(2 + x^3)^(1/3)),x]

[Out]

ArcTan[(1 + (2*x)/(2 + x^3)^(1/3))/Sqrt[3]]/(2*Sqrt[3]) - (Sqrt[3]*ArcTan[(1 + (2*(2 + x))/(2 + x^3)^(1/3))/Sq

rt[3]])/2 - Log[1 + x]/2 + (3*Log[2 + x - (2 + x^3)^(1/3)])/4 - Log[-x + (2 + x^3)^(1/3)]/4

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq

rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 2175

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Dist[1/(2*c), Int[1/(a + b*x^3)^(1/3), x

], x] + Dist[1/(2*c), Int[(c - d*x)/((c + d*x)*(a + b*x^3)^(1/3)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[2*b

*c^3 - a*d^3, 0]

Rule 2176

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[Sqrt[3]*f*(ArcTan

[(1 + 2*Rt[b, 3]*((2*c + d*x)/(d*(a + b*x^3)^(1/3))))/Sqrt[3]]/(Rt[b, 3]*d)), x] + (Simp[(f*Log[c + d*x])/(Rt[

b, 3]*d), x] - Simp[(3*f*Log[Rt[b, 3]*(2*c + d*x) - d*(a + b*x^3)^(1/3)])/(2*Rt[b, 3]*d), x]) /; FreeQ[{a, b,

c, d, e, f}, x] && EqQ[d*e + c*f, 0] && EqQ[2*b*c^3 - a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx &=\frac {1}{2} \int \frac {1}{\sqrt [3]{2+x^3}} \, dx+\frac {1}{2} \int \frac {1-x}{(1+x) \sqrt [3]{2+x^3}} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )-\frac {1}{2} \log (1+x)+\frac {3}{4} \log \left (2+x-\sqrt [3]{2+x^3}\right )-\frac {1}{4} \log \left (-x+\sqrt [3]{2+x^3}\right )\\ \end {align*}

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Mathematica [F]
time = 6.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Integrate[1/((1 + x)*(2 + x^3)^(1/3)),x]

[Out]

Integrate[1/((1 + x)*(2 + x^3)^(1/3)), x]

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((x + 1)*(x^3 + 2)^(1/3)),x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.14, size = 1421, normalized size = 13.16


method	result	size

trager	\(\text {Expression too large to display}\)	\(1421\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x)/(x^3+2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/6*ln((-4550781346817636-6728375859478224*x+4993190285176576*RootOf(_Z^2+_Z+1)^2*x^3+9094739448000192*RootOf(

_Z^2+_Z+1)^2*x^2+5884831407529536*RootOf(_Z^2+_Z+1)^2*x-625895428788672*x^5-68457312523761*x^6-234710785795752

*x^4-469421571591504*x^2+2151515536461060*x^3+8816461926585488*RootOf(_Z^2+_Z+1)*x^3+1055101552116528*RootOf(_

Z^2+_Z+1)*x^2-21283128527537520*RootOf(_Z^2+_Z+1)*x+16295099853018372*x*(x^3+2)^(2/3)-15559137585059152*RootOf

(_Z^2+_Z+1)+4346750471470680*RootOf(_Z^2+_Z+1)^2*x^5+153868976350327*RootOf(_Z^2+_Z+1)*x^6+4547369724000096*Ro

otOf(_Z^2+_Z+1)^2*x^4-868588603920114*RootOf(_Z^2+_Z+1)*x^5+527550776058264*RootOf(_Z^2+_Z+1)*x^4-912549035791

2936*(x^3+2)^(1/3)-928201890361806*(x^3+2)^(2/3)*x^4-540325086981687*(x^3+2)^(1/3)*x^5-1959537324097146*(x^3+2

)^(2/3)*x^3+36303984101745*(x^3+2)^(2/3)*RootOf(_Z^2+_Z+1)^2*x^4+288449229728205*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+

1)^2*x^5-1306943427662820*(x^3+2)^(2/3)*RootOf(_Z^2+_Z+1)^2*x^3-601904942144643*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3

)*x^4+1286927332633530*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x^4-580773949503594*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)*x

