3.72 \(\int \frac{x}{\left (4-d x^3\right ) \sqrt{-1+d x^3}} \, dx\)

Optimal. Leaf size=157 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{d} x+1}{\sqrt{d x^3-1}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac{\tan ^{-1}\left (\sqrt{d x^3-1}\right )}{9\ 2^{2/3} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{d x^3-1}}\right )}{3\ 2^{2/3} \sqrt{3} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d x^3-1}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3} d^{2/3}} \]

[Out]

-ArcTan[(1 + 2^(1/3)*d^(1/3)*x)/Sqrt[-1 + d*x^3]]/(3*2^(2/3)*d^(2/3)) - ArcTan[S
qrt[-1 + d*x^3]]/(9*2^(2/3)*d^(2/3)) - ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*d^(1/3)*x))
/Sqrt[-1 + d*x^3]]/(3*2^(2/3)*Sqrt[3]*d^(2/3)) - ArcTanh[Sqrt[-1 + d*x^3]/Sqrt[3
]]/(3*2^(2/3)*Sqrt[3]*d^(2/3))

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Rubi [A]  time = 0.100935, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{d} x+1}{\sqrt{d x^3-1}}\right )}{3\ 2^{2/3} d^{2/3}}-\frac{\tan ^{-1}\left (\sqrt{d x^3-1}\right )}{9\ 2^{2/3} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{d} x\right )}{\sqrt{d x^3-1}}\right )}{3\ 2^{2/3} \sqrt{3} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d x^3-1}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3} d^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x/((4 - d*x^3)*Sqrt[-1 + d*x^3]),x]

[Out]

-ArcTan[(1 + 2^(1/3)*d^(1/3)*x)/Sqrt[-1 + d*x^3]]/(3*2^(2/3)*d^(2/3)) - ArcTan[S
qrt[-1 + d*x^3]]/(9*2^(2/3)*d^(2/3)) - ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*d^(1/3)*x))
/Sqrt[-1 + d*x^3]]/(3*2^(2/3)*Sqrt[3]*d^(2/3)) - ArcTanh[Sqrt[-1 + d*x^3]/Sqrt[3
]]/(3*2^(2/3)*Sqrt[3]*d^(2/3))

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Rubi in Sympy [A]  time = 4.53822, size = 223, normalized size = 1.42 \[ \frac{\sqrt [3]{2} i \log{\left (\sqrt [3]{2} \sqrt [3]{d} x - i \sqrt{d x^{3} - 1} + 1 \right )}}{12 d^{\frac{2}{3}}} - \frac{\sqrt [3]{2} i \log{\left (\sqrt [3]{2} \sqrt [3]{d} x + i \sqrt{d x^{3} - 1} + 1 \right )}}{12 d^{\frac{2}{3}}} + \frac{\sqrt [3]{2} \sqrt{3} i \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} i \left (- \sqrt{d x^{3} - 1} + i\right )}{3 \sqrt [3]{d} x} \right )}}{18 d^{\frac{2}{3}}} - \frac{\sqrt [3]{2} \sqrt{3} i \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} i \left (\sqrt{d x^{3} - 1} + i\right )}{3 \sqrt [3]{d} x} \right )}}{18 d^{\frac{2}{3}}} - \frac{\sqrt [3]{2} \operatorname{atan}{\left (\sqrt{d x^{3} - 1} \right )}}{18 d^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-d*x**3+4)/(d*x**3-1)**(1/2),x)

[Out]

2**(1/3)*I*log(2**(1/3)*d**(1/3)*x - I*sqrt(d*x**3 - 1) + 1)/(12*d**(2/3)) - 2**
(1/3)*I*log(2**(1/3)*d**(1/3)*x + I*sqrt(d*x**3 - 1) + 1)/(12*d**(2/3)) + 2**(1/
3)*sqrt(3)*I*atan(sqrt(3)/3 + 2**(2/3)*sqrt(3)*I*(-sqrt(d*x**3 - 1) + I)/(3*d**(
1/3)*x))/(18*d**(2/3)) - 2**(1/3)*sqrt(3)*I*atan(sqrt(3)/3 + 2**(2/3)*sqrt(3)*I*
(sqrt(d*x**3 - 1) + I)/(3*d**(1/3)*x))/(18*d**(2/3)) - 2**(1/3)*atan(sqrt(d*x**3
- 1))/(18*d**(2/3))

