Integrand size = 17, antiderivative size = 581 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx=\frac {d x}{c}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}+\frac {\left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x}{\sqrt [4]{a}+\sqrt [4]{c} x^2}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )} a^{3/8} c^{9/8}}-\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x}{\sqrt [4]{a}+\sqrt [4]{c} x^2}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )} a^{3/8} c^{9/8}} \] Output:
d*x/c+1/4*((1+2^(1/2))*a^(1/2)*d+c^(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^(3/8)/c^ (9/8)-1/4*(a^(1/2)*(d-2^(1/2)*d)+c^(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^(3/8)/c^ (9/8)-1/4*((1+2^(1/2))*a^(1/2)*d+c^(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^(3/8)/c^ (9/8)+1/4*(a^(1/2)*(d-2^(1/2)*d)+c^(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^(3/8)/c^ (9/8)+1/4*(a^(1/2)*(d-2^(1/2)*d)+c^(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^(3/8)/c^(9/8)-1 /4*((1+2^(1/2))*a^(1/2)*d+c^(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^(3/8)/c^(9/8)
Time = 1.07 (sec) , antiderivative size = 551, normalized size of antiderivative = 0.95 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx=\frac {8 a c^{5/8} d x+2 \arctan \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 \arctan \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 \arctan \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 \arctan \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}} \] Input:
Integrate[(d + e/x^4)/(c + a/x^8),x]
Output:
(8*a*c^(5/8)*d*x + 2*ArcTan[Cot[Pi/8] + (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(a^ (5/8)*c*e*Cos[Pi/8] - a^(9/8)*Sqrt[c]*d*Sin[Pi/8]) + Log[a^(1/4) + c^(1/4) *x^2 + 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(a^(5/8)*c*e*Cos[Pi/8] - a^(9/8)*Sqr t[c]*d*Sin[Pi/8]) + 2*ArcTan[Cot[Pi/8] - (c^(1/8)*x*Csc[Pi/8])/a^(1/8)]*(- (a^(5/8)*c*e*Cos[Pi/8]) + a^(9/8)*Sqrt[c]*d*Sin[Pi/8]) + Log[a^(1/4) + c^( 1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Sin[Pi/8]]*(-(a^(5/8)*c*e*Cos[Pi/8]) + a^(9 /8)*Sqrt[c]*d*Sin[Pi/8]) - 2*ArcTan[(c^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi /8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8] + a^(5/8)*c*e*Sin[Pi/8]) - 2*ArcTan[(c^( 1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8] + a^(5 /8)*c*e*Sin[Pi/8]) + Log[a^(1/4) + c^(1/4)*x^2 - 2*a^(1/8)*c^(1/8)*x*Cos[P i/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8] + a^(5/8)*c*e*Sin[Pi/8]) - Log[a^(1/4) + c^(1/4)*x^2 + 2*a^(1/8)*c^(1/8)*x*Cos[Pi/8]]*(a^(9/8)*Sqrt[c]*d*Cos[Pi/8 ] + a^(5/8)*c*e*Sin[Pi/8]))/(8*a*c^(13/8))
Time = 1.46 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.49, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {1728, 1827, 1745, 27, 1483, 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.
