Integrand size = 17, antiderivative size = 223 \[ \int \frac {a+b x^4}{c+d x^4} \, dx=\frac {b x}{d}+\frac {(b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{3/4} d^{5/4}}-\frac {(b c-a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{3/4} d^{5/4}}+\frac {(b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{3/4} d^{5/4}}-\frac {(b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{3/4} d^{5/4}} \]
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Time = 0.11 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {396, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {a+b x^4}{c+d x^4} \, dx=\frac {(b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{3/4} d^{5/4}}-\frac {(b c-a d) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} c^{3/4} d^{5/4}}+\frac {(b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{3/4} d^{5/4}}-\frac {(b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {c}+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{3/4} d^{5/4}}+\frac {b x}{d} \]
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Rule 210
Rule 217
Rule 396
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \frac {b x}{d}-\frac {(b c-a d) \int \frac {1}{c+d x^4} \, dx}{d} \\ & = \frac {b x}{d}-\frac {(b c-a d) \int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx}{2 \sqrt {c} d}-\frac {(b c-a d) \int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx}{2 \sqrt {c} d} \\ & = \frac {b x}{d}-\frac {(b c-a d) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 \sqrt {c} d^{3/2}}-\frac {(b c-a d) \int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx}{4 \sqrt {c} d^{3/2}}+\frac {(b c-a d) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{4 \sqrt {2} c^{3/4} d^{5/4}}+\frac {(b c-a d) \int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx}{4 \sqrt {2} c^{3/4} d^{5/4}} \\ & = \frac {b x}{d}+\frac {(b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{3/4} d^{5/4}}-\frac {(b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{3/4} d^{5/4}}-\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{3/4} d^{5/4}}+\frac {(b c-a d) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{3/4} d^{5/4}} \\ & = \frac {b x}{d}+\frac {(b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{3/4} d^{5/4}}-\frac {(b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )}{2 \sqrt {2} c^{3/4} d^{5/4}}+\frac {(b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{3/4} d^{5/4}}-\frac {(b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{4 \sqrt {2} c^{3/4} d^{5/4}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^4}{c+d x^4} \, dx=\frac {8 b c^{3/4} \sqrt [4]{d} x+2 \sqrt {2} (b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )-2 \sqrt {2} (b c-a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} x}{\sqrt [4]{c}}\right )+\sqrt {2} (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )-\sqrt {2} (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} x+\sqrt {d} x^2\right )}{8 c^{3/4} d^{5/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.19
method | result | size |
risch | \(\frac {b x}{d}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}+c \right )}{\sum }\frac {\left (a d -b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 d^{2}}\) | \(42\) |
default | \(\frac {b x}{d}+\frac {\left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}{x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d c}\) | \(120\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.51 \[ \int \frac {a+b x^4}{c+d x^4} \, dx=\frac {d \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac {1}{4}} \log \left (c d \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac {1}{4}} - {\left (b c - a d\right )} x\right ) + i \, d \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac {1}{4}} \log \left (i \, c d \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac {1}{4}} - {\left (b c - a d\right )} x\right ) - i \, d \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac {1}{4}} \log \left (-i \, c d \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac {1}{4}} - {\left (b c - a d\right )} x\right ) - d \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac {1}{4}} \log \left (-c d \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5}}\right )^{\frac {1}{4}} - {\left (b c - a d\right )} x\right ) + 4 \, b x}{4 \, d} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.39 \[ \int \frac {a+b x^4}{c+d x^4} \, dx=\frac {b x}{d} + \operatorname {RootSum} {\left (256 t^{4} c^{3} d^{5} + a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}, \left ( t \mapsto t \log {\left (\frac {4 t c d}{a d - b c} + x \right )} \right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.95 \[ \int \frac {a+b x^4}{c+d x^4} \, dx=\frac {b x}{d} - \frac {\frac {2 \, \sqrt {2} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {d} x - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (b c - a d\right )} \log \left (\sqrt {d} x^{2} + \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b c - a d\right )} \log \left (\sqrt {d} x^{2} - \sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{8 \, d} \]
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Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.10 \[ \int \frac {a+b x^4}{c+d x^4} \, dx=\frac {b x}{d} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c - \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, c d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c - \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{4 \, c d^{2}} - \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c - \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{8 \, c d^{2}} + \frac {\sqrt {2} {\left (\left (c d^{3}\right )^{\frac {1}{4}} b c - \left (c d^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {c}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c}{d}}\right )}{8 \, c d^{2}} \]
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Time = 0.23 (sec) , antiderivative size = 720, normalized size of antiderivative = 3.23 \[ \int \frac {a+b x^4}{c+d x^4} \, dx=\frac {b\,x}{d}-\frac {\mathrm {atan}\left (\frac {\frac {\left (x\,\left (4\,a^2\,d^3-8\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )-\frac {\left (16\,b\,c^2\,d^2-16\,a\,c\,d^3\right )\,\left (a\,d-b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}+\frac {\left (x\,\left (4\,a^2\,d^3-8\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )+\frac {\left (16\,b\,c^2\,d^2-16\,a\,c\,d^3\right )\,\left (a\,d-b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}}{\frac {\left (x\,\left (4\,a^2\,d^3-8\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )-\frac {\left (16\,b\,c^2\,d^2-16\,a\,c\,d^3\right )\,\left (a\,d-b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}\right )\,\left (a\,d-b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}-\frac {\left (x\,\left (4\,a^2\,d^3-8\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )+\frac {\left (16\,b\,c^2\,d^2-16\,a\,c\,d^3\right )\,\left (a\,d-b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}\right )\,\left (a\,d-b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}}\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{2\,{\left (-c\right )}^{3/4}\,d^{5/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (x\,\left (4\,a^2\,d^3-8\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )-\frac {\left (16\,b\,c^2\,d^2-16\,a\,c\,d^3\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}\right )\,\left (a\,d-b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}+\frac {\left (x\,\left (4\,a^2\,d^3-8\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )+\frac {\left (16\,b\,c^2\,d^2-16\,a\,c\,d^3\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}\right )\,\left (a\,d-b\,c\right )}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}}{\frac {\left (x\,\left (4\,a^2\,d^3-8\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )-\frac {\left (16\,b\,c^2\,d^2-16\,a\,c\,d^3\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}-\frac {\left (x\,\left (4\,a^2\,d^3-8\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )+\frac {\left (16\,b\,c^2\,d^2-16\,a\,c\,d^3\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}\right )\,\left (a\,d-b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{3/4}\,d^{5/4}}}\right )\,\left (a\,d-b\,c\right )}{2\,{\left (-c\right )}^{3/4}\,d^{5/4}} \]
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