\(\int \frac {1}{(3 x+3 x^2+x^3) \sqrt [3]{3+3 x+3 x^2+x^3}} \, dx\) [319]

Optimal result

Integrand size = 32, antiderivative size = 90 \[ \int \frac {1}{\left (3 x+3 x^2+x^3\right ) \sqrt [3]{3+3 x+3 x^2+x^3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{3} (1+x)}{\sqrt [3]{2+(1+x)^3}}}{\sqrt {3}}\right )}{3^{5/6}}-\frac {\log \left (1-(1+x)^3\right )}{6 \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3} (1+x)-\sqrt [3]{2+(1+x)^3}\right )}{2 \sqrt [3]{3}} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\left (3 x+3 x^2+x^3\right ) \sqrt [3]{3+3 x+3 x^2+x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{3+3 x+3 x^2+x^3}}{2 \sqrt [3]{3}+2 \sqrt [3]{3} x+\sqrt [3]{3+3 x+3 x^2+x^3}}\right )}{3^{5/6}}+\frac {2 \log \left (\sqrt [3]{3}+\sqrt [3]{3} x-\sqrt [3]{3+3 x+3 x^2+x^3}\right )-\log \left (3^{2/3}+2\ 3^{2/3} x+3^{2/3} x^2+\sqrt [3]{3} (1+x) \sqrt [3]{3+3 x+3 x^2+x^3}+\left (3+3 x+3 x^2+x^3\right )^{2/3}\right )}{6 \sqrt [3]{3}} \] Input:

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

Output:

ArcTan[(Sqrt[3]*(3 + 3*x + 3*x^2 + x^3)^(1/3))/(2*3^(1/3) + 2*3^(1/3)*x + 
(3 + 3*x + 3*x^2 + x^3)^(1/3))]/3^(5/6) + (2*Log[3^(1/3) + 3^(1/3)*x - (3 
+ 3*x + 3*x^2 + x^3)^(1/3)] - Log[3^(2/3) + 2*3^(2/3)*x + 3^(2/3)*x^2 + 3^ 
(1/3)*(1 + x)*(3 + 3*x + 3*x^2 + x^3)^(1/3) + (3 + 3*x + 3*x^2 + x^3)^(2/3 
)])/(6*3^(1/3))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {939, 938, 25, 901}

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.

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

\(\Big \downarrow \) 939

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

\(\Big \downarrow \) 938

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 901

\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{3} (x+1)}{\sqrt [3]{(x+1)^3+2}}+1}{\sqrt {3}}\right )}{3^{5/6}}-\frac {\log \left (1-(x+1)^3\right )}{6 \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3} (x+1)-\sqrt [3]{(x+1)^3+2}\right )}{2 \sqrt [3]{3}}\)

Input:

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

Output:

-(ArcTan[(1 + (2*3^(1/3)*(1 + x))/(2 + (1 + x)^3)^(1/3))/Sqrt[3]]/3^(5/6)) 
 - Log[1 - (1 + x)^3]/(6*3^(1/3)) + Log[3^(1/3)*(1 + x) - (2 + (1 + x)^3)^ 
(1/3)]/(2*3^(1/3))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 938

Int[((a_.) + (b_.)*(u_)^(n_))^(p_.)*((c_.) + (d_.)*(u_)^(n_))^(q_.), x_Symb 
ol] :> Simp[1/Coefficient[u, x, 1]   Subst[Int[(a + b*x^n)^p*(c + d*x^n)^q, 
 x], x, u], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && LinearQ[u, x] && NeQ[u 
, x]
 

rule 939

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[NormalizePseudoBinomial[u, x]^p 
*NormalizePseudoBinomial[v, x]^q, x] /; FreeQ[{p, q}, x] && PseudoBinomialP 
airQ[u, v, x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 11.73 (sec) , antiderivative size = 2515, normalized size of antiderivative = 27.94

method	result	size
trager	\(\text {Expression too large to display}\)	\(2515\)

Input:

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

Output:

-1/9*ln((-45050*RootOf(_Z^3-9)*x^3-66159*(x^3+3*x^2+3*x+3)^(2/3)-135150*Ro 
otOf(_Z^3-9)*x-530901*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2) 
*x^2-530901*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*x-127008* 
RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*(x^3+3 
*x^2+3*x+3)^(1/3)*x^2-254016*(x^3+3*x^2+3*x+3)^(1/3)*RootOf(RootOf(_Z^3-9) 
^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*x-73950*RootOf(_Z^3-9)-2904 
93*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)-42336*(x^3+3*x^2+3 
*x+3)^(2/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+8 
1*_Z^2)*x-135150*RootOf(_Z^3-9)*x^2-176967*RootOf(RootOf(_Z^3-9)^2+9*_Z*Ro 
otOf(_Z^3-9)+81*_Z^2)*x^3-17850*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9 
)+81*_Z^2)*RootOf(_Z^3-9)^3+30051*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3 
-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^2+7650*RootOf(RootOf(_Z^3-9)^2+9*_Z*Root 
Of(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^2+30051*RootOf(RootOf(_Z^3-9)^2+9*_ 
Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x+7650*RootOf(RootOf(_Z^3-9)^ 
2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x-42336*(x^3+3*x^2+3*x+3)^ 
(2/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2 
)-22053*RootOf(_Z^3-9)^2*(x^3+3*x^2+3*x+3)^(1/3)*x^2-44106*(x^3+3*x^2+3*x+ 
3)^(1/3)*RootOf(_Z^3-9)^2*x-127008*(x^3+3*x^2+3*x+3)^(1/3)*RootOf(RootOf(_ 
Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)-70119*RootOf(RootOf(_ 
Z^3-9)^2+9*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2-22053*(x^3+3*x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (71) = 142\).

