Integrand size = 19, antiderivative size = 419 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\frac {d (b c+a d) x}{3 a c (b c-a d)^2 \left (c+d x^3\right )}+\frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}-\frac {2 b^{5/3} (b c-4 a d) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} (b c-a d)^3}-\frac {2 d^{5/3} (4 b c-a d) \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^3}+\frac {2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}+\frac {2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac {b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}-\frac {d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3} \]
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Time = 0.36 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {425, 541, 536, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=-\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (b c-4 a d)}{3 \sqrt {3} a^{5/3} (b c-a d)^3}-\frac {b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}+\frac {2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}-\frac {2 d^{5/3} (4 b c-a d) \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^3}-\frac {d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3}+\frac {2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}+\frac {d x (a d+b c)}{3 a c \left (c+d x^3\right ) (b c-a d)^2} \]
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Rule 31
Rule 206
Rule 210
Rule 425
Rule 536
Rule 541
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}-\frac {\int \frac {-2 b c+3 a d-5 b d x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )^2} \, dx}{3 a (b c-a d)} \\ & = \frac {d (b c+a d) x}{3 a c (b c-a d)^2 \left (c+d x^3\right )}+\frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}-\frac {\int \frac {-6 \left (b^2 c^2-3 a b c d+a^2 d^2\right )-6 b d (b c+a d) x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{9 a c (b c-a d)^2} \\ & = \frac {d (b c+a d) x}{3 a c (b c-a d)^2 \left (c+d x^3\right )}+\frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}+\frac {\left (2 b^2 (b c-4 a d)\right ) \int \frac {1}{a+b x^3} \, dx}{3 a (b c-a d)^3}+\frac {\left (2 d^2 (4 b c-a d)\right ) \int \frac {1}{c+d x^3} \, dx}{3 c (b c-a d)^3} \\ & = \frac {d (b c+a d) x}{3 a c (b c-a d)^2 \left (c+d x^3\right )}+\frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}+\frac {\left (2 b^2 (b c-4 a d)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} (b c-a d)^3}+\frac {\left (2 b^2 (b c-4 a d)\right ) \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}{9 a^{5/3} (b c-a d)^3}+\frac {\left (2 d^2 (4 b c-a d)\right ) \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{9 c^{5/3} (b c-a d)^3}+\frac {\left (2 d^2 (4 b c-a d)\right ) \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}{9 c^{5/3} (b c-a d)^3} \\ & = \frac {d (b c+a d) x}{3 a c (b c-a d)^2 \left (c+d x^3\right )}+\frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}+\frac {2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}+\frac {2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac {\left (b^{5/3} (b c-4 a d)\right ) \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}{9 a^{5/3} (b c-a d)^3}+\frac {\left (b^2 (b c-4 a d)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{4/3} (b c-a d)^3}-\frac {\left (d^{5/3} (4 b c-a d)\right ) \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}{9 c^{5/3} (b c-a d)^3}+\frac {\left (d^2 (4 b c-a d)\right ) \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{4/3} (b c-a d)^3} \\ & = \frac {d (b c+a d) x}{3 a c (b c-a d)^2 \left (c+d x^3\right )}+\frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}+\frac {2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}+\frac {2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac {b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}-\frac {d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3}+\frac {\left (2 b^{5/3} (b c-4 a d)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{5/3} (b c-a d)^3}+\frac {\left (2 d^{5/3} (4 b c-a d)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{3 c^{5/3} (b c-a d)^3} \\ & = \frac {d (b c+a d) x}{3 a c (b c-a d)^2 \left (c+d x^3\right )}+\frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}-\frac {2 b^{5/3} (b c-4 a d) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} (b c-a d)^3}-\frac {2 d^{5/3} (4 b c-a d) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^3}+\frac {2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}+\frac {2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac {b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}-\frac {d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\frac {1}{9} \left (\frac {3 b^2 x}{a (b c-a d)^2 \left (a+b x^3\right )}+\frac {3 d^2 x}{c (b c-a d)^2 \left (c+d x^3\right )}+\frac {2 \sqrt {3} b^{5/3} (b c-4 a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3} (-b c+a d)^3}+\frac {2 \sqrt {3} d^{5/3} (-4 b c+a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{5/3} (b c-a d)^3}+\frac {2 b^{5/3} (-b c+4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3} (-b c+a d)^3}+\frac {2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3} (b c-a d)^3}+\frac {b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3} (-b c+a d)^3}+\frac {d^{5/3} (-4 b c+a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3} (b c-a d)^3}\right ) \]
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Time = 4.07 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {d^{2} \left (\frac {\left (a d -b c \right ) x}{3 c \left (d \,x^{3}+c \right )}+\frac {2 \left (a d -4 b c \right ) \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )}{3 c}\right )}{\left (a d -b c \right )^{3}}+\frac {b^{2} \left (\frac {\left (a d -b c \right ) x}{3 a \left (b \,x^{3}+a \right )}+\frac {2 \left (4 a d -b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a}\right )}{\left (a d -b c \right )^{3}}\) | \(285\) |
risch | \(\text {Expression too large to display}\) | \(1701\) |
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Leaf count of result is larger than twice the leaf count of optimal. 897 vs. \(2 (341) = 682\).
