3.93 \(\int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx\)
Optimal. Leaf size=108 \[ \frac {3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac {1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log (x+1) \]
[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)
________________________________________________________________________________________
Rubi [A] time = 0.09, 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 =
{2149, 239, 2151} \[ \frac {3}{4} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\frac {1}{4} \log \left (\sqrt [3]{x^3+2}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log (x+1) \]
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 239
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 2149
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 2151
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*}
________________________________________________________________________________________
Mathematica [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx \]
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]
________________________________________________________________________________________
fricas [B] time = 2.70, size = 267, normalized size = 2.47 \[ \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {13910019318573948542 \, \sqrt {3} {\left (7114781247 \, x^{4} + 13663058416 \, x^{3} - 46178206896 \, x^{2} - 126842559344 \, x - 77084338088\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} - 27820038637147897084 \, \sqrt {3} {\left (1625757424 \, x^{5} + 16302821713 \, x^{4} + 26102613730 \, x^{3} - 26431113242 \, x^{2} - 80188343316 \, x - 42779182428\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}} + \sqrt {3} {\left (93292570833559435663132301885 \, x^{6} + 382151535711085278859235047618 \, x^{5} + 673924074224408772959625384792 \, x^{4} + 889426563183087468015580290048 \, x^{3} + 888876515195959220955879945824 \, x^{2} + 351260598258508240019971964880 \, x - 47674000995597211057816884304\right )}}{3 \, {\left (78905434814564721745708464883 \, x^{6} + 337746705836458222863347934450 \, x^{5} + 15598952776058587894336070976 \, x^{4} - 895430525315100108684787964824 \, x^{3} + 361667862240477028869533375352 \, x^{2} + 2541802301011632510645972090336 \, x + 1554815286823334092314485968880\right )}}\right ) + \frac {1}{12} \, \log \left (\frac {22 \, x^{6} + 6 \, x^{5} - 48 \, x^{4} + 44 \, x^{3} + 24 \, x^{2} + 3 \, {\left (7 \, x^{4} - 2 \, x^{3} - 32 \, x^{2} - 20 \, x + 4\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} + 3 \, {\left (7 \, x^{5} - 16 \, x^{3} + 34 \, x^{2} + 76 \, x + 32\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}} - 192 \, x - 140}{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1}\right ) \]
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))
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1+x)/(x^3+2)^(1/3),x, algorithm="giac")
[Out]
integrate(1/((x^3 + 2)^(1/3)*(x + 1)), x)
________________________________________________________________________________________
maple [C] time = 5.82, size = 2134, normalized size = 19.76 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x+1)/(x^3+2)^(1/3),x)
[Out]
1/6*RootOf(_Z^2+_Z+1)*ln((23004340956706368*x+234055794617652*x^6+2139938693647104*x^5+802477010117664*x^4-735
6039259411920*x^3+1604954020235328*x^2-938391398532049*RootOf(_Z^2+_Z+1)^2*x^6-3075400381743690*RootOf(_Z^2+_Z
+1)^2*x^5-1160717687406137*RootOf(_Z^2+_Z+1)*x^6-3217341937824168*RootOf(_Z^2+_Z+1)^2*x^4-7959206999356368*Roo
tOf(_Z^2+_Z+1)*x^2+10391133689698608*RootOf(_Z^2+_Z+1)*x-6434683875648336*RootOf(_Z^2+_Z+1)^2*x^2-416361897836
0688*RootOf(_Z^2+_Z+1)^2*x-3532767618003008*x^3*RootOf(_Z^2+_Z+1)^2+12498127505504256*(x^3+2)^(1/3)+1100835623
8241516*RootOf(_Z^2+_Z+1)+16295099853018372*(x^3+2)^(2/3)*x-3979603499678184*RootOf(_Z^2+_Z+1)*x^4-10197714008
127436*x^3*RootOf(_Z^2+_Z+1)-2832707206612248*RootOf(_Z^2+_Z+1)*x^5+10107087250606332*(x^3+2)^(2/3)+6413512798
877184*(x^3+2)^(1/3)*x^2+20062783627256832*(x^3+2)^(1/3)*x+36303984101745*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(2/3)*x^
4+780469570084659*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^5-1306943427662820*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(2/3)*x^3
+3482095004993094*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^4-601904942144643*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^4+16
09243886794551*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^5-3775614346581480*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(2/3)*x^2+3842
311729647552*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^3-3235955176589922*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^3+524931
1303832568*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^4-2613886855325640*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(2/3)*x-8405056908
60402*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^2-1442113972435308*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^2+4061647060486
932*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^3-2161300347926748*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x+7759251414704196*
RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x+3531674097632562*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^2+11688730639030284*RootO
f(_Z^2+_Z+1)*(x^3+2)^(1/3)*x-928201890361806*(x^3+2)^(2/3)*x^4+740020707562752*(x^3+2)^(1/3)*x^5-1959537324097
146*(x^3+2)^(2/3)*x^3+657796184500224*(x^3+2)^(1/3)*x^4+5569211342170836*(x^3+2)^(2/3)*x^2-1644490461250560*(x
