3.145 \(\int \frac{1}{\sqrt [3]{c-3 d x^2} (c+d x^2)} \, dx\)

Optimal. Leaf size=204 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c-3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}} \]

[Out]

ArcTan[Sqrt[c]/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) + ArcTan[(c^(1/6)*(c^(1/3) - 2^(1/3)*(c - 3*d*x^2)^(1/
3)))/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - ArcTanh[(Sqrt[3]*Sqrt[d]*x)/Sqrt[c]]/(2*2^(2/3)*Sqrt[3]*c^(5/6
)*Sqrt[d]) + (Sqrt[3]*ArcTanh[(Sqrt[3]*Sqrt[d]*x)/(c^(1/6)*(c^(1/3) + 2^(1/3)*(c - 3*d*x^2)^(1/3)))])/(2*2^(2/
3)*c^(5/6)*Sqrt[d])

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Rubi [A]  time = 0.0287804, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {393} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c-3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c - 3*d*x^2)^(1/3)*(c + d*x^2)),x]

[Out]

ArcTan[Sqrt[c]/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) + ArcTan[(c^(1/6)*(c^(1/3) - 2^(1/3)*(c - 3*d*x^2)^(1/
3)))/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - ArcTanh[(Sqrt[3]*Sqrt[d]*x)/Sqrt[c]]/(2*2^(2/3)*Sqrt[3]*c^(5/6
)*Sqrt[d]) + (Sqrt[3]*ArcTanh[(Sqrt[3]*Sqrt[d]*x)/(c^(1/6)*(c^(1/3) + 2^(1/3)*(c - 3*d*x^2)^(1/3)))])/(2*2^(2/
3)*c^(5/6)*Sqrt[d])

Rule 393

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b/a), 2]}, Simp[(q*ArcT
an[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (Simp[(q*ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a +
b*x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] - Simp[(q*ArcTanh[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[(
Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{c-3 d x^2} \left (c+d x^2\right )} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}\\ \end{align*}

Mathematica [C]  time = 0.15142, size = 156, normalized size = 0.76 \[ \frac{3 c x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )}{\sqrt [3]{c-3 d x^2} \left (c+d x^2\right ) \left (2 d x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )\right )+3 c F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((c - 3*d*x^2)^(1/3)*(c + d*x^2)),x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}+c}{\frac{1}{\sqrt [3]{-3\,d{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*d*x^2+c)^(1/3)/(d*x^2+c),x)

[Out]

int(1/(-3*d*x^2+c)^(1/3)/(d*x^2+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x^{2} + c\right )}{\left (-3 \, d x^{2} + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*d*x^2+c)^(1/3)/(d*x^2+c),x, algorithm="maxima")

[Out]

integrate(1/((d*x^2 + c)*(-3*d*x^2 + c)^(1/3)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*d*x^2+c)^(1/3)/(d*x^2+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{c - 3 d x^{2}} \left (c + d x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*d*x**2+c)**(1/3)/(d*x**2+c),x)

[Out]

Integral(1/((c - 3*d*x**2)**(1/3)*(c + d*x**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x^{2} + c\right )}{\left (-3 \, d x^{2} + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*d*x^2+c)^(1/3)/(d*x^2+c),x, algorithm="giac")

[Out]

integrate(1/((d*x^2 + c)*(-3*d*x^2 + c)^(1/3)), x)