∫ 1______ (1+x) 3 √ ____

Integrand size = 15, antiderivative size = 108 \[ \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx=\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {1+\frac {2 (2+x)}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )-\frac {1}{2} \log (1+x)+\frac {3}{4} \log \left (2+x-\sqrt [3]{2+x^3}\right )-\frac {1}{4} \log \left (-x+\sqrt [3]{2+x^3}\right ) \]

output

-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)

3.1.93.2 Mathematica [F]

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

input

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

output

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

3.1.93.3 Rubi [A] (veriﬁed)

Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2575, 769, 2576}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules deﬁnitions used are listed below.

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

\(\Big \downarrow \) 2575

\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt [3]{x^3+2}}dx+\frac {1}{2} \int \frac {1-x}{(x+1) \sqrt [3]{x^3+2}}dx\)

\(\Big \downarrow \) 769

\(\displaystyle \frac {1}{2} \int \frac {1-x}{(x+1) \sqrt [3]{x^3+2}}dx+\frac {1}{2} \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3+2}-x\right )\right )\)

\(\Big \downarrow \) 2576

\(\displaystyle \frac {1}{2} \left (-\sqrt {3} \arctan \left (\frac {\frac {2 (x+2)}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )+\frac {3}{2} \log \left (-\sqrt [3]{x^3+2}+x+2\right )-\log (x+1)\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3+2}-x\right )\right )\)

input

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

output

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

rule 769

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]]/(Sqrt[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 2575

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(a + b*x^3)^(1/3), x], x] + Simp[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 2576

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]

3.1.93.4 Maple [C] (veriﬁed)

Time = 4.13 (sec) , antiderivative size = 1421, normalized size of antiderivative = 13.16


method	result	size

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

input

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

output

1/6*ln((-4550781346817636-68457312523761*x^6-6728375859478224*x-6258954287 
88672*x^5+4993190285176576*RootOf(_Z^2+_Z+1)^2*x^3+8816461926585488*RootOf 
(_Z^2+_Z+1)*x^3+1055101552116528*RootOf(_Z^2+_Z+1)*x^2-21283128527537520*R 
ootOf(_Z^2+_Z+1)*x+9094739448000192*RootOf(_Z^2+_Z+1)^2*x^2+58848314075295 
36*RootOf(_Z^2+_Z+1)^2*x-234710785795752*x^4+2151515536461060*x^3-46942157 
1591504*x^2+1326316169500028*RootOf(_Z^2+_Z+1)^2*x^6+4346750471470680*Root 
Of(_Z^2+_Z+1)^2*x^5+4547369724000096*RootOf(_Z^2+_Z+1)^2*x^4+1538689763503 
27*RootOf(_Z^2+_Z+1)*x^6-868588603920114*RootOf(_Z^2+_Z+1)*x^5-15559137585 
059152*RootOf(_Z^2+_Z+1)+16295099853018372*x*(x^3+2)^(2/3)-912549035791293 
6*(x^3+2)^(1/3)+527550776058264*RootOf(_Z^2+_Z+1)*x^4-14648813469281292*x* 
(x^3+2)^(1/3)+10107087250606332*(x^3+2)^(2/3)-4682817420507954*(x^3+2)^(1/ 
3)*x^2-928201890361806*(x^3+2)^(2/3)*x^4-540325086981687*(x^3+2)^(1/3)*x^5 
-1959537324097146*(x^3+2)^(2/3)*x^3-480288966205944*(x^3+2)^(1/3)*x^4+5569 
211342170836*(x^3+2)^(2/3)*x^2+1200722415514860*(x^3+2)^(1/3)*x^3+71155808 
83942020*RootOf(_Z^2+_Z+1)*(x^3+2)^(2/3)+3372637147591320*RootOf(_Z^2+_Z+1 
)*(x^3+2)^(1/3)+36303984101745*RootOf(_Z^2+_Z+1)^2*(x^3+2)^(2/3)*x^4+28844 
9229728205*(x^3+2)^(1/3)*RootOf(_Z^2+_Z+1)^2*x^5-1306943427662820*RootOf(_ 
Z^2+_Z+1)^2*(x^3+2)^(2/3)*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^4-58077394950 
3594*RootOf(_Z^2+_Z+1)*(x^3+2)^(1/3)*x^5-3775614346581480*RootOf(_Z^2+_...

