3.24.78 \(\int \frac {1}{\sqrt [3]{1+x^2} (9+x^2)} \, dx\)

Optimal. Leaf size=190 \[ \frac {1}{12} \tan ^{-1}\left (\frac {x}{2 \sqrt [3]{x^2+1}+1}\right )+\frac {i \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}-\frac {i x}{\sqrt {3}}-\frac {1}{\sqrt {3}}}{\sqrt [3]{x^2+1}}\right )}{8 \sqrt {3}}-\frac {i \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {i x}{\sqrt {3}}-\frac {1}{\sqrt {3}}}{\sqrt [3]{x^2+1}}\right )}{8 \sqrt {3}}-\frac {1}{24} i \tanh ^{-1}\left (\frac {2 i x-2 i x \sqrt [3]{x^2+1}}{x^2-4 \left (x^2+1\right )^{2/3}+2 \sqrt [3]{x^2+1}-1}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 70, normalized size of antiderivative = 0.37, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {394} \begin {gather*} \frac {1}{12} \tan ^{-1}\left (\frac {\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt {3}}+\frac {1}{12} \tan ^{-1}\left (\frac {x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^2)^(1/3)*(9 + x^2)),x]

[Out]

ArcTan[x/3]/12 + ArcTan[(1 - (1 + x^2)^(1/3))^2/(3*x)]/12 - ArcTanh[(Sqrt[3]*(1 - (1 + x^2)^(1/3)))/x]/(4*Sqrt
[3])

Rule 394

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[(q*ArcTan[
(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTan[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(12*Rt[a
, 3]*d), x] - Simp[(q*ArcTanh[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)])/(4*Sqrt[3]*Rt[a, 3]*d)
, x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && PosQ[b/a]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx &=\frac {1}{12} \tan ^{-1}\left (\frac {x}{3}\right )+\frac {1}{12} \tan ^{-1}\left (\frac {\left (1-\sqrt [3]{1+x^2}\right )^2}{3 x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1+x^2}\right )}{x}\right )}{4 \sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 124, normalized size = 0.65 \begin {gather*} -\frac {27 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-x^2,-\frac {x^2}{9}\right )}{\sqrt [3]{x^2+1} \left (x^2+9\right ) \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-x^2,-\frac {x^2}{9}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-x^2,-\frac {x^2}{9}\right )\right )-27 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-x^2,-\frac {x^2}{9}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((1 + x^2)^(1/3)*(9 + x^2)),x]

[Out]

(-27*x*AppellF1[1/2, 1/3, 1, 3/2, -x^2, -1/9*x^2])/((1 + x^2)^(1/3)*(9 + x^2)*(-27*AppellF1[1/2, 1/3, 1, 3/2,
-x^2, -1/9*x^2] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -x^2, -1/9*x^2] + 3*AppellF1[3/2, 4/3, 1, 5/2, -x^2, -1/9*
x^2])))

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IntegrateAlgebraic [F]  time = 3.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((1 + x^2)^(1/3)*(9 + x^2)),x]

[Out]

Could not integrate

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fricas [B]  time = 2.07, size = 1395, normalized size = 7.34

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(1/3)/(x^2+9),x, algorithm="fricas")

[Out]

