3.2.60 \(\int \frac {1}{\sqrt [3]{1+x^2} (9+x^2)} \, dx\) [160]

Optimal. Leaf size=70 \[ \frac {1}{12} \tan ^{-1}\left (\frac {x}{3}\right )+\frac {1}{12} \tan ^{-1}\left (\frac {\left (1-\sqrt [3]{1+x^2}\right )^2}{3 x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1+x^2}\right )}{x}\right )}{4 \sqrt {3}} \]

[Out]

1/12*arctan(1/3*x)+1/12*arctan(1/3*(1-(x^2+1)^(1/3))^2/x)-1/12*arctanh((1-(x^2+1)^(1/3))*3^(1/2)/x)*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 = {403} \begin {gather*} \frac {1}{12} \text {ArcTan}\left (\frac {\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )+\frac {1}{12} \text {ArcTan}\left (\frac {x}{3}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt {3}} \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 403

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] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 4.73, size = 124, normalized size = 1.77 \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]{1+x^2} \left (9+x^2\right ) \left (-27 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-x^2,-\frac {x^2}{9}\right )+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 )\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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 6.44, size = 616, normalized size = 8.80

method	result	size
trager	\(-144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {497664 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} \left (x^{2}+1\right )^{\frac {1}{3}} x -995328 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -6912 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} \left (x^{2}+1\right )^{\frac {1}{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}}+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 -6 \left (x^{2}+1\right )^{\frac {2}{3}}+x^{2}-3}{x^{2}+9}\right )+\RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {497664 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} \left (x^{2}+1\right )^{\frac {1}{3}} x -995328 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -6912 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} \left (x^{2}+1\right )^{\frac {1}{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}}+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 -6 \left (x^{2}+1\right )^{\frac {2}{3}}+x^{2}-3}{x^{2}+9}\right )+\RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {82944 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} \left (x^{2}+1\right )^{\frac {1}{3}} x -165888 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -1728 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} \left (x^{2}+1\right )^{\frac {1}{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 +72 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}+\left (x^{2}+1\right )^{\frac {2}{3}}-\left (x^{2}+1\right )^{\frac {1}{3}}}{x^{2}+9}\right )\)	\(616\)

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*RootOf(20736*_Z^4-144*_Z^2+1)^5*(x^2+1)^(1/3)*x-995328*RootOf(
20736*_Z^4-144*_Z^2+1)^5*x-6912*RootOf(20736*_Z^4-144*_Z^2+1)^3*(x^2+1)^(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)+432*RootOf
(20736*_Z^4-144*_Z^2+1)^2-96*RootOf(20736*_Z^4-144*_Z^2+1)*x-6*(x^2+1)^(2/3)+x^2-3)/(x^2+9))+RootOf(20736*_Z^4
-144*_Z^2+1)*ln((497664*RootOf(20736*_Z^4-144*_Z^2+1)^5*(x^2+1)^(1/3)*x-995328*RootOf(20736*_Z^4-144*_Z^2+1)^5
*x-6912*RootOf(20736*_Z^4-144*_Z^2+1)^3*(x^2+1)^(1/3)*x+20736*RootOf(20736*_Z^4-144*_Z^2+1)^3*x-144*RootOf(207
36*_Z^4-144*_Z^2+1)^2*x^2+864*RootOf(20736*_Z^4-144*_Z^2+1)^2*(x^2+1)^(1/3)+432*RootOf(20736*_Z^4-144*_Z^2+1)^
2-96*RootOf(20736*_Z^4-144*_Z^2+1)*x-6*(x^2+1)^(2/3)+x^2-3)/(x^2+9))+RootOf(20736*_Z^4-144*_Z^2+1)*ln(-(82944*
RootOf(20736*_Z^4-144*_Z^2+1)^5*(x^2+1)^(1/3)*x-165888*RootOf(20736*_Z^4-144*_Z^2+1)^5*x-1728*RootOf(20736*_Z^
4-144*_Z^2+1)^3*(x^2+1)^(1/3)*x+2304*RootOf(20736*_Z^4-144*_Z^2+1)^3*x-24*RootOf(20736*_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+72*RootOf(20
736*_Z^4-144*_Z^2+1)^2+(x^2+1)^(2/3)-(x^2+1)^(1/3))/(x^2+9))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1395 vs. \(2 (49) = 98\).
time = 4.00, size = 1395, normalized size = 19.93 \begin {gather*} \text {Too large to display} \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="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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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