3.1.93.5 Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (84) = 168\).

Time = 1.04 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.47 \[ \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {13910019318573948542 \, \sqrt {3} {\left (7114781247 \, x^{4} + 13663058416 \, x^{3} - 46178206896 \, x^{2} - 126842559344 \, x - 77084338088\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} - 27820038637147897084 \, \sqrt {3} {\left (1625757424 \, x^{5} + 16302821713 \, x^{4} + 26102613730 \, x^{3} - 26431113242 \, x^{2} - 80188343316 \, x - 42779182428\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}} + \sqrt {3} {\left (93292570833559435663132301885 \, x^{6} + 382151535711085278859235047618 \, x^{5} + 673924074224408772959625384792 \, x^{4} + 889426563183087468015580290048 \, x^{3} + 888876515195959220955879945824 \, x^{2} + 351260598258508240019971964880 \, x - 47674000995597211057816884304\right )}}{3 \, {\left (78905434814564721745708464883 \, x^{6} + 337746705836458222863347934450 \, x^{5} + 15598952776058587894336070976 \, x^{4} - 895430525315100108684787964824 \, x^{3} + 361667862240477028869533375352 \, x^{2} + 2541802301011632510645972090336 \, x + 1554815286823334092314485968880\right )}}\right ) + \frac {1}{12} \, \log \left (\frac {22 \, x^{6} + 6 \, x^{5} - 48 \, x^{4} + 44 \, x^{3} + 24 \, x^{2} + 3 \, {\left (7 \, x^{4} - 2 \, x^{3} - 32 \, x^{2} - 20 \, x + 4\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} + 3 \, {\left (7 \, x^{5} - 16 \, x^{3} + 34 \, x^{2} + 76 \, x + 32\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}} - 192 \, x - 140}{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 20 \, x^{3} + 15 \, x^{2} + 6 \, x + 1}\right ) \]

input

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

output

1/6*sqrt(3)*arctan(1/3*(13910019318573948542*sqrt(3)*(7114781247*x^4 + 136 
63058416*x^3 - 46178206896*x^2 - 126842559344*x - 77084338088)*(x^3 + 2)^( 
2/3) - 27820038637147897084*sqrt(3)*(1625757424*x^5 + 16302821713*x^4 + 26 
102613730*x^3 - 26431113242*x^2 - 80188343316*x - 42779182428)*(x^3 + 2)^( 
1/3) + sqrt(3)*(93292570833559435663132301885*x^6 + 3821515357110852788592 
35047618*x^5 + 673924074224408772959625384792*x^4 + 8894265631830874680155 
80290048*x^3 + 888876515195959220955879945824*x^2 + 3512605982585082400199 
71964880*x - 47674000995597211057816884304))/(7890543481456472174570846488 
3*x^6 + 337746705836458222863347934450*x^5 + 15598952776058587894336070976 
*x^4 - 895430525315100108684787964824*x^3 + 361667862240477028869533375352 
*x^2 + 2541802301011632510645972090336*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))

3.1.93.6 Sympy [F]

\[ \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int \frac {1}{\left (x + 1\right ) \sqrt [3]{x^{3} + 2}}\, dx \]

input

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

output

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

3.1.93.7 Maxima [F]

\[ \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]

input

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

output

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

3.1.93.8 Giac [F]

\[ \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x + 1\right )}} \,d x } \]

input

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

output

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

3.1.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int \frac {1}{{\left (x^3+2\right )}^{1/3}\,\left (x+1\right )} \,d x \]

input

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

output

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

3.1.93.10 Reduce [F]

\[ \int \frac {1}{(1+x) \sqrt [3]{2+x^3}} \, dx=\int \frac {1}{\left (x^{3}+2\right )^{\frac {1}{3}} x +\left (x^{3}+2\right )^{\frac {1}{3}}}d x \]

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

3.1.93.1 Optimal result

3.1.93.2 Mathematica [F]

3.1.93.3 Rubi [A] (veriﬁed)

3.1.93.4 Maple [C] (veriﬁed)

3.1.93.5 Fricas [B] (veriﬁcation not implemented)

3.1.93.6 Sympy [F]

3.1.93.7 Maxima [F]

3.1.93.8 Giac [F]

3.1.93.9 Mupad [F(-1)]

3.1.93.10 Reduce [F]