| \(\displaystyle \int \frac {d+\frac {e}{x^4}}{\frac {a}{x^8}+c} \, dx\) |
| \(\Big \downarrow \) 1728 |
| \(\displaystyle \int \frac {x^4 \left (d x^4+e\right )}{a+c x^8}dx\) |
| \(\Big \downarrow \) 1827 |
| \(\displaystyle \frac {d x}{c}-\frac {\int \frac {a d-c e x^4}{c x^8+a}dx}{c}\) |
| \(\Big \downarrow \) 1745 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\int \frac {\sqrt {a} \left (\sqrt {2} a^{3/4} d-\sqrt [4]{c} \left (\sqrt {a} d+\sqrt {c} e\right ) x^2\right )}{\sqrt [4]{c} \left (x^4-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {a} \left (\sqrt [4]{c} \left (\sqrt {a} d+\sqrt {c} e\right ) x^2+\sqrt {2} a^{3/4} d\right )}{\sqrt [4]{c} \left (x^4+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}dx}{2 \sqrt {2} a^{3/4} \sqrt [4]{c}}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\int \frac {\sqrt {2} a^{3/4} d-\sqrt [4]{c} \left (\sqrt {a} d+\sqrt {c} e\right ) x^2}{x^4-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt [4]{c} \left (\sqrt {a} d+\sqrt {c} e\right ) x^2+\sqrt {2} a^{3/4} d}{x^4+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {c^{3/8} \int \frac {\sqrt [4]{a} \left (\sqrt {2 \left (2-\sqrt {2}\right )} a^{5/8} d+\sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) x\right )}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2-\sqrt {2}} a^{3/8}}+\frac {c^{3/8} \int \frac {\sqrt [4]{a} \left (\sqrt {2 \left (2-\sqrt {2}\right )} a^{5/8} d-\sqrt [8]{c} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) x\right )}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2-\sqrt {2}} a^{3/8}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\frac {c^{3/8} \int \frac {\sqrt [4]{a} \left (\sqrt {2 \left (2+\sqrt {2}\right )} a^{5/8} d-\sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) x\right )}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2+\sqrt {2}} a^{3/8}}+\frac {c^{3/8} \int \frac {\sqrt [4]{a} \left (\sqrt {2 \left (2+\sqrt {2}\right )} a^{5/8} d+\sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) x\right )}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2+\sqrt {2}} a^{3/8}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2-\sqrt {2}\right )} a^{5/8} d+\sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) x}{x^2-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2-\sqrt {2}\right )} a^{5/8} d-\sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\right ) x}{x^2+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2+\sqrt {2}\right )} a^{5/8} d-\sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) x}{x^2-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \int \frac {\sqrt {2 \left (2+\sqrt {2}\right )} a^{5/8} d+\sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) x}{x^2+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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+\frac {1}{2} \sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\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 {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} 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\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (-\frac {1}{2} \sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\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 (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\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 {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} 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\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\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 {1}{2} \sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\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 [4]{c} \left (\frac {1}{2} \sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} 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\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\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 {1}{2} \sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\frac {1}{2} \sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {\sqrt [4]{c} \left (-\frac {1}{2} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\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-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 )\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (-\frac {1}{2} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} 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-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 )\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\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+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {\sqrt [4]{c} \left (\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}} \sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\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 [4]{c} \left (\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}} \sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} 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\right )}{2 \sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\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-\sqrt {\frac {2+\sqrt {2}}{2-\sqrt {2}}} \sqrt [8]{c} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [4]{c} \left (\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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-\sqrt {\frac {2+\sqrt {2}}{2-\sqrt {2}}} \sqrt [8]{c} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d x}{c}-\frac {\frac {\frac {\sqrt [4]{c} \left (\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}} \sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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} \sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\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 (\sqrt {\frac {2-\sqrt {2}}{2+\sqrt {2}}} \sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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} \sqrt [8]{c} \left (\sqrt {a} \left (d-\sqrt {2} d\right )+\sqrt {c} e\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}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\frac {\sqrt [4]{c} \left (-\sqrt {\frac {2+\sqrt {2}}{2-\sqrt {2}}} \sqrt [8]{c} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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} \sqrt [8]{c} \left (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\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 (\left (1+\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [4]{c} x^2+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}\right )-\sqrt {\frac {2+\sqrt {2}}{2-\sqrt {2}}} \sqrt [8]{c} \left (\left (1-\sqrt {2}\right ) \sqrt {a} d+\sqrt {c} 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 )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}}{c}\) |
Input:
Int[(d + e/x^4)/(c + a/x^8),x]
Output:
(d*x)/c - (((c^(1/4)*(Sqrt[(2 - Sqrt[2])/(2 + Sqrt[2])]*c^(1/8)*((1 + Sqrt [2])*Sqrt[a]*d + Sqrt[c]*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^(1/8)*(Sqrt[a]*(d - Sq rt[2]*d) + Sqrt[c]*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^(1/4)*(Sqrt[(2 - Sqrt [2])/(2 + Sqrt[2])]*c^(1/8)*((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*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^(1/8)*(Sqrt[a]*(d - Sqrt[2]*d) + Sqrt[c]*e)*Log[a^(1/4) + Sqr t[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^(1/4)*Sqrt[c]) + ((c^(1/4)*(-(Sqrt[(2 + Sqrt[2])/(2 - Sqrt[2])]*c^(1/8)*((1 - Sqrt[2])*Sqrt[a]*d + Sqrt[c]*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^(1/8)*((1 + Sqrt[2])*Sqrt[a]*d + Sqrt[c]*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^(1/4)*(-(Sqrt[(2 + Sqrt[2])/(2 - Sqrt[2])]*c^(1/8)*((1 - Sqrt[2 ])*Sqrt[a]*d + Sqrt[c]*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^(1/8)*((1 + Sqrt[2])*Sqrt[ a]*d + Sqrt[c]*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)))/(2*Sqrt[2]*a^(1/4)*Sqrt[c]))/ 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[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> Int[x^(n*(2*p + q))*(e + d/x^n)^q*(c + a/x^(2*n))^p, x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]
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]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^( p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*(m + n*(2*p + 1) + 1)) In t[(f*x)^(m - n)*(a + c*x^(2*n))^p*(a*e*(m - n + 1) - c*d*(m + n*(2*p + 1) + 1)*x^n), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.08
| method | result | size |
| default | \(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +a \right )}{\sum }\frac {\left (\textit {\_R}^{4} c e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c^{2}}\) | \(45\) |
| risch | \(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +a \right )}{\sum }\frac {\left (\textit {\_R}^{4} c e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c^{2}}\) | \(45\) |
Input:
int((d+e/x^4)/(c+a/x^8),x,method=_RETURNVERBOSE)
Output:
d*x/c+1/8/c^2*sum((_R^4*c*e-a*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. 2730 vs. \(2 (409) = 818\).
Time = 0.36 (sec) , antiderivative size = 2730, normalized size of antiderivative = 4.70 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx=\text {Too large to display} \] Input:
integrate((d+e/x^4)/(c+a/x^8),x, algorithm="fricas")
Output:
-1/8*(c*sqrt(-sqrt((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d ^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/( a*c^4)))*log((a^3*d^6 - 5*a^2*c*d^4*e^2 - 5*a*c^2*d^2*e^4 + c^3*e^6)*x + ( a^2*c^6*e*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^ 3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + a^3*c*d^5 - 6*a^2*c^2*d^3*e^2 + a*c^3*d* e^4)*sqrt(-sqrt((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4* e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/(a*c ^4)))) - c*sqrt(-sqrt((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^ 2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3 )/(a*c^4)))*log((a^3*d^6 - 5*a^2*c*d^4*e^2 - 5*a*c^2*d^2*e^4 + c^3*e^6)*x - (a^2*c^6*e*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a *c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + a^3*c*d^5 - 6*a^2*c^2*d^3*e^2 + a*c^3 *d*e^4)*sqrt(-sqrt((a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d ^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) + 4*a*d^3*e - 4*c*d*e^3)/( a*c^4)))) - c*sqrt(-sqrt(-(a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^ 2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - 4*a*d^3*e + 4*c*d *e^3)/(a*c^4)))*log((a^3*d^6 - 5*a^2*c*d^4*e^2 - 5*a*c^2*d^2*e^4 + c^3*e^6 )*x + (a^2*c^6*e*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a*c^3*d^2*e^6 + c^4*e^8)/(a^3*c^9)) - a^3*c*d^5 + 6*a^2*c^2*d^3*e^2 - a *c^3*d*e^4)*sqrt(-sqrt(-(a*c^4*sqrt(-(a^4*d^8 - 12*a^3*c*d^6*e^2 + 38*a...
