Optimal. Leaf size=753 \[ \frac {d x}{c}+\frac {\sqrt {2-\sqrt {2}} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac {\sqrt {2+\sqrt {2}} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac {\sqrt {2-\sqrt {2}} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}+\frac {\sqrt {2+\sqrt {2}} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac {\left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}} \]
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Rubi [A]
time = 0.99, antiderivative size = 753, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {1408, 1517,
1429, 1183, 648, 632, 210, 642} \begin {gather*} \frac {\sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right ) \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right )}{8 a^{3/8} c^{9/8}}-\frac {\sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right ) \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right )}{8 a^{3/8} c^{9/8}}-\frac {\sqrt {2-\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right ) \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right )}{8 a^{3/8} c^{9/8}}+\frac {\sqrt {2+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right ) \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right )}{8 a^{3/8} c^{9/8}}-\frac {\left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \log \left (-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \log \left (\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {d x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 1408
Rule 1429
Rule 1517
Rubi steps
\begin {align*} \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx &=\int \frac {x^4 \left (e+d x^4\right )}{a+c x^8} \, dx\\ &=\frac {d x}{c}-\frac {\int \frac {a d-c e x^4}{a+c x^8} \, dx}{c}\\ &=\frac {d x}{c}-\frac {\int \frac {\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}+\left (-a d-\sqrt {a} \sqrt {c} e\right ) x^2}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+x^4} \, dx}{2 \sqrt {2} a^{3/4} c^{5/4}}-\frac {\int \frac {\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}+\left (a d+\sqrt {a} \sqrt {c} e\right ) x^2}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+x^4} \, dx}{2 \sqrt {2} a^{3/4} c^{5/4}}\\ &=\frac {d x}{c}-\frac {\int \frac {\frac {\sqrt {2 \left (2-\sqrt {2}\right )} a^{11/8} d}{c^{3/8}}-\left (\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (a d+\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right ) x}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{4 \sqrt {2 \left (2-\sqrt {2}\right )} a^{9/8} c^{7/8}}-\frac {\int \frac {\frac {\sqrt {2 \left (2-\sqrt {2}\right )} a^{11/8} d}{c^{3/8}}+\left (\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (a d+\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right ) x}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{4 \sqrt {2 \left (2-\sqrt {2}\right )} a^{9/8} c^{7/8}}-\frac {\int \frac {\frac {\sqrt {2 \left (2+\sqrt {2}\right )} a^{11/8} d}{c^{3/8}}-\left (\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (-a d-\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right ) x}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{9/8} c^{7/8}}-\frac {\int \frac {\frac {\sqrt {2 \left (2+\sqrt {2}\right )} a^{11/8} d}{c^{3/8}}+\left (\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (-a d-\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right ) x}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{9/8} c^{7/8}}\\ &=\frac {d x}{c}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \int \frac {1}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \int \frac {1}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \int \frac {-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \int \frac {\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (a d+\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right ) \int \frac {\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{9/8} c^{7/8}}-\frac {\left (-\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}+\frac {\sqrt [4]{a} \left (a d+\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right ) \int \frac {-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{9/8} c^{7/8}}-\frac {\left (\frac {2 \sqrt {2 \left (2+\sqrt {2}\right )} a^{11/8} d}{c^{3/8}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (-a d-\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right )}{\sqrt [8]{c}}\right ) \int \frac {1}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{9/8} c^{7/8}}-\frac {\left (\frac {2 \sqrt {2 \left (2+\sqrt {2}\right )} a^{11/8} d}{c^{3/8}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (-\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}+\frac {\sqrt [4]{a} \left (-a d-\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right )}{\sqrt [8]{c}}\right ) \int \frac {1}{\frac {\sqrt [4]{a}}{\sqrt [4]{c}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+x^2} \, dx}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{9/8} c^{7/8}}\\ &=\frac {d x}{c}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \text {Subst}\left (\int \frac {1}{-\frac {\left (2+\sqrt {2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}-x^2} \, dx,x,-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \text {Subst}\left (\int \frac {1}{-\frac {\left (2+\sqrt {2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}-x^2} \, dx,x,\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {\left (\frac {2 \sqrt {2 \left (2+\sqrt {2}\right )} a^{11/8} d}{c^{3/8}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (-a d-\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right )}{\sqrt [8]{c}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {\left (2-\sqrt {2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}-x^2} \, dx,x,\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{9/8} c^{7/8}}+\frac {\left (\frac {2 \sqrt {2 \left (2+\sqrt {2}\right )} a^{11/8} d}{c^{3/8}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (-\frac {\sqrt {2} a^{5/4} d}{\sqrt [4]{c}}+\frac {\sqrt [4]{a} \left (-a d-\sqrt {a} \sqrt {c} e\right )}{\sqrt [4]{c}}\right )}{\sqrt [8]{c}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {\left (2-\sqrt {2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}-x^2} \, dx,x,-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}+2 x\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{9/8} c^{7/8}}\\ &=\frac {d x}{c}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\sqrt {2+\sqrt {2}} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\sqrt {2+\sqrt {2}} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{3/8} c^{9/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{a}+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 