Integrand size = 21, antiderivative size = 162 \[ \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx=-\frac {d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac {(4 b c-3 a d) \arctan \left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac {(4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}} \]
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Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {390, 385, 218, 214, 211} \[ \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx=\frac {(4 b c-3 a d) \arctan \left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac {(4 b c-3 a d) \text {arctanh}\left (\frac {x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}-\frac {d x \left (a+b x^4\right )^{3/4}}{4 c \left (c+d x^4\right ) (b c-a d)} \]
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 390
Rubi steps \begin{align*} \text {integral}& = -\frac {d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac {(4 b c-3 a d) \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{4 c (b c-a d)} \\ & = -\frac {d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac {(4 b c-3 a d) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 c (b c-a d)} \\ & = -\frac {d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac {(4 b c-3 a d) \text {Subst}\left (\int \frac {1}{\sqrt {c}-\sqrt {b c-a d} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} (b c-a d)}+\frac {(4 b c-3 a d) \text {Subst}\left (\int \frac {1}{\sqrt {c}+\sqrt {b c-a d} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^{3/2} (b c-a d)} \\ & = -\frac {d x \left (a+b x^4\right )^{3/4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac {(4 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}}+\frac {(4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{8 c^{7/4} (b c-a d)^{5/4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.40 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.55 \[ \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx=\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) \left (-\frac {(2-2 i) c^{3/4} d x \left (a+b x^4\right )^{3/4}}{(b c-a d) \left (c+d x^4\right )}+\frac {(4 b c-3 a d) \arctan \left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}-\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{(b c-a d)^{5/4}}+\frac {(4 b c-3 a d) \text {arctanh}\left (\frac {\frac {(1-i) \sqrt [4]{b c-a d} x^2}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}+\frac {(1+i) \sqrt [4]{c} \sqrt [4]{a+b x^4}}{\sqrt [4]{b c-a d}}}{2 x}\right )}{(b c-a d)^{5/4}}\right )}{c^{7/4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(134)=268\).
Time = 4.40 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.12
method | result | size |
pseudoelliptic | \(\frac {-\frac {3 \left (a d -\frac {4 b c}{3}\right ) \sqrt {2}\, \left (d \,x^{4}+c \right ) \ln \left (\frac {-\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} \left (b \,x^{4}+a \right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a d -b c}{c}}\, x^{2}+\sqrt {b \,x^{4}+a}}\right )}{32}+\frac {3 \left (a d -\frac {4 b c}{3}\right ) \sqrt {2}\, \left (d \,x^{4}+c \right ) \arctan \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x -\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}\right )}{16}-\frac {3 \left (a d -\frac {4 b c}{3}\right ) \sqrt {2}\, \left (d \,x^{4}+c \right ) \arctan \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x +\sqrt {2}\, \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{4}} x}\right )}{16}+\frac {d \left (b \,x^{4}+a \right )^{\frac {3}{4}} x c \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}}}{4}}{c^{2} \left (a d -b c \right ) \left (d \,x^{4}+c \right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{4}}}\) | \(343\) |
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Timed out. \[ \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx=\int \frac {1}{\sqrt [4]{a + b x^{4}} \left (c + d x^{4}\right )^{2}}\, dx \]
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\[ \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} {\left (d x^{4} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} {\left (d x^{4} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )^2} \, dx=\int \frac {1}{{\left (b\,x^4+a\right )}^{1/4}\,{\left (d\,x^4+c\right )}^2} \,d x \]
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