^5-3775614346581480*(x^3+2)^(2/3)*RootOf(_Z^2+_Z+1)^2*x^2-3235955176589922*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^3

+1420057746354240*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x^3-3304587786698814*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)*x^4-2

613886855325640*(x^3+2)^(2/3)*RootOf(_Z^2+_Z+1)^2*x-1442113972435308*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^2-31063

7632014990*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x^2-4286079775383252*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)*x^3+77592514

14704196*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x-798782482324260*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x+257120106922963

2*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+7575270505902288*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)*x-480288966205944*(x^3+

2)^(1/3)*x^4+5569211342170836*(x^3+2)^(2/3)*x^2+1200722415514860*(x^3+2)^(1/3)*x^3+7115580883942020*RootOf(_Z^

2+_Z+1)*(x^3+2)^(2/3)+3372637147591320*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)+10107087250606332*(x^3+2)^(2/3)+1326316

169500028*RootOf(_Z^2+_Z+1)^2*x^6-4682817420507954*(x^3+2)^(1/3)*x^2-14648813469281292*(x^3+2)^(1/3)*x)/(1+x)^

6)+1/6*RootOf(_Z^2+_Z+1)*ln((-15559137585059152-28889172364235904*x-1460422667173568*RootOf(_Z^2+_Z+1)^2*x^3-2

660055572351856*RootOf(_Z^2+_Z+1)^2*x^2-1721212429168848*RootOf(_Z^2+_Z+1)^2*x-6486689165117784*x^5-1560371964

117680*x^6-5349846734117760*x^4-10699693468235520*x^2+2362848974235344*x^3-4302097415889084*RootOf(_Z^2+_Z+1)*

x^3-12224216591943552*RootOf(_Z^2+_Z+1)*x^2-14334419696176608*RootOf(_Z^2+_Z+1)*x+5921961582988536*x*(x^3+2)^(

2/3)-4550781346817636*RootOf(_Z^2+_Z+1)-1271350089726990*RootOf(_Z^2+_Z+1)^2*x^5-1782698252991768*RootOf(_Z^2+

_Z+1)*x^6-1330027786175928*RootOf(_Z^2+_Z+1)^2*x^4-6243995989986342*RootOf(_Z^2+_Z+1)*x^5-6112108295971776*Roo

tOf(_Z^2+_Z+1)*x^4+9125490357912936*(x^3+2)^(1/3)-289992964115418*(x^3+2)^(2/3)*x^4-240144483102972*(x^3+2)^(1

/3)*x^5-30525575170044*(x^3+2)^(2/3)*x^3+36303984101745*(x^3+2)^(2/3)*RootOf(_Z^2+_Z+1)^2*x^4-1068918799812864

*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x^5-1306943427662820*(x^3+2)^(2/3)*RootOf(_Z^2+_Z+1)^2*x^3+674512910348133*

RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^4-4769022337626624*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x^4-1109367662334771*(x

^3+2)^(1/3)*RootOf(_Z^2+_Z+1)*x^5-3775614346581480*(x^3+2)^(2/3)*RootOf(_Z^2+_Z+1)^2*x^2+622068321264282*RootO

f(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^3-5262369476001792*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x^3-7593321158119494*(x^3+2)

^(1/3)*RootOf(_Z^2+_Z+1)*x^4-2613886855325640*(x^3+2)^(2/3)*RootOf(_Z^2+_Z+1)^2*x-6109114720727652*RootOf(_Z^2

+_Z+1)*(x^3+2)^(2/3)*x^2+1151143322875392*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x^2-10749171666899904*(x^3+2)^(1/3

)*RootOf(_Z^2+_Z+1)*x^3-12987025125355476*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x+2960082830251008*(x^3+2)^(1/3)*Roo

tOf(_Z^2+_Z+1)^2*x+8405161812612978*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+25184166805434588*(x^3+2)^(1/3)*RootOf

(_Z^2+_Z+1)*x-3001806038787150*(x^3+2)^(1/3)*x^4+3235710968024664*(x^3+2)^(2/3)*x^2-5043034145162412*(x^3+2)^(

