\(\int \frac {1}{\sqrt [3]{1+b x^2} (9+b x^2)} \, dx\) [161]

Optimal result

Integrand size = 21, antiderivative size = 104 \[ \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}}+\frac {\arctan \left (\frac {\left (1-\sqrt [3]{1+b x^2}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1+b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}} \]

[Out]

1/12*arctan(1/3*(1-(b*x^2+1)^(1/3))^2/x/b^(1/2))/b^(1/2)+1/12*arctan(1/3*x*b^(1/2))/b^(1/2)-1/12*arctanh((1-(b
*x^2+1)^(1/3))*3^(1/2)/x/b^(1/2))*3^(1/2)/b^(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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {403} \[ \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx=\frac {\arctan \left (\frac {\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}} \]

[In]

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

[Out]

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

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*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{1+b x^2}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1+b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 4.79 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx=-\frac {27 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-b x^2,-\frac {b x^2}{9}\right )}{\sqrt [3]{1+b x^2} \left (9+b x^2\right ) \left (-27 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-b x^2,-\frac {b x^2}{9}\right )+2 b x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-b x^2,-\frac {b x^2}{9}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-b x^2,-\frac {b x^2}{9}\right )\right )\right )} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {1}{\left (b \,x^{2}+1\right )^{\frac {1}{3}} \left (b \,x^{2}+9\right )}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx=\int \frac {1}{\sqrt [3]{b x^{2} + 1} \left (b x^{2} + 9\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + 9\right )} {\left (b x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + 9\right )} {\left (b x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+1\right )}^{1/3}\,\left (b\,x^2+9\right )} \,d x \]

[In]

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

[Out]

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