Integrand size = 17, antiderivative size = 578 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=-\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} \sqrt [8]{c}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} \sqrt [8]{c}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x}{\sqrt [4]{a}+\sqrt [4]{c} x^2}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )} a^{7/8} c^{5/8}}+\frac {\left (d+\sqrt {2} d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x}{\sqrt [4]{a}+\sqrt [4]{c} x^2}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{7/8} \sqrt [8]{c}} \] Output:
-1/4*(d+2^(1/2)*d-a^(1/2)*e/c^(1/2))*arctan(((2-2^(1/2))^(1/2)*a^(1/8)-2*c ^(1/8)*x)/(2+2^(1/2))^(1/2)/a^(1/8))/(4+2*2^(1/2))^(1/2)/a^(7/8)/c^(1/8)+1 /4*((1-2^(1/2))*c^(1/2)*d-a^(1/2)*e)*arctan(((2+2^(1/2))^(1/2)*a^(1/8)-2*c ^(1/8)*x)/(2-2^(1/2))^(1/2)/a^(1/8))/(4-2*2^(1/2))^(1/2)/a^(7/8)/c^(5/8)+1 /4*(d+2^(1/2)*d-a^(1/2)*e/c^(1/2))*arctan(((2-2^(1/2))^(1/2)*a^(1/8)+2*c^( 1/8)*x)/(2+2^(1/2))^(1/2)/a^(1/8))/(4+2*2^(1/2))^(1/2)/a^(7/8)/c^(1/8)-1/4 *((1-2^(1/2))*c^(1/2)*d-a^(1/2)*e)*arctan(((2+2^(1/2))^(1/2)*a^(1/8)+2*c^( 1/8)*x)/(2-2^(1/2))^(1/2)/a^(1/8))/(4-2*2^(1/2))^(1/2)/a^(7/8)/c^(5/8)-1/4 *((1-2^(1/2))*c^(1/2)*d-a^(1/2)*e)*arctanh((2-2^(1/2))^(1/2)*a^(1/8)*c^(1/ 8)*x/(a^(1/4)+c^(1/4)*x^2))/(4-2*2^(1/2))^(1/2)/a^(7/8)/c^(5/8)+1/4*(d+2^( 1/2)*d-a^(1/2)*e/c^(1/2))*arctanh((2+2^(1/2))^(1/2)*a^(1/8)*c^(1/8)*x/(a^( 1/4)+c^(1/4)*x^2))/(4+2*2^(1/2))^(1/2)/a^(7/8)/c^(1/8)
Time = 0.69 (sec) , antiderivative size = 534, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=\frac {-2 \sqrt [8]{a} \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{c} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \left (\sqrt {a} e \cos \left (\frac {\pi }{8}\right )+\sqrt {c} d \sin \left (\frac {\pi }{8}\right )\right )+2 \sqrt [8]{a} \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{c} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\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 (\sqrt [4]{a}+\sqrt [4]{c} x^2-2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac {\pi }{8}\right )\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 (\sqrt [4]{a}+\sqrt [4]{c} x^2+2 \sqrt [8]{a} \sqrt [8]{c} x \sin \left (\frac {\pi }{8}\right )\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 (\sqrt [4]{a}+\sqrt [4]{c} x^2-2 \sqrt [8]{a} \sqrt [8]{c} x \cos \left (\frac {\pi }{8}\right )\right ) \left (-\sqrt {c} d \cos \left (\frac {\pi }{8}\right )+\sqrt {a} e \sin \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{a} \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 (-\sqrt {c} d \cos \left (\frac {\pi }{8}\right )+\sqrt {a} e \sin \left (\frac {\pi }{8}\right )\right )+2 \arctan \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 \arctan \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 )}{8 a c^{5/8}} \] Input:
Integrate[(d + e*x^4)/(a + c*x^8),x]
Output:
(-2*a^(1/8)*ArcTan[Cot[Pi/8] - (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(Sqrt[a]*e*C os[Pi/8] + Sqrt[c]*d*Sin[Pi/8]) + 2*a^(1/8)*ArcTan[Cot[Pi/8] + (c^(1/8)*x* Csc[Pi/8])/a^(1/8)]*(Sqrt[a]*e*Cos[Pi/8] + Sqrt[c]*d*Sin[Pi/8]) - a^(1/8)* Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(Sqrt[a]*e*Cos[ Pi/8] + Sqrt[c]*d*Sin[Pi/8]) + a^(1/8)*Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/ 8)*c^(1/8)*x*Sin[Pi/8]]*(Sqrt[a]*e*Cos[Pi/8] + Sqrt[c]*d*Sin[Pi/8]) + a^(1 /8)*Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(-(Sqrt[c]* d*Cos[Pi/8]) + Sqrt[a]*e*Sin[Pi/8]) - a^(1/8)*Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(-(Sqrt[c]*d*Cos[Pi/8]) + Sqrt[a]*e*Sin[Pi/ 8]) + 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*(a^(1/8)*Sqrt[c] *d*Cos[Pi/8] - a^(5/8)*e*Sin[Pi/8]) + 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/ 8) + Tan[Pi/8]]*(a^(1/8)*Sqrt[c]*d*Cos[Pi/8] - a^(5/8)*e*Sin[Pi/8]))/(8*a* c^(5/8))
Time = 1.50 (sec) , antiderivative size = 849, normalized size of antiderivative = 1.47, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {1745, 27, 1483, 27, 27, 1142, 25, 27, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {d+e x^4}{a+c x^8} \, dx\) |
| \(\Big \downarrow \) 1745 |