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Mathematica [C]  time = 0.249286, size = 135, normalized size = 0.86 \[ -\frac{10 x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};d x^3,\frac{d x^3}{4}\right )}{\left (d x^3-4\right ) \sqrt{d x^3-1} \left (3 d x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};d x^3,\frac{d x^3}{4}\right )+2 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};d x^3,\frac{d x^3}{4}\right )\right )+20 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};d x^3,\frac{d x^3}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/((4 - d*x^3)*Sqrt[-1 + d*x^3]),x]

[Out]

(-10*x^2*AppellF1[2/3, 1/2, 1, 5/3, d*x^3, (d*x^3)/4])/((-4 + d*x^3)*Sqrt[-1 + d
*x^3]*(20*AppellF1[2/3, 1/2, 1, 5/3, d*x^3, (d*x^3)/4] + 3*d*x^3*(AppellF1[5/3,
1/2, 2, 8/3, d*x^3, (d*x^3)/4] + 2*AppellF1[5/3, 3/2, 1, 8/3, d*x^3, (d*x^3)/4])
))

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Maple [C]  time = 0.153, size = 240, normalized size = 1.5 \[ -{\frac{i}{9}}\sqrt{2}\sum _{{\it \_alpha}={\it RootOf} \left ( d{{\it \_Z}}^{3}-4 \right ) }{\frac{1}{{\it \_alpha}}\sqrt{-{\frac{i}{2}} \left ( 2\,x+{\frac{1}{\sqrt [3]{d}}}+{i\sqrt{3}{\frac{1}{\sqrt [3]{d}}}} \right ) \sqrt [3]{d}}\sqrt{{1 \left ( x-{\frac{1}{\sqrt [3]{d}}} \right ) \left ( -3\,{\frac{1}{\sqrt [3]{d}}}-{i\sqrt{3}{\frac{1}{\sqrt [3]{d}}}} \right ) ^{-1}}}\sqrt{{\frac{i}{2}} \left ( 2\,x+{\frac{1}{\sqrt [3]{d}}}-{i\sqrt{3}{\frac{1}{\sqrt [3]{d}}}} \right ) \sqrt [3]{d}} \left ( -2\,{{\it \_alpha}}^{2}d+i\sqrt{3}{\it \_alpha}\,{d}^{{\frac{2}{3}}}+{\it \_alpha}\,{d}^{{\frac{2}{3}}}-i\sqrt{3}\sqrt [3]{d}+\sqrt [3]{d} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{-i \left ( x+{\frac{1}{2}{\frac{1}{\sqrt [3]{d}}}}+{{\frac{i}{2}}\sqrt{3}{\frac{1}{\sqrt [3]{d}}}} \right ) \sqrt{3}\sqrt [3]{d}}},{\frac{i}{3}}{{\it \_alpha}}^{2}\sqrt{3}{d}^{{\frac{2}{3}}}-{\frac{i}{6}}{\it \_alpha}\,\sqrt{3}\sqrt [3]{d}+{\frac{{\it \_alpha}}{2}\sqrt [3]{d}}-{\frac{i}{6}}\sqrt{3}-{\frac{1}{2}},\sqrt{{-i\sqrt{3}{\frac{1}{\sqrt [3]{d}}} \left ( -{\frac{3}{2}{\frac{1}{\sqrt [3]{d}}}}-{{\frac{i}{2}}\sqrt{3}{\frac{1}{\sqrt [3]{d}}}} \right ) ^{-1}}} \right ){d}^{-{\frac{4}{3}}}{\frac{1}{\sqrt{d{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-d*x^3+4)/(d*x^3-1)^(1/2),x)

[Out]

