\(\int \frac {1}{(\sqrt {3}+x) \sqrt [3]{1+x^2}} \, dx\) [712]

Optimal result

Integrand size = 19, antiderivative size = 104 \[ \int \frac {1}{\left (\sqrt {3}+x\right ) \sqrt [3]{1+x^2}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \left (\sqrt {3}-x\right )}{3 \sqrt [3]{1+x^2}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (\sqrt {3}+x\right )}{2\ 2^{2/3}}+\frac {\log \left (\sqrt {3}-x-\sqrt [3]{2} \sqrt {3} \sqrt [3]{1+x^2}\right )}{2\ 2^{2/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {765} \[ \int \frac {1}{\left (\sqrt {3}+x\right ) \sqrt [3]{1+x^2}} \, dx=-\frac {\arctan \left (\frac {2^{2/3} \left (\sqrt {3}-x\right )}{3 \sqrt [3]{x^2+1}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (-\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+1}-x+\sqrt {3}\right )}{2\ 2^{2/3}}-\frac {\log \left (x+\sqrt {3}\right )}{2\ 2^{2/3}} \]

[In]

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

[Out]

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

Rule 765

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[6*c^2*(e^2/d^2), 3]}, Simp[
(-Sqrt[3])*c*e*(ArcTan[1/Sqrt[3] + 2*c*((d - e*x)/(Sqrt[3]*d*q*(a + c*x^2)^(1/3)))]/(d^2*q^2)), x] + (-Simp[3*
c*e*(Log[d + e*x]/(2*d^2*q^2)), x] + Simp[3*c*e*(Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)]/(2*d^2*q^2)), x])] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \left (\sqrt {3}-x\right )}{3 \sqrt [3]{1+x^2}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (\sqrt {3}+x\right )}{2\ 2^{2/3}}+\frac {\log \left (\sqrt {3}-x-\sqrt [3]{2} \sqrt {3} \sqrt [3]{1+x^2}\right )}{2\ 2^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\left (\sqrt {3}+x\right ) \sqrt [3]{1+x^2}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {2^{2/3} x-\sqrt {3} \left (2^{2/3}+\sqrt [3]{1+x^2}\right )}{3 \sqrt [3]{1+x^2}}\right )+2 \log \left (-3 2^{2/3}+2^{2/3} \sqrt {3} x+6 \sqrt [3]{1+x^2}\right )-\log \left (-\sqrt [3]{2} x^2+\sqrt [3]{2} \sqrt {3} x \left (2+\sqrt [3]{2} \sqrt [3]{1+x^2}\right )-3 \left (\sqrt [3]{2}+2^{2/3} \sqrt [3]{1+x^2}+2 \left (1+x^2\right )^{2/3}\right )\right )}{6\ 2^{2/3}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 23.86 (sec) , antiderivative size = 2447, normalized size of antiderivative = 23.53

method	result	size
trager	\(\text {Expression too large to display}\)	\(2447\)

[In]

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

[Out]