| \(\displaystyle \frac {\int \frac {\sqrt {2} \sqrt [4]{a} d-\sqrt [4]{c} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x^2}{\sqrt [4]{c} \left (x^4-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {\int \frac {\sqrt [4]{c} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x^2+\sqrt {2} \sqrt [4]{a} d}{\sqrt [4]{c} \left (x^4+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {2} \sqrt [4]{a} d-\sqrt [4]{c} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x^2}{x^4-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\int \frac {\sqrt [4]{c} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x^2+\sqrt {2} \sqrt [4]{a} d}{x^4+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {c^{3/8} \int \frac {\sqrt [4]{a} \left (\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} d+\sqrt [8]{c} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x\right )}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2-\sqrt {2}} a^{3/8}}+\frac {c^{3/8} \int \frac {\sqrt [4]{a} \left (\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} d-\sqrt [8]{c} \left (\left (1-\sqrt {2}\right ) d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x\right )}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2-\sqrt {2}} a^{3/8}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {c^{3/8} \int \frac {\sqrt [4]{a} \left (\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} d-\sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x\right )}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2+\sqrt {2}} a^{3/8}}+\frac {c^{3/8} \int \frac {\sqrt [4]{a} \left (\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} d+\sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x\right )}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2+\sqrt {2}} a^{3/8}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} c^{3/8} d-\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) x}{c^{3/8} \left (x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} d+\sqrt [8]{c} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} d-\sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} d+\sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} c^{3/8} d-\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) x}{x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c}}+\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2-\sqrt {2}\right )} \sqrt [8]{a} d+\sqrt [8]{c} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} d-\sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2+\sqrt {2}\right )} \sqrt [8]{a} d+\sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) x}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}+\frac {1}{2} \sqrt [8]{c} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int -\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [8]{c}}-\frac {1}{2} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}-\frac {1}{2} \sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int -\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}-\frac {1}{2} \sqrt [8]{c} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [8]{c}}-\frac {1}{2} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}-\frac {1}{2} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [8]{c}}-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [8]{c}}}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {c}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \left (-\frac {1}{2} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{-\left (2 x-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )^2-\frac {\left (2+\sqrt {2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}}d\left (2 x-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {c}}\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [8]{c}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{-\left (2 x+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )^2-\frac {\left (2+\sqrt {2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}}d\left (2 x+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt [8]{c}}}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{-\left (2 x-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )^2-\frac {\left (2-\sqrt {2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}}d\left (2 x-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {c}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{-\left (2 x+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )^2-\frac {\left (2-\sqrt {2}\right ) \sqrt [4]{a}}{\sqrt [4]{c}}}d\left (2 x+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {c}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \left (\frac {\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} \left (2 x-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{c^{3/8}}-\frac {1}{2} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} \left (2 x+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )-\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [8]{c}}}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx-\frac {\sqrt {\frac {2+\sqrt {2}}{2-\sqrt {2}}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} \left (2 x-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{c^{3/8}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [8]{c} x+\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx-\frac {\sqrt {\frac {2+\sqrt {2}}{2-\sqrt {2}}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} \left (2 x+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{c^{3/8}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt [4]{c} \left (\frac {\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} \left (2 x-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{c^{3/8}}+\frac {1}{2} \sqrt [8]{c} \left (-\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [4]{c} x^2-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}\right )\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}} \left (\left (1+\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} \left (2 x+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )-\frac {1}{2} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [4]{c} x^2+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (-\frac {\sqrt {\frac {2+\sqrt {2}}{2-\sqrt {2}}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} \left (2 x-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{c^{3/8}}-\frac {1}{2} \sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [4]{c} x^2-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt [8]{c} \left (\sqrt {2} d+d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [4]{c} x^2+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}\right )-\frac {\sqrt {\frac {2+\sqrt {2}}{2-\sqrt {2}}} \left (\left (1-\sqrt {2}\right ) \sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} \left (2 x+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}}{\sqrt [8]{c}}\right )}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{c^{3/8}}\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} a^{3/4} \sqrt {c}}\) |
Input:
Int[(d + e*x^4)/(a + c*x^8),x]
Output:
((c^(1/4)*((Sqrt[(2 - Sqrt[2])/(2 + Sqrt[2])]*((1 + Sqrt[2])*Sqrt[c]*d - S qrt[a]*e)*ArcTan[(c^(1/8)*(-((Sqrt[2 - Sqrt[2]]*a^(1/8))/c^(1/8)) + 2*x))/ (Sqrt[2 + Sqrt[2]]*a^(1/8))])/c^(3/8) + (c^(1/8)*(d - Sqrt[2]*d - (Sqrt[a] *e)/Sqrt[c])*Log[a^(1/4) - Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x ^2])/2))/(2*Sqrt[2 - Sqrt[2]]*a^(1/8)) + (Sqrt[(2 - Sqrt[2])/(2 + Sqrt[2]) ]*((1 + Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(c^(1/8)*((Sqrt[2 - Sqrt[2] ]*a^(1/8))/c^(1/8) + 2*x))/(Sqrt[2 + Sqrt[2]]*a^(1/8))] - (((1 - Sqrt[2])* Sqrt[c]*d - Sqrt[a]*e)*Log[a^(1/4) + Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/2)/(2*Sqrt[2 - Sqrt[2]]*a^(1/8)*c^(1/8)))/(2*Sqrt[2]*a^(3/4 )*Sqrt[c]) + ((c^(1/4)*(-((Sqrt[(2 + Sqrt[2])/(2 - Sqrt[2])]*((1 - Sqrt[2] )*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(c^(1/8)*(-((Sqrt[2 + Sqrt[2]]*a^(1/8))/c^ (1/8)) + 2*x))/(Sqrt[2 - Sqrt[2]]*a^(1/8))])/c^(3/8)) - (c^(1/8)*(d + Sqrt [2]*d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/4) - Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/ 8)*x + c^(1/4)*x^2])/2))/(2*Sqrt[2 + Sqrt[2]]*a^(1/8)) + (c^(1/4)*(-((Sqrt [(2 + Sqrt[2])/(2 - Sqrt[2])]*((1 - Sqrt[2])*Sqrt[c]*d - Sqrt[a]*e)*ArcTan [(c^(1/8)*((Sqrt[2 + Sqrt[2]]*a^(1/8))/c^(1/8) + 2*x))/(Sqrt[2 - Sqrt[2]]* a^(1/8))])/c^(3/8)) + (c^(1/8)*(d + Sqrt[2]*d - (Sqrt[a]*e)/Sqrt[c])*Log[a ^(1/4) + Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/2))/(2*Sqrt[2 + Sqrt[2]]*a^(1/8)))/(2*Sqrt[2]*a^(3/4)*Sqrt[c])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{ q = Rt[a/c, 4]}, Simp[1/(2*Sqrt[2]*c*q^3) Int[(Sqrt[2]*d*q - (d - e*q^2)* x^(n/2))/(q^2 - Sqrt[2]*q*x^(n/2) + x^n), x], x] + Simp[1/(2*Sqrt[2]*c*q^3) Int[(Sqrt[2]*d*q + (d - e*q^2)*x^(n/2))/(q^2 + Sqrt[2]*q*x^(n/2) + x^n), x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && PosQ[a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.06
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c}\) | \(34\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c}\) | \(34\) |
Input:
int((e*x^4+d)/(c*x^8+a),x,method=_RETURNVERBOSE)
Output:
1/8/c*sum((_R^4*e+d)/_R^7*ln(x-_R),_R=RootOf(_Z^8*c+a))
Leaf count of result is larger than twice the leaf count of optimal. 2749 vs. \(2 (391) = 782\).