551, normalized size = 0.73 \begin {gather*} \frac {8 a c^{5/8} d x+2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{c} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \left (a^{5/8} c e \cos \left (\frac {\pi }{8}\right )-a^{9/8} \sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{c} x^2+2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac {\pi }{8}\right )\right ) \left (a^{5/8} c e \cos \left (\frac {\pi }{8}\right )-a^{9/8} \sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+2 \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{c} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \left (-a^{5/8} c e \cos \left (\frac {\pi }{8}\right )+a^{9/8} \sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{c} x^2-2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac {\pi }{8}\right )\right ) \left (-a^{5/8} c e \cos \left (\frac {\pi }{8}\right )+a^{9/8} \sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )-2 \tan ^{-1}\left (\frac {\sqrt [8]{c} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \left (a^{9/8} \sqrt {c} d \cos \left (\frac {\pi }{8}\right )+a^{5/8} c e \sin \left (\frac {\pi }{8}\right )\right )-2 \tan ^{-1}\left (\frac {\sqrt [8]{c} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \left (a^{9/8} \sqrt {c} d \cos \left (\frac {\pi }{8}\right )+a^{5/8} c e \sin \left (\frac {\pi }{8}\right )\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{c} x^2-2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac {\pi }{8}\right )\right ) \left (a^{9/8} \sqrt {c} d \cos \left (\frac {\pi }{8}\right )+a^{5/8} c e \sin \left (\frac {\pi }{8}\right )\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{c} x^2+2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac {\pi }{8}\right )\right ) \left (a^{9/8} \sqrt {c} d \cos \left (\frac {\pi }{8}\right )+a^{5/8} c e \sin \left (\frac {\pi }{8}\right )\right )}{8 a c^{13/8}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.04, size = 45, normalized size = 0.06
method | result | size |
default | \(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} c e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c^{2}}\) | \(45\) |
risch | \(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} c e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c^{2}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3552 vs.
\(2 (537) = 1074\).
time = 2.06, size = 3552, normalized size = 4.72 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A]
time = 3.65, size = 647, normalized size = 0.86 \begin {gather*} \frac {d x}{c} - \frac {{\left (c \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e + a d \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{8 \, a c} - \frac {{\left (c \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e + a d \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{8 \, a c} + \frac {{\left (c \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e - a d \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{8 \, a c} + \frac {{\left (c \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e - a d \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{8 \, a c} - \frac {{\left (c \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e + a d \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{16 \, a c} + \frac {{\left (c \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e + a d \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{16 \, a c} + \frac {{\left (c \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e - a d \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{16 \, a c} - \frac {{\left (c \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e - a d \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}\right )} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{16 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.22, size = 2520, normalized size = 3.35 \begin {gather*} \frac {\mathrm {atan}\left (\frac {a^3\,d^6\,x-c^3\,e^6\,x-a\,c^2\,d^2\,e^4\,x+a^2\,c\,d^4\,e^2\,x+\frac {2\,d\,e\,x\,\left (a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}\right )}{a\,c^4}}{a^2\,c^6\,e\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{5/4}-a^3\,c\,d^5\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}+2\,a^2\,c^2\,d^3\,e^2\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}+3\,a\,c^3\,d\,e^4\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}}\right )\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}}{4}-\frac {\mathrm {atan}\left (\frac {c^3\,e^6\,x-a^3\,d^6\,x+a\,c^2\,d^2\,e^4\,x-a^2\,c\,d^4\,e^2\,x+\frac {2\,d\,e\,x\,\left (a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}\right )}{a\,c^4}}{a^2\,c^6\,e\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{5/4}-a^3\,c\,d^5\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}+2\,a^2\,c^2\,d^3\,e^2\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}+3\,a\,c^3\,d\,e^4\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}}\right )\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}}{4}+\frac {d\,x}{c}+\mathrm {atan}\left (\frac {-a^3\,d^6\,x\,1{}\mathrm {i}+c^3\,e^6\,x\,1{}\mathrm {i}+a\,c^2\,d^2\,e^4\,x\,1{}\mathrm {i}-a^2\,c\,d^4\,e^2\,x\,1{}\mathrm {i}+\frac {d\,e\,x\,\left (a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}\right )\,2{}\mathrm {i}}{a\,c^4}}{a^2\,c^6\,e\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{5/4}-a^3\,c\,d^5\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}+2\,a^2\,c^2\,d^3\,e^2\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}+3\,a\,c^3\,d\,e^4\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}}\right )\,{\left (\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}-4\,a^2\,c^6\,d\,e^3+4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{4096\,a^3\,c^9}\right )}^{1/4}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {a^3\,d^6\,x\,1{}\mathrm {i}-c^3\,e^6\,x\,1{}\mathrm {i}-a\,c^2\,d^2\,e^4\,x\,1{}\mathrm {i}+a^2\,c\,d^4\,e^2\,x\,1{}\mathrm {i}+\frac {d\,e\,x\,\left (a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}\right )\,2{}\mathrm {i}}{a\,c^4}}{a^2\,c^6\,e\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{5/4}-a^3\,c\,d^5\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}+2\,a^2\,c^2\,d^3\,e^2\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}+3\,a\,c^3\,d\,e^4\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{a^3\,c^9}\right )}^{1/4}}\right )\,{\left (-\frac {a^2\,d^4\,\sqrt {-a^3\,c^9}+c^2\,e^4\,\sqrt {-a^3\,c^9}+4\,a^2\,c^6\,d\,e^3-4\,a^3\,c^5\,d^3\,e-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^9}}{4096\,a^3\,c^9}\right )}^{1/4}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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