3.1.94 \(\int \frac {1}{(1-x^3) \sqrt [3]{a+b x^3}} \, dx\) [94]

Optimal. Leaf size=98 \[ \frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \]

[Out]

1/6*ln(-x^3+1)/(a+b)^(1/3)-1/2*ln((a+b)^(1/3)*x-(b*x^3+a)^(1/3))/(a+b)^(1/3)+1/3*arctan(1/3*(1+2*(a+b)^(1/3)*x
/(b*x^3+a)^(1/3))*3^(1/2))/(a+b)^(1/3)*3^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {384} \begin {gather*} \frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (x \sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}+\frac {\tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - x^3)*(a + b*x^3)^(1/3)),x]

[Out]

ArcTan[(1 + (2*(a + b)^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*(a + b)^(1/3)) + Log[1 - x^3]/(6*(a + b)^
(1/3)) - Log[(a + b)^(1/3)*x - (a + b*x^3)^(1/3)]/(2*(a + b)^(1/3))

Rule 384

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

Rubi steps

\begin {align*} \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx &=\text {Subst}\left (\int \frac {1}{1-(a+b) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{a+b} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {2+\sqrt [3]{a+b} x}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )\\ &=-\frac {\log \left (1-\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt [3]{a+b}+2 (a+b)^{2/3} x}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}\\ &=-\frac {\log \left (1-\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac {\log \left (1+\frac {(a+b)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{a+b}}\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}-\frac {\log \left (1-\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac {\log \left (1+\frac {(a+b)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.87, size = 189, normalized size = 1.93 \begin {gather*} \frac {-2 \sqrt {-6+6 i \sqrt {3}} \tan ^{-1}\left (\frac {3 \sqrt [3]{a+b} x}{\sqrt {3} \sqrt [3]{a+b} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{a+b x^3}}\right )+\left (1+i \sqrt {3}\right ) \left (2 \log \left (2 \sqrt [3]{a+b} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{a+b x^3}\right )-\log \left (\left (-\sqrt [3]{a+b} x+\sqrt [3]{a+b x^3}\right ) \left (2 i \sqrt [3]{a+b} x+\left (i+\sqrt {3}\right ) \sqrt [3]{a+b x^3}\right )\right )\right )}{12 \sqrt [3]{a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x^3)*(a + b*x^3)^(1/3)),x]

[Out]

(-2*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[(3*(a + b)^(1/3)*x)/(Sqrt[3]*(a + b)^(1/3)*x - (3*I + Sqrt[3])*(a + b*x^3)
^(1/3))] + (1 + I*Sqrt[3])*(2*Log[2*(a + b)^(1/3)*x + (1 + I*Sqrt[3])*(a + b*x^3)^(1/3)] - Log[(-((a + b)^(1/3
)*x) + (a + b*x^3)^(1/3))*((2*I)*(a + b)^(1/3)*x + (I + Sqrt[3])*(a + b*x^3)^(1/3))]))/(12*(a + b)^(1/3))

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((1 - x^3)*(a + b*x^3)^(1/3)),x]')

[Out]

cought exception: maximum recursion depth exceeded

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-x^{3}+1\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^3+1)/(b*x^3+a)^(1/3),x)

[Out]

int(1/(-x^3+1)/(b*x^3+a)^(1/3),x)

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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^3+1)/(b*x^3+a)^(1/3),x, algorithm="maxima")

[Out]