Time = 0.37 (sec) , antiderivative size = 2749, normalized size of antiderivative = 4.76 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=\text {Too large to display} \] Input:
integrate((e*x^4+d)/(c*x^8+a),x, algorithm="fricas")
Output:
1/8*sqrt(-sqrt(-(a^3*c^2*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^ 4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e - 4*a*d*e^3)/(a ^3*c^2)))*log((c^3*d^6 - 5*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 + a^3*e^6)*x + (a^5*c^3*e*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a^3 *c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + a*c^3*d^5 - 6*a^2*c^2*d^3*e^2 + a^3*c*d *e^4)*sqrt(-sqrt(-(a^3*c^2*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2* d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e - 4*a*d*e^3)/ (a^3*c^2)))) - 1/8*sqrt(-sqrt(-(a^3*c^2*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3*e - 4*a*d*e^3)/(a^3*c^2)))*log((c^3*d^6 - 5*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 + a^3*e^6)*x - (a^5*c^3*e*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d ^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + a*c^3*d^5 - 6*a^2*c^2*d^ 3*e^2 + a^3*c*d*e^4)*sqrt(-sqrt(-(a^3*c^2*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^ 2 + 38*a^2*c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 4*c*d^3* e - 4*a*d*e^3)/(a^3*c^2)))) - 1/8*sqrt(-sqrt((a^3*c^2*sqrt(-(c^4*d^8 - 12* a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5) ) - 4*c*d^3*e + 4*a*d*e^3)/(a^3*c^2)))*log((c^3*d^6 - 5*a*c^2*d^4*e^2 - 5* a^2*c*d^2*e^4 + a^3*e^6)*x + (a^5*c^3*e*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - a*c^3*d^5 + 6*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*sqrt(-sqrt((a^3*c^2*sqrt(-(c^4*d^8 -...
Timed out. \[ \int \frac {d+e x^4}{a+c x^8} \, dx=\text {Timed out} \] Input:
integrate((e*x**4+d)/(c*x**8+a),x)
Output:
Timed out
\[ \int \frac {d+e x^4}{a+c x^8} \, dx=\int { \frac {e x^{4} + d}{c x^{8} + a} \,d x } \] Input:
integrate((e*x^4+d)/(c*x^8+a),x, algorithm="maxima")
Output:
integrate((e*x^4 + d)/(c*x^8 + a), x)
Time = 0.17 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x^4}{a+c x^8} \, dx =\text {Too large to display} \] Input:
integrate((e*x^4+d)/(c*x^8+a),x, algorithm="giac")
Output:
-1/8*(e*sqrt(-sqrt(2) + 2)*(a/c)^(5/8) - d*sqrt(sqrt(2) + 2)*(a/c)^(1/8))* arctan((2*x + sqrt(-sqrt(2) + 2)*(a/c)^(1/8))/(sqrt(sqrt(2) + 2)*(a/c)^(1/ 8)))/a - 1/8*(e*sqrt(-sqrt(2) + 2)*(a/c)^(5/8) - d*sqrt(sqrt(2) + 2)*(a/c) ^(1/8))*arctan((2*x - sqrt(-sqrt(2) + 2)*(a/c)^(1/8))/(sqrt(sqrt(2) + 2)*( a/c)^(1/8)))/a + 1/8*(e*sqrt(sqrt(2) + 2)*(a/c)^(5/8) + d*sqrt(-sqrt(2) + 2)*(a/c)^(1/8))*arctan((2*x + sqrt(sqrt(2) + 2)*(a/c)^(1/8))/(sqrt(-sqrt(2 ) + 2)*(a/c)^(1/8)))/a + 1/8*(e*sqrt(sqrt(2) + 2)*(a/c)^(5/8) + d*sqrt(-sq rt(2) + 2)*(a/c)^(1/8))*arctan((2*x - sqrt(sqrt(2) + 2)*(a/c)^(1/8))/(sqrt (-sqrt(2) + 2)*(a/c)^(1/8)))/a - 1/16*(e*sqrt(-sqrt(2) + 2)*(a/c)^(5/8) - d*sqrt(sqrt(2) + 2)*(a/c)^(1/8))*log(x^2 + x*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + (a/c)^(1/4))/a + 1/16*(e*sqrt(-sqrt(2) + 2)*(a/c)^(5/8) - d*sqrt(sqrt(2 ) + 2)*(a/c)^(1/8))*log(x^2 - x*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + (a/c)^(1/4 ))/a + 1/16*(e*sqrt(sqrt(2) + 2)*(a/c)^(5/8) + d*sqrt(-sqrt(2) + 2)*(a/c)^ (1/8))*log(x^2 + x*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + (a/c)^(1/4))/a - 1/16* (e*sqrt(sqrt(2) + 2)*(a/c)^(5/8) + d*sqrt(-sqrt(2) + 2)*(a/c)^(1/8))*log(x ^2 - x*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + (a/c)^(1/4))/a