1/144*sqrt(3)*log(4*(x^6 + 1647*x^4 + 891*x^2 + 18*(3*x^4 + 32*sqrt(3)*x^3 + 126*x^2 + 27)*(x^2 + 1)^(2/3) + 1
08*sqrt(3)*(x^5 + 10*x^3 + 9*x) + 6*(81*x^4 + 162*x^2 + sqrt(3)*(x^5 + 210*x^3 + 81*x) + 81)*(x^2 + 1)^(1/3) -
243)/(x^6 + 27*x^4 + 243*x^2 + 729)) - 1/144*sqrt(3)*log(4*(x^6 + 1647*x^4 + 891*x^2 + 18*(3*x^4 - 32*sqrt(3)
*x^3 + 126*x^2 + 27)*(x^2 + 1)^(2/3) - 108*sqrt(3)*(x^5 + 10*x^3 + 9*x) + 6*(81*x^4 + 162*x^2 - sqrt(3)*(x^5 +
210*x^3 + 81*x) + 81)*(x^2 + 1)^(1/3) - 243)/(x^6 + 27*x^4 + 243*x^2 + 729)) - 1/36*arctan((384*x^11 - 130320
*x^9 + 2379456*x^7 - 629856*x^5 - 1259712*x^3 + 36*(388*x^9 - 27864*x^7 + 303264*x^5 + 17496*x^3 + sqrt(3)*(x^
10 + 549*x^8 - 8046*x^6 + 129762*x^4 - 19683*x^2 + 59049) - 236196*x)*(x^2 + 1)^(2/3) + sqrt(3)*(x^12 - 234*x^
10 + 229311*x^8 - 1214028*x^6 + 6816879*x^4 + 6022998*x^2 + 531441) + 2*(x^12 + 50616*x^10 - 1869399*x^8 - 377
3304*x^6 - 6908733*x^4 + 72*(x^10 + 1620*x^8 - 63666*x^6 - 43740*x^4 + 59049*x^2 + 12*sqrt(3)*(11*x^9 - 261*x^
7 - 6075*x^5 - 2187*x^3))*(x^2 + 1)^(2/3) + 6*sqrt(3)*(43*x^11 + 14055*x^9 - 563922*x^7 - 1307826*x^5 - 898857
*x^3 + 177147*x) + 6*(453*x^10 + 21141*x^8 - 1483758*x^6 - 1404054*x^4 - 885735*x^2 + sqrt(3)*(x^11 + 8985*x^9
- 349110*x^7 + 118098*x^5 + 32805*x^3 - 177147*x) + 531441)*(x^2 + 1)^(1/3) + 1594323)*sqrt((x^6 + 1647*x^4 +
891*x^2 + 18*(3*x^4 - 32*sqrt(3)*x^3 + 126*x^2 + 27)*(x^2 + 1)^(2/3) - 108*sqrt(3)*(x^5 + 10*x^3 + 9*x) + 6*(
81*x^4 + 162*x^2 - sqrt(3)*(x^5 + 210*x^3 + 81*x) + 81)*(x^2 + 1)^(1/3) - 243)/(x^6 + 27*x^4 + 243*x^2 + 729))
+ 12*(x^11 - 6423*x^9 + 225018*x^7 - 1106622*x^5 - 1541835*x^3 + 3*sqrt(3)*(37*x^10 - 675*x^8 + 34722*x^6 - 9
7686*x^4 + 59049*x^2 + 59049) - 177147*x)*(x^2 + 1)^(1/3) - 8503056*x)/(x^12 - 48978*x^10 + 2332071*x^8 - 1641
9996*x^6 - 24151041*x^4 - 9565938*x^2 + 4782969)) + 1/36*arctan(-(384*x^11 - 130320*x^9 + 2379456*x^7 - 629856
*x^5 - 1259712*x^3 + 36*(388*x^9 - 27864*x^7 + 303264*x^5 + 17496*x^3 - sqrt(3)*(x^10 + 549*x^8 - 8046*x^6 + 1
29762*x^4 - 19683*x^2 + 59049) - 236196*x)*(x^2 + 1)^(2/3) - sqrt(3)*(x^12 - 234*x^10 + 229311*x^8 - 1214028*x
^6 + 6816879*x^4 + 6022998*x^2 + 531441) + 2*(x^12 + 50616*x^10 - 1869399*x^8 - 3773304*x^6 - 6908733*x^4 + 72
*(x^10 + 1620*x^8 - 63666*x^6 - 43740*x^4 + 59049*x^2 - 12*sqrt(3)*(11*x^9 - 261*x^7 - 6075*x^5 - 2187*x^3))*(
x^2 + 1)^(2/3) - 6*sqrt(3)*(43*x^11 + 14055*x^9 - 563922*x^7 - 1307826*x^5 - 898857*x^3 + 177147*x) + 6*(453*x
^10 + 21141*x^8 - 1483758*x^6 - 1404054*x^4 - 885735*x^2 - sqrt(3)*(x^11 + 8985*x^9 - 349110*x^7 + 118098*x^5
+ 32805*x^3 - 177147*x) + 531441)*(x^2 + 1)^(1/3) + 1594323)*sqrt((x^6 + 1647*x^4 + 891*x^2 + 18*(3*x^4 + 32*s
qrt(3)*x^3 + 126*x^2 + 27)*(x^2 + 1)^(2/3) + 108*sqrt(3)*(x^5 + 10*x^3 + 9*x) + 6*(81*x^4 + 162*x^2 + sqrt(3)*
(x^5 + 210*x^3 + 81*x) + 81)*(x^2 + 1)^(1/3) - 243)/(x^6 + 27*x^4 + 243*x^2 + 729)) + 12*(x^11 - 6423*x^9 + 22
5018*x^7 - 1106622*x^5 - 1541835*x^3 - 3*sqrt(3)*(37*x^10 - 675*x^8 + 34722*x^6 - 97686*x^4 + 59049*x^2 + 5904
9) - 177147*x)*(x^2 + 1)^(1/3) - 8503056*x)/(x^12 - 48978*x^10 + 2332071*x^8 - 16419996*x^6 - 24151041*x^4 - 9
565938*x^2 + 4782969)) - 1/36*arctan(6*(11*x^5 + 30*x^3 + 6*(23*x^3 + 27*x)*(x^2 + 1)^(2/3) + (x^5 - 240*x^3 -
81*x)*(x^2 + 1)^(1/3) - 81*x)/(x^6 - 1971*x^4 - 1701*x^2 - 729))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(1/3)/(x^2+9),x, algorithm="giac")

