3.2.15 \(\int \frac {1}{(a+b x^3) (c+d x^3)} \, dx\) [115]

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

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {400, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {b^{2/3} \text {ArcTan}\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 {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 {d^{2/3} \text {ArcTan}\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)}+\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)} \end {gather*}

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)*Log[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))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 400

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 224, normalized size = 0.78 \begin {gather*} \frac {\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 {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}}-\frac {2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac {2 d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{2/3}}+\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 {d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{2/3}}}{-6 b c+6 a d} \end {gather*}

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*Sqrt[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)*Log[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 = 0.29, size = 207, normalized size = 0.72

method	result	size
default	\(-\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a d -b c}+\frac {\left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right ) d}{a d -b c}\)	\(207\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{5} d^{3}-3 a^{4} b c \,d^{2}+3 a^{3} b^{2} c^{2} d -b^{3} a^{2} c^{3}\right ) \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{5} d^{5}+3 a^{4} b c \,d^{4}-2 a^{3} b^{2} c^{2} d^{3}-2 a^{2} b^{3} c^{3} d^{2}+3 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \textit {\_R}^{3}-2 b^{2} d^{2}\right ) x +\left (-a^{6} c \,d^{5}+3 a^{5} b \,c^{2} d^{4}-2 a^{4} b^{2} c^{3} d^{3}-2 a^{3} b^{3} c^{4} d^{2}+3 a^{2} b^{4} c^{5} d -a \,c^{6} b^{5}\right ) \textit {\_R}^{4}+\left (a^{2} b \,d^{3}-2 a \,b^{2} c \,d^{2}+b^{3} c^{2} d \right ) \textit {\_R} \right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{3} c^{2} d^{3}-3 a^{2} b \,c^{3} d^{2}+3 a \,b^{2} c^{4} d -b^{3} c^{5}\right ) \textit {\_Z}^{3}-d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{5} d^{5}+3 a^{4} b c \,d^{4}-2 a^{3} b^{2} c^{2} d^{3}-2 a^{2} b^{3} c^{3} d^{2}+3 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \textit {\_R}^{3}-2 b^{2} d^{2}\right ) x +\left (-a^{6} c \,d^{5}+3 a^{5} b \,c^{2} d^{4}-2 a^{4} b^{2} c^{3} d^{3}-2 a^{3} b^{3} c^{4} d^{2}+3 a^{2} b^{4} c^{5} d -a \,c^{6} b^{5}\right ) \textit {\_R}^{4}+\left (a^{2} b \,d^{3}-2 a \,b^{2} c \,d^{2}+b^{3} c^{2} d \right ) \textit {\_R} \right )\right )}{3}\)	\(490\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)/(d*x^3+c),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 293, normalized size = 1.02 \begin {gather*} \frac {\sqrt {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 )}{3 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\sqrt {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 )}{3 \, {\left (b c \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {\log \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 \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {\log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} \end {gather*}

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]

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

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Fricas [A]
time = 2.06, size = 254, normalized size = 0.88 \begin {gather*} -\frac {2 \, \sqrt {3} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 2 \, \sqrt {3} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} c x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) - \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 ) - \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 \, \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 \, \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 (b c - a d\right )}} \end {gather*}

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]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.70, size = 278, normalized size = 0.97 \begin {gather*} -\frac {b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \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}} \log \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}} \log \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}} \log \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 )}} \end {gather*}

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

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Mupad [B]
time = 9.01, size = 1364, normalized size = 4.74 \begin {gather*} \ln \left (\frac {{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (9\,a^2\,b^4\,d^6+9\,b^6\,c^2\,d^4-18\,a\,b^5\,c\,d^5-9\,b^3\,d^3\,\left (x+a\,c\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}\right )}{3}-6\,b^5\,d^5\,x\right )\,{\left (-\frac {b^2}{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 )}^{1/3}+\ln \left (\frac {{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (9\,a^2\,b^4\,d^6+9\,b^6\,c^2\,d^4-18\,a\,b^5\,c\,d^5-9\,b^3\,d^3\,\left (x+a\,c\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}\right )}{3}-6\,b^5\,d^5\,x\right )\,{\left (-\frac {d^2}{-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 )}^{1/3}+\frac {\ln \left (6\,b^5\,d^5\,x+\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (81\,b^3\,d^3\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4+\frac {81\,a\,b^3\,c\,d^3\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}}{2}\right )\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{36}-9\,a^2\,b^4\,d^6-9\,b^6\,c^2\,d^4+18\,a\,b^5\,c\,d^5\right )}{6}\right )\,{\left (-\frac {b^2}{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 )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (6\,b^5\,d^5\,x-\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (81\,b^3\,d^3\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4-\frac {81\,a\,b^3\,c\,d^3\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}}{2}\right )\,{\left (-\frac {b^2}{a^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{36}-9\,a^2\,b^4\,d^6-9\,b^6\,c^2\,d^4+18\,a\,b^5\,c\,d^5\right )}{6}\right )\,{\left (-\frac {b^2}{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 )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+\frac {\ln \left (6\,b^5\,d^5\,x+\frac {\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (81\,b^3\,d^3\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4+\frac {81\,a\,b^3\,c\,d^3\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}}{2}\right )\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{36}-9\,a^2\,b^4\,d^6-9\,b^6\,c^2\,d^4+18\,a\,b^5\,c\,d^5\right )}{6}\right )\,{\left (-\frac {d^2}{-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 )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (6\,b^5\,d^5\,x-\frac {\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\,\left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (81\,b^3\,d^3\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4-\frac {81\,a\,b^3\,c\,d^3\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^4\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}}{2}\right )\,{\left (\frac {d^2}{c^2\,{\left (a\,d-b\,c\right )}^3}\right )}^{2/3}}{36}-9\,a^2\,b^4\,d^6-9\,b^6\,c^2\,d^4+18\,a\,b^5\,c\,d^5\right )}{6}\right )\,{\left (-\frac {d^2}{-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 )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^3)*(c + d*x^3)),x)