Time = 11.78 (sec) , antiderivative size = 2510, normalized size of antiderivative = 4.34 \[ \int \frac {d+e x^4}{a+c x^8} \, dx=\text {Too large to display} \] Input:
int((d + e*x^4)/(a + c*x^8),x)
Output:
(atan((c^3*d^6*x - a^3*e^6*x + a*c^2*d^4*e^2*x - a^2*c*d^2*e^4*x + (2*d*e* x*(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2)))/(a^3*c^2))/(a*c^3*d^5* ((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(1/4) + a^5*c ^3*e*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3 *e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c^5))^(5/4) - 2*a^2*c^2*d^3*e^2*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^7*c ^5))^(1/4) - 3*a^3*c*d*e^4*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5) ^(1/2) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2 ))/(a^7*c^5))^(1/4)))*((a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2 ) - 4*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a ^7*c^5))^(1/4))/4 - (atan((a^3*e^6*x - c^3*d^6*x - a*c^2*d^4*e^2*x + a^2*c *d^2*e^4*x + (2*d*e*x*(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2)))/(a ^3*c^2))/(a*c^3*d^5*(-(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5)^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(a^ 7*c^5))^(1/4) + a^5*c^3*e*(-(a^2*e^4*(-a^7*c^5)^(1/2) + c^2*d^4*(-a^7*c^5) ^(1/2) + 4*a^4*c^4*d^3*e - 4*a^5*c^3*d*e^3 - 6*a*c*d^2*e^2*(-a^7*c^5)^(...
Time = 0.22 (sec) , antiderivative size = 1033, normalized size of antiderivative = 1.79 \[ \int \frac {d+e x^4}{a+c x^8} \, dx =\text {Too large to display} \] Input:
int((e*x^4+d)/(c*x^8+a),x)
Output:
(c**(3/8)*a**(1/8)*(2*sqrt(a)*sqrt(sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a** (1/8)*sqrt( - sqrt(2) + 2) - 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*e - 2*sqrt(a)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sq rt(2) + 2) - 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*e - 2*sq rt(c)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) - 2*c **(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*d - 2*sqrt(a)*sqrt(sqrt( 2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*c**(1/4)* x)/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*e + 2*sqrt(a)*sqrt(sqrt(2) + 2)* atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*c**(1/4)*x)/(c**(1/8)*a** (1/8)*sqrt(sqrt(2) + 2)))*e + 2*sqrt(c)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a **(1/8)*sqrt( - sqrt(2) + 2) + 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(sqrt( 2) + 2)))*d - 2*sqrt(a)*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/ 8)*sqrt(sqrt(2) + 2) - 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*e - 2*sqrt(a)*sqrt( - sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt( 2) + 2) - 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*e - 2*sq rt(c)*sqrt( - sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) - 2*c **(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*d + 2*sqrt(a)*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*c**(1/ 4)*x)/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*e + 2*sqrt(a)*sqrt( - sqrt (2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*c**(1/4)*x)/(c**...