[Out]

integrate(1/((x^2 + 9)*(x^2 + 1)^(1/3)), x)

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maple [C]  time = 6.52, size = 622, normalized size = 3.27

method	result	size
trager	\(-144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} \ln \left (-\frac {-497664 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x +995328 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x +6912 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x -20736 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}-864 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}+6 \left (x^{2}+1\right )^{\frac {2}{3}}-432 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}+96 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x -x^{2}+3}{x^{2}+9}\right )+\RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {82944 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -165888 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -1728 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +2304 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x -24 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}+144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}+8 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x +\left (x^{2}+1\right )^{\frac {2}{3}}+72 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-\left (x^{2}+1\right )^{\frac {1}{3}}}{x^{2}+9}\right )+\RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-497664 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x +995328 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x +6912 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x -20736 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}-864 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}+6 \left (x^{2}+1\right )^{\frac {2}{3}}-432 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}+96 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x -x^{2}+3}{x^{2}+9}\right )\)	\(622\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)^(1/3)/(x^2+9),x,method=_RETURNVERBOSE)

[Out]

-144*RootOf(20736*_Z^4-144*_Z^2+1)^3*ln(-(-497664*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^5*x+995328*RootO
f(20736*_Z^4-144*_Z^2+1)^5*x+6912*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^3*x-20736*RootOf(20736*_Z^4-144*
_Z^2+1)^3*x+144*RootOf(20736*_Z^4-144*_Z^2+1)^2*x^2-864*RootOf(20736*_Z^4-144*_Z^2+1)^2*(x^2+1)^(1/3)+6*(x^2+1
)^(2/3)-432*RootOf(20736*_Z^4-144*_Z^2+1)^2+96*RootOf(20736*_Z^4-144*_Z^2+1)*x-x^2+3)/(x^2+9))+RootOf(20736*_Z
^4-144*_Z^2+1)*ln(-(82944*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^5*x-165888*RootOf(20736*_Z^4-144*_Z^2+1)
^5*x-1728*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^3*x+2304*RootOf(20736*_Z^4-144*_Z^2+1)^3*x-24*RootOf(207
36*_Z^4-144*_Z^2+1)^2*x^2+144*RootOf(20736*_Z^4-144*_Z^2+1)^2*(x^2+1)^(1/3)+8*(x^2+1)^(1/3)*RootOf(20736*_Z^4-
144*_Z^2+1)*x+(x^2+1)^(2/3)+72*RootOf(20736*_Z^4-144*_Z^2+1)^2-(x^2+1)^(1/3))/(x^2+9))+RootOf(20736*_Z^4-144*_
Z^2+1)*ln(-(-497664*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^5*x+995328*RootOf(20736*_Z^4-144*_Z^2+1)^5*x+6
912*(x^2+1)^(1/3)*RootOf(20736*_Z^4-144*_Z^2+1)^3*x-20736*RootOf(20736*_Z^4-144*_Z^2+1)^3*x+144*RootOf(20736*_
Z^4-144*_Z^2+1)^2*x^2-864*RootOf(20736*_Z^4-144*_Z^2+1)^2*(x^2+1)^(1/3)+6*(x^2+1)^(2/3)-432*RootOf(20736*_Z^4-
144*_Z^2+1)^2+96*RootOf(20736*_Z^4-144*_Z^2+1)*x-x^2+3)/(x^2+9))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(1/3)/(x^2+9),x, algorithm="maxima")

[Out]

integrate(1/((x^2 + 9)*(x^2 + 1)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (x^2+1\right )}^{1/3}\,\left (x^2+9\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^2 + 1)^(1/3)*(x^2 + 9)),x)

[Out]

int(1/((x^2 + 1)^(1/3)*(x^2 + 9)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x^{2} + 1} \left (x^{2} + 9\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+1)**(1/3)/(x**2+9),x)

[Out]

Integral(1/((x**2 + 1)**(1/3)*(x**2 + 9)), x)

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