3.18 \(\int \frac{1}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=288 \[ -\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} (b c-a d)}+\frac{d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)}-\frac{d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)}+\frac{d^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.320915, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} (b c-a d)}+\frac{d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)}-\frac{d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)}+\frac{d^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 68.2094, size = 260, normalized size = 0.9 \[ \frac{d^{\frac{2}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} \left (a d - b c\right )} - \frac{d^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} d^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} \left (a d - b c\right )} - \frac{b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} \left (a d - b c\right )} + \frac{b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**3+a)/(d*x**3+c),x)

[Out]

d**(2/3)*log(c**(1/3) + d**(1/3)*x)/(3*c**(2/3)*(a*d - b*c)) - d**(2/3)*log(c**(
2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*c**(2/3)*(a*d - b*c)) - sqrt(3)*d
**(2/3)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(2/3)*(a*d -
b*c)) - b**(2/3)*log(a**(1/3) + b**(1/3)*x)/(3*a**(2/3)*(a*d - b*c)) + b**(2/3)*
log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(2/3)*(a*d - b*c)) + s
qrt(3)*b**(2/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(2/3)
*(a*d - b*c))

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Mathematica [A]  time = 0.219728, size = 224, normalized size = 0.78 \[ \frac{\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}-\frac{2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{2/3}}+\frac{2 d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{2/3}}-\frac{2 \sqrt{3} d^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{2/3}}}{6 a d-6 b c} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^3)*(c + d*x^3)),x]

[Out]

((2*Sqrt[3]*b^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) - (2*Sq
rt[3]*d^(2/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(2/3) - (2*b^(2/3)*
Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (2*d^(2/3)*Log[c^(1/3) + d^(1/3)*x])/c^(2/3)
+ (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3) - (d^(2/3)*L
og[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(2/3))/(-6*b*c + 6*a*d)

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Maple [A]  time = 0.009, size = 222, normalized size = 0.8 \[ -{\frac{1}{3\,ad-3\,bc}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{6\,ad-6\,bc}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}}{3\,ad-3\,bc}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{3\,ad-3\,bc}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,ad-6\,bc}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,ad-3\,bc}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279112, size = 377, normalized size = 1.31 \[ \frac{\sqrt{3}{\left (\sqrt{3} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) + \sqrt{3} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} + c^{2} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}\right ) + 6 \, \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 6 \, \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}{3 \, c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}\right )\right )}}{18 \,{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*(sqrt(3)*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^
2*(-b^2/a^2)^(2/3)) + sqrt(3)*(d^2/c^2)^(1/3)*log(d^2*x^2 - c*d*x*(d^2/c^2)^(1/3
) + c^2*(d^2/c^2)^(2/3)) - 2*sqrt(3)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/
3)) - 2*sqrt(3)*(d^2/c^2)^(1/3)*log(d*x + c*(d^2/c^2)^(1/3)) + 6*(-b^2/a^2)^(1/3
)*arctan(1/3*(2*sqrt(3)*b*x + sqrt(3)*a*(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3)))
+ 6*(d^2/c^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*d*x - sqrt(3)*c*(d^2/c^2)^(1/3))/(c*(
d^2/c^2)^(1/3))))/(b*c - a*d)

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Sympy [A]  time = 104.613, size = 447, normalized size = 1.55 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{5} d^{3} - 81 a^{4} b c d^{2} + 81 a^{3} b^{2} c^{2} d - 27 a^{2} b^{3} c^{3}\right ) + b^{2}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{4} a^{7} c^{2} d^{5} - 243 t^{4} a^{6} b c^{3} d^{4} + 162 t^{4} a^{5} b^{2} c^{4} d^{3} + 162 t^{4} a^{4} b^{3} c^{5} d^{2} - 243 t^{4} a^{3} b^{4} c^{6} d + 81 t^{4} a^{2} b^{5} c^{7} - 3 t a^{4} d^{4} + 3 t a^{3} b c d^{3} + 3 t a b^{3} c^{3} d - 3 t b^{4} c^{4}}{a^{2} b d^{3} + b^{3} c^{2} d} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} c^{2} d^{3} - 81 a^{2} b c^{3} d^{2} + 81 a b^{2} c^{4} d - 27 b^{3} c^{5}\right ) - d^{2}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{4} a^{7} c^{2} d^{5} - 243 t^{4} a^{6} b c^{3} d^{4} + 162 t^{4} a^{5} b^{2} c^{4} d^{3} + 162 t^{4} a^{4} b^{3} c^{5} d^{2} - 243 t^{4} a^{3} b^{4} c^{6} d + 81 t^{4} a^{2} b^{5} c^{7} - 3 t a^{4} d^{4} + 3 t a^{3} b c d^{3} + 3 t a b^{3} c^{3} d - 3 t b^{4} c^{4}}{a^{2} b d^{3} + b^{3} c^{2} d} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**3+a)/(d*x**3+c),x)

[Out]

RootSum(_t**3*(27*a**5*d**3 - 81*a**4*b*c*d**2 + 81*a**3*b**2*c**2*d - 27*a**2*b
**3*c**3) + b**2, Lambda(_t, _t*log(x + (81*_t**4*a**7*c**2*d**5 - 243*_t**4*a**
6*b*c**3*d**4 + 162*_t**4*a**5*b**2*c**4*d**3 + 162*_t**4*a**4*b**3*c**5*d**2 -
243*_t**4*a**3*b**4*c**6*d + 81*_t**4*a**2*b**5*c**7 - 3*_t*a**4*d**4 + 3*_t*a**
3*b*c*d**3 + 3*_t*a*b**3*c**3*d - 3*_t*b**4*c**4)/(a**2*b*d**3 + b**3*c**2*d))))
+ RootSum(_t**3*(27*a**3*c**2*d**3 - 81*a**2*b*c**3*d**2 + 81*a*b**2*c**4*d - 2
7*b**3*c**5) - d**2, Lambda(_t, _t*log(x + (81*_t**4*a**7*c**2*d**5 - 243*_t**4*
a**6*b*c**3*d**4 + 162*_t**4*a**5*b**2*c**4*d**3 + 162*_t**4*a**4*b**3*c**5*d**2
- 243*_t**4*a**3*b**4*c**6*d + 81*_t**4*a**2*b**5*c**7 - 3*_t*a**4*d**4 + 3*_t*
a**3*b*c*d**3 + 3*_t*a*b**3*c**3*d - 3*_t*b**4*c**4)/(a**2*b*d**3 + b**3*c**2*d)
)))

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GIAC/XCAS [A]  time = 0.227861, size = 375, normalized size = 1.3 \[ -\frac{b \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b c - a^{2} d\right )}} + \frac{d \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} - a c d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a b c - \sqrt{3} a^{2} d} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{2} - \sqrt{3} a c d} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b c - a^{2} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{2} - a c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="giac")

[Out]

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