-1/18*3^(1/2)*(ln(-(2*RootOf(_Z^3-6*3^(1/2))^3*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*
_Z^2)*x^3-144*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*RootOf(_Z^3-6*3^(1/
2))^2*(x^2+1)^(2/3)*x+60*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+
36*_Z^2)^2*x^3-54*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)*x+9*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)*x^
2-864*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*RootOf(_Z^3-6*3^(1/2))*(x^2+1)^(1/3
)*x+144*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*(x
^2+1)^(1/3)*x^2+18*RootOf(_Z^3-6*3^(1/2))^3*x*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_
Z^2)+540*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)^2*RootOf(_Z^3-6*3^(1/2))^2*x+432
*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(2/3)-R
ootOf(_Z^3-6*3^(1/2))*3^(1/2)*x^3-162*(x^2+1)^(2/3)*x-30*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/
2))*_Z+36*_Z^2)*3^(1/2)*x^3+27*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)+432*RootOf(RootOf(_Z^3-6*3^(1/2)
)^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*(x^2+1)^(1/3)-9*RootOf(_Z^3-6*3^(1/2))
*3^(1/2)*x+63*RootOf(_Z^3-6*3^(1/2))*x^2-270*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z
^2)*3^(1/2)*x+162*3^(1/2)*(x^2+1)^(2/3)+1890*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z
^2)*x^2+63*RootOf(_Z^3-6*3^(1/2))+1890*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2))/(
3^(1/2)*x+3)^3)*RootOf(_Z^3-6*3^(1/2))+6*ln(-(2*RootOf(_Z^3-6*3^(1/2))^3*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*Roo
tOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3-144*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3
^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(2/3)*x+60*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*
RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)^2*x^3-54*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)*x+9*3^(1/2)*RootOf(_Z^3-6*3
^(1/2))^2*(x^2+1)^(1/3)*x^2-864*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*RootOf(_Z
^3-6*3^(1/2))*(x^2+1)^(1/3)*x+144*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)
*RootOf(_Z^3-6*3^(1/2))*(x^2+1)^(1/3)*x^2+18*RootOf(_Z^3-6*3^(1/2))^3*x*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*Root
Of(_Z^3-6*3^(1/2))*_Z+36*_Z^2)+540*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)^2*Root
Of(_Z^3-6*3^(1/2))^2*x+432*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*RootOf(_Z^3-6*
3^(1/2))^2*(x^2+1)^(2/3)-RootOf(_Z^3-6*3^(1/2))*3^(1/2)*x^3-162*(x^2+1)^(2/3)*x-30*RootOf(RootOf(_Z^3-6*3^(1/2
))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*x^3+27*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)+432*Ro
otOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*(x^2+1)^(1/3
)-9*RootOf(_Z^3-6*3^(1/2))*3^(1/2)*x+63*RootOf(_Z^3-6*3^(1/2))*x^2-270*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootO
f(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*x+162*3^(1/2)*(x^2+1)^(2/3)+1890*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootO
f(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^2+63*RootOf(_Z^3-6*3^(1/2))+1890*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3
-6*3^(1/2))*_Z+36*_Z^2))/(3^(1/2)*x+3)^3)*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)
-6*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*ln(-(8*RootOf(_Z^3-6*3^(1/2))^3*RootOf
(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3+144*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootO
f(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(2/3)*x+60*RootOf(_Z^3-6*3^(1/2))^2*Roo
tOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)^2*x^3+90*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1
/3)*x-15*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)*x^2+864*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-
6*3^(1/2))*_Z+36*_Z^2)*RootOf(_Z^3-6*3^(1/2))*(x^2+1)^(1/3)*x-144*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^
3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*(x^2+1)^(1/3)*x^2+72*RootOf(_Z^3-6*3^(1/2))^3*x*RootOf
(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)+540*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z
^3-6*3^(1/2))*_Z+36*_Z^2)^2*RootOf(_Z^3-6*3^(1/2))^2*x-432*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(
1/2))*_Z+36*_Z^2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(2/3)+12*RootOf(_Z^3-6*3^(1/2))*3^(1/2)*x^3+270*(x^2+1)^(2/
3)*x+90*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*x^3-45*3^(1/2)*RootOf(_Z^
3-6*3^(1/2))^2*(x^2+1)^(1/3)-432*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*
RootOf(_Z^3-6*3^(1/2))*(x^2+1)^(1/3)+108*RootOf(_Z^3-6*3^(1/2))*3^(1/2)*x-252*RootOf(_Z^3-6*3^(1/2))*x^2+810*R
ootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*3^(1/2)*x-270*3^(1/2)*(x^2+1)^(2/3)-1890*R
ootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^2-252*RootOf(_Z^3-6*3^(1/2))-1890*RootOf
(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2))/(3^(1/2)*x+3)^3))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (74) = 148\).

Time = 3.21 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.71 \[ \int \frac {1}{\left (\sqrt {3}+x\right ) \sqrt [3]{1+x^2}} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} {\left (x^{4} + 8 \, \sqrt {3} x^{3} - 18 \, x^{2} - 27\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} + 99 \, x^{4} + 243 \, x^{2} + 12 \, \sqrt {3} {\left (x^{5} + 10 \, x^{3} + 9 \, x\right )} + 81\right )} + 4 \, {\left (21 \, x^{4} + 54 \, x^{2} + \sqrt {3} {\left (x^{5} - 42 \, x^{3} - 27 \, x\right )} + 81\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{6} - 225 \, x^{4} - 405 \, x^{2} - 243\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{2} - 2 \, \sqrt {3} x + 3\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} + 18 \, x^{2} - 4 \, \sqrt {3} {\left (x^{3} + 3 \, x\right )} + 9\right )} + 2 \, {\left (9 \, x^{2} - \sqrt {3} {\left (x^{3} + 9 \, x\right )} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{x^{4} - 6 \, x^{2} + 9}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (x^{2} - 2 \, \sqrt {3} x + 3\right )} + 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {3} x - 3\right )}}{x^{2} - 3}\right ) \]

[In]

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

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(x^4 + 8*sqrt(3)*x^3 - 18*x^2 - 27)*(x^2 + 1)^(2/3)
+ 4^(1/3)*(x^6 + 99*x^4 + 243*x^2 + 12*sqrt(3)*(x^5 + 10*x^3 + 9*x) + 81) + 4*(21*x^4 + 54*x^2 + sqrt(3)*(x^5
- 42*x^3 - 27*x) + 81)*(x^2 + 1)^(1/3))/(x^6 - 225*x^4 - 405*x^2 - 243)) - 1/24*4^(2/3)*log((3*4^(2/3)*(x^2 -
2*sqrt(3)*x + 3)*(x^2 + 1)^(2/3) + 4^(1/3)*(x^4 + 18*x^2 - 4*sqrt(3)*(x^3 + 3*x) + 9) + 2*(9*x^2 - sqrt(3)*(x
^3 + 9*x) + 9)*(x^2 + 1)^(1/3))/(x^4 - 6*x^2 + 9)) + 1/12*4^(2/3)*log((4^(1/3)*(x^2 - 2*sqrt(3)*x + 3) + 2*(x^
2 + 1)^(1/3)*(sqrt(3)*x - 3))/(x^2 - 3))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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