3.712 \(\int \frac {1}{(\sqrt {3}+x) \sqrt [3]{1+x^2}} \, dx\)
Optimal. Leaf size=104 \[ \frac {\log \left (-\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+1}-x+\sqrt {3}\right )}{2\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} \left (\sqrt {3}-x\right )}{3 \sqrt [3]{x^2+1}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (x+\sqrt {3}\right )}{2\ 2^{2/3}} \]
[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)
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Rubi [A] time = 0.01, antiderivative size = 104, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used =
{751} \[ \frac {\log \left (-\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^2+1}-x+\sqrt {3}\right )}{2\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} \left (\sqrt {3}-x\right )}{3 \sqrt [3]{x^2+1}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (x+\sqrt {3}\right )}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
[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 751
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*} \int \frac {1}{\left (\sqrt {3}+x\right ) \sqrt [3]{1+x^2}} \, dx &=-\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*}
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Mathematica [C] time = 0.07, size = 102, normalized size = 0.98 \[ -\frac {3 \sqrt [3]{\frac {x-i}{x+\sqrt {3}}} \sqrt [3]{\frac {x+i}{x+\sqrt {3}}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {-i+\sqrt {3}}{x+\sqrt {3}},\frac {i+\sqrt {3}}{x+\sqrt {3}}\right )}{2 \sqrt [3]{x^2+1}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((Sqrt[3] + x)*(1 + x^2)^(1/3)),x]
[Out]
(-3*((-I + x)/(Sqrt[3] + x))^(1/3)*((I + x)/(Sqrt[3] + x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (-I + Sqrt[3])/(
Sqrt[3] + x), (I + Sqrt[3])/(Sqrt[3] + x)])/(2*(1 + x^2)^(1/3))
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fricas [B] time = 10.36, size = 282, normalized size = 2.71 \[ -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} {\left (x^{4} + 8 \, \sqrt {3} x^{3} - 18 \, x^{2} - 27\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} + 99 \, x^{4} + 243 \, x^{2} + 12 \, \sqrt {3} {\left (x^{5} + 10 \, x^{3} + 9 \, x\right )} + 81\right )} + 4 \, {\left (21 \, x^{4} + 54 \, x^{2} + \sqrt {3} {\left (x^{5} - 42 \, x^{3} - 27 \, x\right )} + 81\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{6} - 225 \, x^{4} - 405 \, x^{2} - 243\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{2} - 2 \, \sqrt {3} x + 3\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} + 18 \, x^{2} - 4 \, \sqrt {3} {\left (x^{3} + 3 \, x\right )} + 9\right )} + 2 \, {\left (9 \, x^{2} - \sqrt {3} {\left (x^{3} + 9 \, x\right )} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{x^{4} - 6 \, x^{2} + 9}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (x^{2} - 2 \, \sqrt {3} x + 3\right )} + 2 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (\sqrt {3} x - 3\right )}}{x^{2} - 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[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))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + \sqrt {3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maple [C] time = 47.00, size = 2422, normalized size = 23.29 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^2+1)^(1/3)/(x+3^(1/2)),x)
[Out]
-1/18*3^(1/2)*(6*ln((12*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+3
6*_Z^2)^2*x^3-144*3^(1/2)*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)*(x^2+1)^(2/3)*x+10*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+540*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^
2)*(x^2+1)^(1/3)*x-90*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*
_Z+36*_Z^2)*(x^2+1)^(1/3)*x^2-54*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+108*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+18*3^(1/2)*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3+90*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+432*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)*(x^2+1)^(2/3)-270*(x^2+1)^(2/3)*x+15*3^(
1/2)*RootOf(_Z^3-6*3^(1/2))*x^3-270*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z
^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)+27*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)+162*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-378*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z
^3-6*3^(1/2))*_Z+36*_Z^2)*x^2+135*RootOf(_Z^3-6*3^(1/2))*3^(1/2)*x+270*3^(1/2)*(x^2+1)^(2/3)-315*x^2*RootOf(_Z
^3-6*3^(1/2))-378*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)-315*RootOf(_Z^3-6*3^(1/
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)+ln((12*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^3-144*3^(1/2)*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)*(x^2+1)^(2/3)*x+10*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+540*RootOf(_Z^3-6*3
^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)*x-90*3^(1/2)*RootOf
(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)*x^2-54*Roo
tOf(_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+108*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+18*3^(1/2)*RootOf(RootOf(_Z
^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3+90*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+432*RootOf(_Z^3-6*3^(1/2))^2*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*R
ootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(2/3)-270*(x^2+1)^(2/3)*x+15*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*x^3-270*
3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1
/3)+27*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)+162*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-378*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^2+135*Ro
otOf(_Z^3-6*3^(1/2))*3^(1/2)*x+270*3^(1/2)*(x^2+1)^(2/3)-315*x^2*RootOf(_Z^3-6*3^(1/2))-378*RootOf(RootOf(_Z^3
-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)-315*RootOf(_Z^3-6*3^(1/2)))/(3^(1/2)*x+3)^3)*RootOf(_Z^3-6*
3^(1/2))-RootOf(_Z^3-6*3^(1/2))*ln((48*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+144*3^(1/2)*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)*(x^2+1)^(2/3)*x+10*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^3-324*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(
1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)*x+54*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_
Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)*x^2+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+432*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-24*3^(1/2)*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^3+9
0*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-432*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)*(x^2+1)^(2/3)+162*(x^2+1
)^(2/3)*x-5*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*x^3+162*3^(1/2)*RootOf(_Z^3-6*3^(1/2))*RootOf(RootOf(_Z^3-6*3^(1/2)
)^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*(x^2+1)^(1/3)-45*3^(1/2)*RootOf(_Z^3-6*3^(1/2))^2*(x^2+1)^(1/3)-216*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+1512*RootOf(RootOf(_Z^3-6*3^(1/2
))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)*x^2-45*RootOf(_Z^3-6*3^(1/2))*3^(1/2)*x-162*3^(1/2)*(x^2+1)^(2/3)+31
5*x^2*RootOf(_Z^3-6*3^(1/2))+1512*RootOf(RootOf(_Z^3-6*3^(1/2))^2+6*RootOf(_Z^3-6*3^(1/2))*_Z+36*_Z^2)+315*Roo
tOf(_Z^3-6*3^(1/2)))/(3^(1/2)*x+3)^3))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + \sqrt {3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (x^2+1\right )}^{1/3}\,\left (x+\sqrt {3}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x + \sqrt {3}\right ) \sqrt [3]{x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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