Time = 52.87 (sec) , antiderivative size = 897, normalized size of antiderivative = 2.14 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\frac {3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{4} + 2 \, \sqrt {3} {\left ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\right )} \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 ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \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 ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\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 ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{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 ({\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} b c^{2} d + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 4 \, a^{2} b c d^{2}\right )} x^{3}\right )} \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 ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \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 ) + 3 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{9 \, {\left (a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + {\left (a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{6} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{3}\right )}} \]
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Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (341) = 682\).
Time = 0.30 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.87 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\frac {2 \, \sqrt {3} {\left (b^{2} c - 4 \, a b d\right )} \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 )}{9 \, {\left (a b^{3} c^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 3 \, a^{2} b^{2} c^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a^{3} b c d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} d^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} {\left (4 \, b c d - a d^{2}\right )} \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 )}{9 \, {\left (b^{3} c^{4} \left (\frac {c}{d}\right )^{\frac {1}{3}} - 3 \, a b^{2} c^{3} d \left (\frac {c}{d}\right )^{\frac {1}{3}} + 3 \, a^{2} b c^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a^{3} c d^{3} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {{\left (b^{2} c - 4 \, a b d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, {\left (a b^{3} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{2} b^{2} c^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b c d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{4} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {{\left (4 \, b c d - a d^{2}\right )} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{9 \, {\left (b^{3} c^{4} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 3 \, a b^{2} c^{3} d \left (\frac {c}{d}\right )^{\frac {2}{3}} + 3 \, a^{2} b c^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a^{3} c d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {2 \, {\left (b^{2} c - 4 \, a b d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, {\left (a b^{3} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{2} b^{2} c^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b c d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{4} d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {2 \, {\left (4 \, b c d - a d^{2}\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{9 \, {\left (b^{3} c^{4} \left (\frac {c}{d}\right )^{\frac {2}{3}} - 3 \, a b^{2} c^{3} d \left (\frac {c}{d}\right )^{\frac {2}{3}} + 3 \, a^{2} b c^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a^{3} c d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {{\left (b^{2} c d + a b d^{2}\right )} x^{4} + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{3 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{6} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{3}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=-\frac {2 \, {\left (b^{3} c - 4 \, a b^{2} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac {2 \, {\left (4 \, b c d^{2} - a d^{3}\right )} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{9 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} + \frac {2 \, {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} c - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b d\right )} \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 (\sqrt {3} a^{2} b^{3} c^{3} - 3 \, \sqrt {3} a^{3} b^{2} c^{2} d + 3 \, \sqrt {3} a^{4} b c d^{2} - \sqrt {3} a^{5} d^{3}\right )}} + \frac {2 \, {\left (4 \, \left (-c d^{2}\right )^{\frac {1}{3}} b c d - \left (-c d^{2}\right )^{\frac {1}{3}} a d^{2}\right )} \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 (\sqrt {3} b^{3} c^{5} - 3 \, \sqrt {3} a b^{2} c^{4} d + 3 \, \sqrt {3} a^{2} b c^{3} d^{2} - \sqrt {3} a^{3} c^{2} d^{3}\right )}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} c - 4 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} + \frac {{\left (4 \, \left (-c d^{2}\right )^{\frac {1}{3}} b c d - \left (-c d^{2}\right )^{\frac {1}{3}} a d^{2}\right )} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{9 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} + \frac {b^{2} c d x^{4} + a b d^{2} x^{4} + b^{2} c^{2} x + a^{2} d^{2} x}{3 \, {\left (b d x^{6} + b c x^{3} + a d x^{3} + a c\right )} {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )}} \]
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Time = 28.74 (sec) , antiderivative size = 3637, normalized size of antiderivative = 8.68 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\text {Too large to display} \]
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