Time = 4.32 (sec) , antiderivative size = 458, normalized size of antiderivative = 5.09 \[ \int \frac {1}{\left (3 x+3 x^2+x^3\right ) \sqrt [3]{3+3 x+3 x^2+x^3}} \, dx=-\frac {1}{54} \cdot 3^{\frac {2}{3}} \log \left (\frac {3 \cdot 3^{\frac {2}{3}} {\left (7 \, x^{4} + 28 \, x^{3} + 42 \, x^{2} + 30 \, x + 9\right )} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} + 3^{\frac {1}{3}} {\left (31 \, x^{6} + 186 \, x^{5} + 465 \, x^{4} + 666 \, x^{3} + 603 \, x^{2} + 324 \, x + 81\right )} + 9 \, {\left (5 \, x^{5} + 25 \, x^{4} + 50 \, x^{3} + 54 \, x^{2} + 33 \, x + 9\right )} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}}}{x^{6} + 6 \, x^{5} + 15 \, x^{4} + 18 \, x^{3} + 9 \, x^{2}}\right ) + \frac {1}{27} \cdot 3^{\frac {2}{3}} \log \left (\frac {2 \cdot 3^{\frac {2}{3}} {\left (x^{3} + 3 \, x^{2} + 3 \, x\right )} - 9 \cdot 3^{\frac {1}{3}} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 1\right )} + 9 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} {\left (x + 1\right )}}{x^{3} + 3 \, x^{2} + 3 \, x}\right ) - \frac {1}{9} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (12 \cdot 3^{\frac {2}{3}} {\left (7 \, x^{7} + 49 \, x^{6} + 147 \, x^{5} + 240 \, x^{4} + 225 \, x^{3} + 117 \, x^{2} + 27 \, x\right )} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 1143 \, x^{8} + 4572 \, x^{7} + 11070 \, x^{6} + 18414 \, x^{5} + 22032 \, x^{4} + 18900 \, x^{3} + 11178 \, x^{2} + 4131 \, x + 729\right )} - 18 \, {\left (31 \, x^{8} + 248 \, x^{7} + 868 \, x^{6} + 1782 \, x^{5} + 2400 \, x^{4} + 2196 \, x^{3} + 1332 \, x^{2} + 486 \, x + 81\right )} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 3\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (251 \, x^{9} + 2259 \, x^{8} + 9036 \, x^{7} + 21546 \, x^{6} + 34398 \, x^{5} + 38556 \, x^{4} + 30348 \, x^{3} + 16038 \, x^{2} + 5103 \, x + 729\right )}}\right ) \] Input:

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

Output:

-1/54*3^(2/3)*log((3*3^(2/3)*(7*x^4 + 28*x^3 + 42*x^2 + 30*x + 9)*(x^3 + 3 
*x^2 + 3*x + 3)^(2/3) + 3^(1/3)*(31*x^6 + 186*x^5 + 465*x^4 + 666*x^3 + 60 
3*x^2 + 324*x + 81) + 9*(5*x^5 + 25*x^4 + 50*x^3 + 54*x^2 + 33*x + 9)*(x^3 
 + 3*x^2 + 3*x + 3)^(1/3))/(x^6 + 6*x^5 + 15*x^4 + 18*x^3 + 9*x^2)) + 1/27 
*3^(2/3)*log((2*3^(2/3)*(x^3 + 3*x^2 + 3*x) - 9*3^(1/3)*(x^3 + 3*x^2 + 3*x 
 + 3)^(1/3)*(x^2 + 2*x + 1) + 9*(x^3 + 3*x^2 + 3*x + 3)^(2/3)*(x + 1))/(x^ 
3 + 3*x^2 + 3*x)) - 1/9*3^(1/6)*arctan(1/3*3^(1/6)*(12*3^(2/3)*(7*x^7 + 49 
*x^6 + 147*x^5 + 240*x^4 + 225*x^3 + 117*x^2 + 27*x)*(x^3 + 3*x^2 + 3*x + 
3)^(2/3) - 3^(1/3)*(127*x^9 + 1143*x^8 + 4572*x^7 + 11070*x^6 + 18414*x^5 
+ 22032*x^4 + 18900*x^3 + 11178*x^2 + 4131*x + 729) - 18*(31*x^8 + 248*x^7 
 + 868*x^6 + 1782*x^5 + 2400*x^4 + 2196*x^3 + 1332*x^2 + 486*x + 81)*(x^3 
+ 3*x^2 + 3*x + 3)^(1/3))/(251*x^9 + 2259*x^8 + 9036*x^7 + 21546*x^6 + 343 
98*x^5 + 38556*x^4 + 30348*x^3 + 16038*x^2 + 5103*x + 729))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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