1/3)*x^3-7115580883942020*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)+12498127505504256*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)+29

91506366664312*(x^3+2)^(2/3)-387924770967979*RootOf(_Z^2+_Z+1)^2*x^6+5523323111368356*(x^3+2)^(1/3)*x^2+168101

13817208040*(x^3+2)^(1/3)*x)/(1+x)^6)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)/(x^3+2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 2)^(1/3)*(x + 1)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (84) = 168\).
time = 1.04, size = 267, normalized size = 2.47 \begin {gather*} \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {13910019318573948542 \, \sqrt {3} {\left (7114781247 \, x^{4} + 13663058416 \, x^{3} - 46178206896 \, x^{2} - 126842559344 \, x - 77084338088\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} - 27820038637147897084 \, \sqrt {3} {\left (1625757424 \, x^{5} + 16302821713 \, x^{4} + 26102613730 \, x^{3} - 26431113242 \, x^{2} - 80188343316 \, x - 42779182428\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}} + \sqrt {3} {\left (93292570833559435663132301885 \, x^{6} + 382151535711085278859235047618 \, x^{5} + 673924074224408772959625384792 \, x^{4} + 889426563183087468015580290048 \, x^{3} + 888876515195959220955879945824 \, x^{2} + 351260598258508240019971964880 \, x - 47674000995597211057816884304\right )}}{3 \, {\left (78905434814564721745708464883 \, x^{6} + 337746705836458222863347934450 \, x^{5} + 15598952776058587894336070976 \, x^{4} - 895430525315100108684787964824 \, x^{3} + 361667862240477028869533375352 \, x^{2} + 2541802301011632510645972090336 \, x + 1554815286823334092314485968880\right )}}\right ) + \frac {1}{12} \, \log \left (\frac {22 \, x^{6} + 6 \, x^{5} - 48 \, x^{4} + 44 \, x^{3} + 24 \, x^{2} + 3 \, {\left (7 \, x^{4} - 2 \, x^{3} - 32 \, x^{2} - 20 \, x + 4\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} + 3 \, {\left (7 \, x^{5} - 16 \, x^{3} + 34 \, x^{2} + 76 \, x + 32\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}} - 192 \, x - 140}{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)/(x^3+2)^(1/3),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(1/3*(13910019318573948542*sqrt(3)*(7114781247*x^4 + 13663058416*x^3 - 46178206896*x^2 - 126

842559344*x - 77084338088)*(x^3 + 2)^(2/3) - 27820038637147897084*sqrt(3)*(1625757424*x^5 + 16302821713*x^4 +

26102613730*x^3 - 26431113242*x^2 - 80188343316*x - 42779182428)*(x^3 + 2)^(1/3) + sqrt(3)*(932925708335594356

63132301885*x^6 + 382151535711085278859235047618*x^5 + 673924074224408772959625384792*x^4 + 889426563183087468

015580290048*x^3 + 888876515195959220955879945824*x^2 + 351260598258508240019971964880*x - 4767400099559721105

7816884304))/(78905434814564721745708464883*x^6 + 337746705836458222863347934450*x^5 + 15598952776058587894336

070976*x^4 - 895430525315100108684787964824*x^3 + 361667862240477028869533375352*x^2 + 25418023010116325106459

72090336*x + 1554815286823334092314485968880)) + 1/12*log((22*x^6 + 6*x^5 - 48*x^4 + 44*x^3 + 24*x^2 + 3*(7*x^

4 - 2*x^3 - 32*x^2 - 20*x + 4)*(x^3 + 2)^(2/3) + 3*(7*x^5 - 16*x^3 + 34*x^2 + 76*x + 32)*(x^3 + 2)^(1/3) - 192

*x - 140)/(x^6 + 6*x^5 + 15*x^4 + 20*x^3 + 15*x^2 + 6*x + 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)/(x**3+2)**(1/3),x)

[Out]

Integral(1/((x + 1)*(x**3 + 2)**(1/3)), x)

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)/(x^3+2)^(1/3),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^3 + 2)^(1/3)*(x + 1)),x)

[Out]

int(1/((x^3 + 2)^(1/3)*(x + 1)), x)

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