Optimal. Leaf size=754 \[ \frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\sqrt {2-\sqrt {2}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\sqrt {2+\sqrt {2}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\left (-\frac {\sqrt {a} e}{\sqrt {c}}+\sqrt {2} d+d\right ) \log \left (\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} \sqrt [8]{c}} \]
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Rubi [A] time = 1.25, antiderivative size = 754, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1415, 1169, 634, 618, 204, 628} \[ \frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\sqrt {2-\sqrt {2}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\sqrt {2+\sqrt {2}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{8 a^{7/8} c^{5/8}}+\frac {\left (-\frac {\sqrt {a} e}{\sqrt {c}}+\sqrt {2} d+d\right ) \log \left (\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} \sqrt [8]{c}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1169
Rule 1415
Rubi steps
\begin {align*} \int \frac {d+e x^4}{a+c x^8} \, dx &=\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}+\left (-d+\frac {\sqrt {a} e}{\sqrt {c}}\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} \sqrt [4]{c}}+\frac {\int \frac {\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}+\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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} \sqrt [4]{c}}\\ &=\frac {\sqrt [8]{c} \int \frac {\frac {\sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} d}{c^{3/8}}-\left (\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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}}+\frac {\sqrt [8]{c} \int \frac {\frac {\sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} d}{c^{3/8}}+\left (\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\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}}+\frac {\sqrt [8]{c} \int \frac {\frac {\sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} d}{c^{3/8}}-\left (\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (-d+\frac {\sqrt {a} e}{\sqrt {c}}\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}}+\frac {\sqrt [8]{c} \int \frac {\frac {\sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} d}{c^{3/8}}+\left (\frac {\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c}}-\frac {\sqrt [4]{a} \left (-d+\frac {\sqrt {a} e}{\sqrt {c}}\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}}\\ &=-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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} a^{3/4} c^{3/4}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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} a^{3/4} c^{3/4}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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} a^{3/4} c^{3/4}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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} a^{3/4} c^{3/4}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {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^{7/8} \sqrt [8]{c}}\\ &=\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {c}}\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^{7/8} \sqrt [8]{c}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \operatorname {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} a^{3/4} c^{3/4}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \operatorname {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} a^{3/4} c^{3/4}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \operatorname {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} a^{3/4} c^{3/4}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \operatorname {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} a^{3/4} c^{3/4}}\\ &=-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} 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^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {c}}\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^{7/8} \sqrt [8]{c}}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 534, normalized size = 0.71 \[ \frac {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 (\sqrt [8]{a} \sqrt {c} d \cos \left (\frac {\pi }{8}\right )-a^{5/8} 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 (\sqrt [8]{a} \sqrt {c} d \cos \left (\frac {\pi }{8}\right )-a^{5/8} e \sin \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{a} \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+\sqrt [8]{a} \log \left (2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+\sqrt [8]{a} \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (\sqrt {a} e \sin \left (\frac {\pi }{8}\right )-\sqrt {c} d \cos \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{a} \log \left (2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac {\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x^2\right ) \left (\sqrt {a} e \sin \left (\frac {\pi }{8}\right )-\sqrt {c} d \cos \left (\frac {\pi }{8}\right )\right )-2 \sqrt [8]{a} \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right ) \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{c} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt [8]{a} \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right ) \tan ^{-1}\left (\frac {\sqrt [8]{c} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac {\pi }{8}\right )\right )}{8 a c^{5/8}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.71, size = 3406, normalized size = 4.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 601, normalized size = 0.80 \[ -\frac {{\left (\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e - 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} - \frac {{\left (\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e - 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} + \frac {{\left (\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e + 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} + \frac {{\left (\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e + 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} - \frac {{\left (\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e - 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} + \frac {{\left (\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e - 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} + \frac {{\left (\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e + 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} - \frac {{\left (\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {5}{8}} e + 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} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 34, normalized size = 0.05 \[ \frac {\left (\RootOf \left (\textit {\_Z}^{8} c +a \right )^{4} e +d \right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8} c +a \right )+x \right )}{8 c \RootOf \left (\textit {\_Z}^{8} c +a \right )^{7}} \]
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 \[ \int \frac {e x^{4} + d}{c x^{8} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.78, size = 2510, normalized size = 3.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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