^3+2)^(1/3)*x^3+7115580883942020*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)+9125490357912936*RootOf(_Z^2+_Z+1)*(x^3+2)^(1
/3)+15559137585059152)/(x+1)^6)-1/6*ln((8449588288647072*x+456382083491740*x^6+1897245518515662*x^5+1564738571
971680*x^4-691092869287492*x^3+3129477143943360*x^2-938391398532049*RootOf(_Z^2+_Z+1)^2*x^6-3075400381743690*R
ootOf(_Z^2+_Z+1)^2*x^5-716065109657961*RootOf(_Z^2+_Z+1)*x^6-3217341937824168*RootOf(_Z^2+_Z+1)^2*x^4-49101607
51940304*RootOf(_Z^2+_Z+1)*x^2-18718371646419984*RootOf(_Z^2+_Z+1)*x-6434683875648336*RootOf(_Z^2+_Z+1)^2*x^2-
4163618978360688*RootOf(_Z^2+_Z+1)^2*x-3532767618003008*x^3*RootOf(_Z^2+_Z+1)^2+3372637147591320*(x^3+2)^(1/3)
-11008356238241516*RootOf(_Z^2+_Z+1)+5921961582988536*(x^3+2)^(2/3)*x-2455080375970152*RootOf(_Z^2+_Z+1)*x^4+3
132178772121420*x^3*RootOf(_Z^2+_Z+1)-3318093556875132*RootOf(_Z^2+_Z+1)*x^5+2991506366664312*(x^3+2)^(2/3)+20
41333010384220*(x^3+2)^(1/3)*x^2+6212752640299800*(x^3+2)^(1/3)*x+36303984101745*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(
2/3)*x^4+780469570084659*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^5-1306943427662820*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(2
/3)*x^3+3482095004993094*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^4+674512910348133*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)
*x^4-48304746625233*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^5-3775614346581480*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(2/3)*x^2
+3842311729647552*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^3+622068321264282*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^3+17
14878706153620*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^4-2613886855325640*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(2/3)*x-840505
690860402*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^2-6109114720727652*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x^2+362297639
8808172*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^3-2161300347926748*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x-1298702512535
5476*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x-5212685479353366*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^2-16011331334883780*
RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x-289992964115418*(x^3+2)^(2/3)*x^4-88753609147140*(x^3+2)^(1/3)*x^5-305255751
70044*(x^3+2)^(2/3)*x^3-1109420114339250*(x^3+2)^(1/3)*x^4+3235710968024664*(x^3+2)^(2/3)*x^2-1863825792089940
*(x^3+2)^(1/3)*x^3-7115580883942020*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)-9125490357912936*RootOf(_Z^2+_Z+1)*(x^3+2)
^(1/3)+4550781346817636)/(x+1)^6)*RootOf(_Z^2+_Z+1)-1/6*ln((8449588288647072*x+456382083491740*x^6+18972455185
15662*x^5+1564738571971680*x^4-691092869287492*x^3+3129477143943360*x^2-938391398532049*RootOf(_Z^2+_Z+1)^2*x^
6-3075400381743690*RootOf(_Z^2+_Z+1)^2*x^5-716065109657961*RootOf(_Z^2+_Z+1)*x^6-3217341937824168*RootOf(_Z^2+
_Z+1)^2*x^4-4910160751940304*RootOf(_Z^2+_Z+1)*x^2-18718371646419984*RootOf(_Z^2+_Z+1)*x-6434683875648336*Root
Of(_Z^2+_Z+1)^2*x^2-4163618978360688*RootOf(_Z^2+_Z+1)^2*x-3532767618003008*x^3*RootOf(_Z^2+_Z+1)^2+3372637147
591320*(x^3+2)^(1/3)-11008356238241516*RootOf(_Z^2+_Z+1)+5921961582988536*(x^3+2)^(2/3)*x-2455080375970152*Roo
tOf(_Z^2+_Z+1)*x^4+3132178772121420*x^3*RootOf(_Z^2+_Z+1)-3318093556875132*RootOf(_Z^2+_Z+1)*x^5+2991506366664
312*(x^3+2)^(2/3)+2041333010384220*(x^3+2)^(1/3)*x^2+6212752640299800*(x^3+2)^(1/3)*x+36303984101745*RootOf(_Z
^2+_Z+1)^2*(x^3+2)^(2/3)*x^4+780469570084659*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^5-1306943427662820*RootOf(_Z^
2+_Z+1)^2*(x^3+2)^(2/3)*x^3+3482095004993094*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^4+674512910348133*RootOf(_Z^2
+_Z+1)*(x^3+2)^(2/3)*x^4-48304746625233*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^5-3775614346581480*RootOf(_Z^2+_Z+1)
^2*(x^3+2)^(2/3)*x^2+3842311729647552*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^3+622068321264282*RootOf(_Z^2+_Z+1)*
(x^3+2)^(2/3)*x^3+1714878706153620*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^4-2613886855325640*RootOf(_Z^2+_Z+1)^2*(x
^3+2)^(2/3)*x-840505690860402*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(1/3)*x^2-6109114720727652*RootOf(_Z^2+_Z+1)*(x^3+2)
^(2/3)*x^2+3622976398808172*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^3-2161300347926748*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(
1/3)*x-12987025125355476*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)*x-5212685479353366*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^
2-16011331334883780*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x-289992964115418*(x^3+2)^(2/3)*x^4-88753609147140*(x^3+2)
^(1/3)*x^5-30525575170044*(x^3+2)^(2/3)*x^3-1109420114339250*(x^3+2)^(1/3)*x^4+3235710968024664*(x^3+2)^(2/3)*
x^2-1863825792089940*(x^3+2)^(1/3)*x^3-7115580883942020*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)-9125490357912936*RootO
f(_Z^2+_Z+1)*(x^3+2)^(1/3)+4550781346817636)/(x+1)^6)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \]
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)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \]
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)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \]
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)
________________________________________________________________________________________