-1/9*I*2^(1/2)*sum(1/_alpha/d^(4/3)*(-1/2*I*(2*x+1/d^(1/3)+I*3^(1/2)/d^(1/3))*d^
(1/3))^(1/2)*((x-1/d^(1/3))/(-3/d^(1/3)-I*3^(1/2)/d^(1/3)))^(1/2)*(1/2*I*(2*x+1/
d^(1/3)-I*3^(1/2)/d^(1/3))*d^(1/3))^(1/2)/(d*x^3-1)^(1/2)*(-2*_alpha^2*d+I*3^(1/
2)*_alpha*d^(2/3)+_alpha*d^(2/3)-I*3^(1/2)*d^(1/3)+d^(1/3))*EllipticPi(1/3*3^(1/
2)*(-I*(x+1/2/d^(1/3)+1/2*I*3^(1/2)/d^(1/3))*3^(1/2)*d^(1/3))^(1/2),1/3*I*_alpha
^2*3^(1/2)*d^(2/3)-1/6*I*_alpha*3^(1/2)*d^(1/3)+1/2*_alpha*d^(1/3)-1/6*I*3^(1/2)
-1/2,(-I*3^(1/2)/d^(1/3)/(-3/2/d^(1/3)-1/2*I*3^(1/2)/d^(1/3)))^(1/2)),_alpha=Roo
tOf(_Z^3*d-4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x}{\sqrt{d x^{3} - 1}{\left (d x^{3} - 4\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(d*x^3 - 1)*(d*x^3 - 4)),x, algorithm="maxima")

[Out]

-integrate(x/(sqrt(d*x^3 - 1)*(d*x^3 - 4)), x)