-integrate(1/((b*x^3 + a)^(1/3)*(x^3 - 1)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (78) = 156\).
time = 118.52, size = 1252, normalized size = 12.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/18*(3*sqrt(1/3)*(a + b)*sqrt((-a - b)^(1/3)/(a + b))*log(-((a^3 - 27*a^2*b - 108*a*b^2 - 81*b^3)*x^9 - 3*(1
0*a^3 + 54*a^2*b + 45*a*b^2)*x^6 - 3*(17*a^3 + 18*a^2*b)*x^3 - a^3 + 9*((2*a^2 + 3*a*b)*x^7 - (a^2 + 3*a*b)*x^
4 - a^2*x)*(b*x^3 + a)^(2/3)*(-a - b)^(1/3) + 9*((a^2 + 9*a*b + 9*b^2)*x^8 + (7*a^2 + 9*a*b)*x^5 + a^2*x^2)*(b
*x^3 + a)^(1/3)*(-a - b)^(2/3) + 3*sqrt(1/3)*(3*((4*a^2 + 21*a*b + 18*b^2)*x^7 + (13*a^2 + 15*a*b)*x^4 + a^2*x
)*(b*x^3 + a)^(2/3)*(-a - b)^(2/3) + 3*((a^3 - 2*a^2*b - 12*a*b^2 - 9*b^3)*x^8 - 5*(a^3 + 4*a^2*b + 3*a*b^2)*x
^5 - 5*(a^3 + a^2*b)*x^2)*(b*x^3 + a)^(1/3) + ((a^3 + 27*a^2*b + 54*a*b^2 + 27*b^3)*x^9 + 3*(8*a^3 + 18*a^2*b
+ 9*a*b^2)*x^6 + 3*a^3*x^3 - a^3)*(-a - b)^(1/3))*sqrt((-a - b)^(1/3)/(a + b)))/(x^9 - 3*x^6 + 3*x^3 - 1)) - 2
*(-a - b)^(2/3)*log(-(3*(b*x^3 + a)^(1/3)*(a + b)*(-a - b)^(1/3)*x^2 + 3*(b*x^3 + a)^(2/3)*(a + b)*x + (a*x^3
- a)*(-a - b)^(2/3))/(x^3 - 1)) + (-a - b)^(2/3)*log((3*((2*a + 3*b)*x^4 + a*x)*(b*x^3 + a)^(2/3)*(-a - b)^(2/
3) + 3*((a^2 + 4*a*b + 3*b^2)*x^5 + 2*(a^2 + a*b)*x^2)*(b*x^3 + a)^(1/3) - ((a^2 + 9*a*b + 9*b^2)*x^6 + (7*a^2
+ 9*a*b)*x^3 + a^2)*(-a - b)^(1/3))/(x^6 - 2*x^3 + 1)))/(a + b), 1/18*(6*sqrt(1/3)*(a + b)*sqrt(-(-a - b)^(1/
3)/(a + b))*arctan(sqrt(1/3)*(6*((2*a^2 + 3*a*b)*x^7 - (a^2 + 3*a*b)*x^4 - a^2*x)*(b*x^3 + a)^(2/3)*(-a - b)^(
2/3) - 6*((a^3 + 10*a^2*b + 18*a*b^2 + 9*b^3)*x^8 + (7*a^3 + 16*a^2*b + 9*a*b^2)*x^5 + (a^3 + a^2*b)*x^2)*(b*x
^3 + a)^(1/3) - ((a^3 - 9*a^2*b - 36*a*b^2 - 27*b^3)*x^9 - 3*(4*a^3 + 18*a^2*b + 15*a*b^2)*x^6 - 3*(5*a^3 + 6*
a^2*b)*x^3 - a^3)*(-a - b)^(1/3))*sqrt(-(-a - b)^(1/3)/(a + b))/((a^3 + 27*a^2*b + 54*a*b^2 + 27*b^3)*x^9 + 3*
(8*a^3 + 18*a^2*b + 9*a*b^2)*x^6 + 3*a^3*x^3 - a^3)) - 2*(-a - b)^(2/3)*log(-(3*(b*x^3 + a)^(1/3)*(a + b)*(-a
- b)^(1/3)*x^2 + 3*(b*x^3 + a)^(2/3)*(a + b)*x + (a*x^3 - a)*(-a - b)^(2/3))/(x^3 - 1)) + (-a - b)^(2/3)*log((
3*((2*a + 3*b)*x^4 + a*x)*(b*x^3 + a)^(2/3)*(-a - b)^(2/3) + 3*((a^2 + 4*a*b + 3*b^2)*x^5 + 2*(a^2 + a*b)*x^2)
*(b*x^3 + a)^(1/3) - ((a^2 + 9*a*b + 9*b^2)*x^6 + (7*a^2 + 9*a*b)*x^3 + a^2)*(-a - b)^(1/3))/(x^6 - 2*x^3 + 1)
))/(a + b)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{x^{3} \sqrt [3]{a + b x^{3}} - \sqrt [3]{a + b x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**3+1)/(b*x**3+a)**(1/3),x)

[Out]

-Integral(1/(x**3*(a + b*x**3)**(1/3) - (a + b*x**3)**(1/3)), x)

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Giac [F] N/A
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^3+1)/(b*x^3+a)^(1/3),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{\left (x^3-1\right )\,{\left (b\,x^3+a\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((x^3 - 1)*(a + b*x^3)^(1/3)),x)

[Out]

-int(1/((x^3 - 1)*(a + b*x^3)^(1/3)), x)

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