[Out]

log(((-b^2/(a^2*(a*d - b*c)^3))^(1/3)*(9*a^2*b^4*d^6 + 9*b^6*c^2*d^4 - 18*a*b^5*c*d^5 - 9*b^3*d^3*(x + a*c*(-b
^2/(a^2*(a*d - b*c)^3))^(1/3))*(a*d + b*c)*(a*d - b*c)^4*(-b^2/(a^2*(a*d - b*c)^3))^(2/3)))/3 - 6*b^5*d^5*x)*(
-b^2/(27*a^5*d^3 - 27*a^2*b^3*c^3 + 81*a^3*b^2*c^2*d - 81*a^4*b*c*d^2))^(1/3) + log(((d^2/(c^2*(a*d - b*c)^3))
^(1/3)*(9*a^2*b^4*d^6 + 9*b^6*c^2*d^4 - 18*a*b^5*c*d^5 - 9*b^3*d^3*(x + a*c*(d^2/(c^2*(a*d - b*c)^3))^(1/3))*(
a*d + b*c)*(a*d - b*c)^4*(d^2/(c^2*(a*d - b*c)^3))^(2/3)))/3 - 6*b^5*d^5*x)*(-d^2/(27*b^3*c^5 - 27*a^3*c^2*d^3
+ 81*a^2*b*c^3*d^2 - 81*a*b^2*c^4*d))^(1/3) + (log(6*b^5*d^5*x + ((3^(1/2)*1i - 1)*(-b^2/(a^2*(a*d - b*c)^3))
^(1/3)*(((3^(1/2)*1i - 1)^2*(81*b^3*d^3*x*(a*d + b*c)*(a*d - b*c)^4 + (81*a*b^3*c*d^3*(3^(1/2)*1i - 1)*(a*d +
b*c)*(a*d - b*c)^4*(-b^2/(a^2*(a*d - b*c)^3))^(1/3))/2)*(-b^2/(a^2*(a*d - b*c)^3))^(2/3))/36 - 9*a^2*b^4*d^6 -
9*b^6*c^2*d^4 + 18*a*b^5*c*d^5))/6)*(-b^2/(27*a^5*d^3 - 27*a^2*b^3*c^3 + 81*a^3*b^2*c^2*d - 81*a^4*b*c*d^2))^
(1/3)*(3^(1/2)*1i - 1))/2 - (log(6*b^5*d^5*x - ((3^(1/2)*1i + 1)*(-b^2/(a^2*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1
i + 1)^2*(81*b^3*d^3*x*(a*d + b*c)*(a*d - b*c)^4 - (81*a*b^3*c*d^3*(3^(1/2)*1i + 1)*(a*d + b*c)*(a*d - b*c)^4*
(-b^2/(a^2*(a*d - b*c)^3))^(1/3))/2)*(-b^2/(a^2*(a*d - b*c)^3))^(2/3))/36 - 9*a^2*b^4*d^6 - 9*b^6*c^2*d^4 + 18
*a*b^5*c*d^5))/6)*(-b^2/(27*a^5*d^3 - 27*a^2*b^3*c^3 + 81*a^3*b^2*c^2*d - 81*a^4*b*c*d^2))^(1/3)*(3^(1/2)*1i +
1))/2 + (log(6*b^5*d^5*x + ((3^(1/2)*1i - 1)*(d^2/(c^2*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i - 1)^2*(81*b^3*d^3
*x*(a*d + b*c)*(a*d - b*c)^4 + (81*a*b^3*c*d^3*(3^(1/2)*1i - 1)*(a*d + b*c)*(a*d - b*c)^4*(d^2/(c^2*(a*d - b*c
)^3))^(1/3))/2)*(d^2/(c^2*(a*d - b*c)^3))^(2/3))/36 - 9*a^2*b^4*d^6 - 9*b^6*c^2*d^4 + 18*a*b^5*c*d^5))/6)*(-d^
2/(27*b^3*c^5 - 27*a^3*c^2*d^3 + 81*a^2*b*c^3*d^2 - 81*a*b^2*c^4*d))^(1/3)*(3^(1/2)*1i - 1))/2 - (log(6*b^5*d^
5*x - ((3^(1/2)*1i + 1)*(d^2/(c^2*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i + 1)^2*(81*b^3*d^3*x*(a*d + b*c)*(a*d -
b*c)^4 - (81*a*b^3*c*d^3*(3^(1/2)*1i + 1)*(a*d + b*c)*(a*d - b*c)^4*(d^2/(c^2*(a*d - b*c)^3))^(1/3))/2)*(d^2/(
c^2*(a*d - b*c)^3))^(2/3))/36 - 9*a^2*b^4*d^6 - 9*b^6*c^2*d^4 + 18*a*b^5*c*d^5))/6)*(-d^2/(27*b^3*c^5 - 27*a^3
*c^2*d^3 + 81*a^2*b*c^3*d^2 - 81*a*b^2*c^4*d))^(1/3)*(3^(1/2)*1i + 1))/2

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