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Fricas [A]  time = 0.460739, size = 2641, normalized size = 16.82 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(d*x^3 - 1)*(d*x^3 - 4)),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*(1/432)^(1/6)*(d^(-4))^(1/6)*arctan(6*(4*sqrt(3)*(1/2)^(2/3)*(d^5*x^
7 + d^4*x^4 - 2*d^3*x)*(d^(-4))^(2/3) - sqrt(3)*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5
- 8*d^2*x^2)*(d^(-4))^(1/3) + (648*sqrt(3)*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5
- sqrt(3)*(1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt(d*x
^3 - 1))/(d^3*x^9 + 66*d^2*x^6 - 72*d*x^3 - 24*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4 -
2*d^3*x)*(d^(-4))^(2/3) - 6*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4
))^(1/3) + 6*(648*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(5*d^4*x^6 +
20*d^3*x^3 - 16*d^2)*sqrt(d^(-4)) + (1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x)
*(d^(-4))^(1/6))*sqrt(d*x^3 - 1) + (d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)*sqrt((
d^3*x^9 - 60*d^2*x^6 - 24*(1/2)^(2/3)*(d^5*x^7 - 5*d^4*x^4 + 4*d^3*x)*(d^(-4))^(
2/3) + 12*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) - 12*(648
*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(d^4*x^6 + 16*d^3*x^3 - 8*d^2)
*sqrt(d^(-4)) - (1/432)^(1/6)*(d^3*x^7 - 2*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt
(d*x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)) + 32)) + 1/9*sqrt(3)*(
1/432)^(1/6)*(d^(-4))^(1/6)*arctan(-6*(4*sqrt(3)*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4
- 2*d^3*x)*(d^(-4))^(2/3) - sqrt(3)*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2
)*(d^(-4))^(1/3) - (648*sqrt(3)*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(3)*(
1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt(d*x^3 - 1))/(d^
3*x^9 + 66*d^2*x^6 - 72*d*x^3 - 24*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4 - 2*d^3*x)*(d^
(-4))^(2/3) - 6*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) - 6
*(648*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(5*d^4*x^6 + 20*d^3*x^3 -
16*d^2)*sqrt(d^(-4)) + (1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x)*(d^(-4))^(1
/6))*sqrt(d*x^3 - 1) + (d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)*sqrt((d^3*x^9 - 60
*d^2*x^6 - 24*(1/2)^(2/3)*(d^5*x^7 - 5*d^4*x^4 + 4*d^3*x)*(d^(-4))^(2/3) + 12*(1
/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) + 12*(648*(1/432)^(5/
6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(d^4*x^6 + 16*d^3*x^3 - 8*d^2)*sqrt(d^(-4)
) - (1/432)^(1/6)*(d^3*x^7 - 2*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt(d*x^3 - 1)
+ 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)) + 32)) + 1/18*(1/432)^(1/6)*(d^(-4
))^(1/6)*log((d^3*x^9 + 66*d^2*x^6 - 72*d*x^3 + 48*(1/2)^(2/3)*(d^5*x^7 + d^4*x^
4 - 2*d^3*x)*(d^(-4))^(2/3) + 12*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(
d^(-4))^(1/3) + 6*(1296*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 + sqrt(1/3)*(5*d^4*
x^6 + 20*d^3*x^3 - 16*d^2)*sqrt(d^(-4)) + 2*(1/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4
- 8*d*x)*(d^(-4))^(1/6))*sqrt(d*x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3
- 64)) - 1/18*(1/432)^(1/6)*(d^(-4))^(1/6)*log((d^3*x^9 + 66*d^2*x^6 - 72*d*x^3
+ 48*(1/2)^(2/3)*(d^5*x^7 + d^4*x^4 - 2*d^3*x)*(d^(-4))^(2/3) + 12*(1/2)^(1/3)*(
d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) - 6*(1296*(1/432)^(5/6)*d^5*(d^(
-4))^(5/6)*x^5 + sqrt(1/3)*(5*d^4*x^6 + 20*d^3*x^3 - 16*d^2)*sqrt(d^(-4)) + 2*(1
/432)^(1/6)*(d^3*x^7 + 16*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sqrt(d*x^3 - 1) + 32)
/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)) - 1/36*(1/432)^(1/6)*(d^(-4))^(1/6)*log
((d^3*x^9 - 60*d^2*x^6 - 24*(1/2)^(2/3)*(d^5*x^7 - 5*d^4*x^4 + 4*d^3*x)*(d^(-4))
^(2/3) + 12*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^(-4))^(1/3) + 12*(6
48*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(d^4*x^6 + 16*d^3*x^3 - 8*d^
2)*sqrt(d^(-4)) - (1/432)^(1/6)*(d^3*x^7 - 2*d^2*x^4 - 8*d*x)*(d^(-4))^(1/6))*sq
rt(d*x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64)) + 1/36*(1/432)^(1/6)
*(d^(-4))^(1/6)*log((d^3*x^9 - 60*d^2*x^6 - 24*(1/2)^(2/3)*(d^5*x^7 - 5*d^4*x^4
+ 4*d^3*x)*(d^(-4))^(2/3) + 12*(1/2)^(1/3)*(d^4*x^8 + 7*d^3*x^5 - 8*d^2*x^2)*(d^
(-4))^(1/3) - 12*(648*(1/432)^(5/6)*d^5*(d^(-4))^(5/6)*x^5 - sqrt(1/3)*(d^4*x^6
+ 16*d^3*x^3 - 8*d^2)*sqrt(d^(-4)) - (1/432)^(1/6)*(d^3*x^7 - 2*d^2*x^4 - 8*d*x)
*(d^(-4))^(1/6))*sqrt(d*x^3 - 1) + 32)/(d^3*x^9 - 12*d^2*x^6 + 48*d*x^3 - 64))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{d x^{3} \sqrt{d x^{3} - 1} - 4 \sqrt{d x^{3} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-d*x**3+4)/(d*x**3-1)**(1/2),x)

[Out]

-Integral(x/(d*x**3*sqrt(d*x**3 - 1) - 4*sqrt(d*x**3 - 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x}{\sqrt{d x^{3} - 1}{\left (d x^{3} - 4\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x/(sqrt(d*x^3 - 1)*(d*x^3 - 4)),x, algorithm="giac")

[Out]

integrate(-x/(sqrt(d*x^3 - 1)*(d*x^3 - 4)), x)