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]

1/4*arctan(c^(1/6)*(c^(1/3)-2^(1/3)*(-3*d*x^2+c)^(1/3))/x/d^(1/2))*2^(1/3)/c^(5/6)/d^(1/2)+1/4*arctan(1/x/d^(1
/2)*c^(1/2))*2^(1/3)/c^(5/6)/d^(1/2)-1/12*arctanh(x*3^(1/2)*d^(1/2)/c^(1/2))*2^(1/3)/c^(5/6)*3^(1/2)/d^(1/2)+1
/4*arctanh(x*3^(1/2)*d^(1/2)/c^(1/6)/(c^(1/3)+2^(1/3)*(-3*d*x^2+c)^(1/3)))*3^(1/2)*2^(1/3)/c^(5/6)/d^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 204, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.16, 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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x^{2} + c\right )} {\left (-3 \, d x^{2} + c\right )}^{\frac {1}{3}}}\,{d x} \]

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)

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maple [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-3 d \,x^{2}+c \right )^{\frac {1}{3}} \left (d \,x^{2}+c \right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x^{2} + c\right )} {\left (-3 \, d x^{2} + c\right )}^{\frac {1}{3}}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{c - 3 d x^{2}} \left (c + d x^{2}\right )}\, dx \]

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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