3.160 \(\int \frac{1}{\sqrt [3]{1+x^2} (9+x^2)} \, dx\)
Optimal. Leaf size=70 \[ \frac{1}{12} \tan ^{-1}\left (\frac{\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt{3}}+\frac{1}{12} \tan ^{-1}\left (\frac{x}{3}\right ) \]
[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])
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Rubi [A] time = 0.0088051, antiderivative size = 70, normalized size of antiderivative = 1.,
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} \[ \frac{1}{12} \tan ^{-1}\left (\frac{\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt{3}}+\frac{1}{12} \tan ^{-1}\left (\frac{x}{3}\right ) \]
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*}
Mathematica [C] time = 0.0934875, size = 124, normalized size = 1.77 \[ -\frac{27 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,-\frac{x^2}{9}\right )}{\sqrt [3]{x^2+1} \left (x^2+9\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,-\frac{x^2}{9}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,-\frac{x^2}{9}\right )\right )-27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,-\frac{x^2}{9}\right )\right )} \]
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, -x^2/9])/((1 + x^2)^(1/3)*(9 + x^2)*(-27*AppellF1[1/2, 1/3, 1, 3/2, -x
^2, -x^2/9] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -x^2, -x^2/9] + 3*AppellF1[3/2, 4/3, 1, 5/2, -x^2, -x^2/9])))
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}+9}{\frac{1}{\sqrt [3]{{x}^{2}+1}}}}\, dx \end{align*}
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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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} + 9\right )}{\left (x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
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] time = 14.6608, size = 4397, normalized size = 62.81 \begin{align*} \text{result too large to display} \end{align*}
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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{x^{2} + 1} \left (x^{2} + 9\right )}\, dx \end{align*}
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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} + 9\right )}{\left (x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
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)