Timed out. \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx=\text {Timed out} \] Input:
integrate((d+e/x**4)/(c+a/x**8),x)
Output:
Timed out
\[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx=\int { \frac {d + \frac {e}{x^{4}}}{c + \frac {a}{x^{8}}} \,d x } \] Input:
integrate((d+e/x^4)/(c+a/x^8),x, algorithm="maxima")
Output:
d*x/c + integrate((c*e*x^4 - a*d)/(c*x^8 + a), x)/c
Time = 0.19 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.10 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx =\text {Too large to display} \] Input:
integrate((d+e/x^4)/(c+a/x^8),x, algorithm="giac")
Output:
d*x/c - 1/8*(c*e*sqrt(-sqrt(2) + 2)*(a/c)^(5/8) + a*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*c) - 1/8*(c*e*sqrt(-sqrt(2) + 2)*(a/c)^(5/8) + a*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*c) + 1/8*(c*e*sqrt(sqrt(2) + 2)*(a/c)^( 5/8) - a*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*c) + 1/8*(c*e*sqrt(sqrt (2) + 2)*(a/c)^(5/8) - a*d*sqrt(-sqrt(2) + 2)*(a/c)^(1/8))*arctan((2*x - s qrt(sqrt(2) + 2)*(a/c)^(1/8))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*c) - 1/ 16*(c*e*sqrt(-sqrt(2) + 2)*(a/c)^(5/8) + a*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*c) + 1/16*(c *e*sqrt(-sqrt(2) + 2)*(a/c)^(5/8) + a*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*c) + 1/16*(c*e*sq rt(sqrt(2) + 2)*(a/c)^(5/8) - a*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*c) - 1/16*(c*e*sqrt(s qrt(2) + 2)*(a/c)^(5/8) - a*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*c)
Time = 1.30 (sec) , antiderivative size = 2520, normalized size of antiderivative = 4.34 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx=\text {Too large to display} \] Input:
int((d + e/x^4)/(c + a/x^8),x)
Output:
(atan((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 + (2*d*e* x*(a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^9)^(1/2) + 4*a^2*c^6*d*e^3 - 4*a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1/2)))/(a*c^4))/(a^2*c^6*e*(- (a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^9)^(1/2) + 4*a^2*c^6*d*e^3 - 4 *a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1/2))/(a^3*c^9))^(5/4) - a^3*c* d^5*(-(a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^9)^(1/2) + 4*a^2*c^6*d*e ^3 - 4*a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1/2))/(a^3*c^9))^(1/4) + 2*a^2*c^2*d^3*e^2*(-(a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^9)^(1/2) + 4*a^2*c^6*d*e^3 - 4*a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1/2))/(a^3* c^9))^(1/4) + 3*a*c^3*d*e^4*(-(a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^ 9)^(1/2) + 4*a^2*c^6*d*e^3 - 4*a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1 /2))/(a^3*c^9))^(1/4)))*(-(a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^9)^( 1/2) + 4*a^2*c^6*d*e^3 - 4*a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1/2)) /(a^3*c^9))^(1/4))/4 - (atan((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 + (2*d*e*x*(a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^9)^(1 /2) - 4*a^2*c^6*d*e^3 + 4*a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1/2))) /(a*c^4))/(a^2*c^6*e*((a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^9)^(1/2) - 4*a^2*c^6*d*e^3 + 4*a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1/2))/(a^ 3*c^9))^(5/4) - a^3*c*d^5*((a^2*d^4*(-a^3*c^9)^(1/2) + c^2*e^4*(-a^3*c^9)^ (1/2) - 4*a^2*c^6*d*e^3 + 4*a^3*c^5*d^3*e - 6*a*c*d^2*e^2*(-a^3*c^9)^(1...
Time = 0.25 (sec) , antiderivative size = 1135, normalized size of antiderivative = 1.95 \[ \int \frac {d+\frac {e}{x^4}}{c+\frac {a}{x^8}} \, dx =\text {Too large to display} \] Input:
int((d+e/x^4)/(c+a/x^8),x)
Output:
(2*c**(3/8)*a**(5/8)*sqrt(sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqr t( - sqrt(2) + 2) - 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*c *e - 2*c**(3/8)*a**(5/8)*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)))*c*e + 2*c**(7/8)*a**(1/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqr t(2) + 2) - 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*a*d - 2*c **(3/8)*a**(5/8)*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)))*c*e + 2*c**(3/8)*a**(5/8)*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqr t(2) + 2) + 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2)))*c*e - 2*c **(7/8)*a**(1/8)*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)))*a*d - 2*c**(3 /8)*a**(5/8)*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(sqr t(2) + 2) - 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)))*c*e - 2*c**(3/8)*a**(5/8)*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)))*c*e + 2 *c**(7/8)*a**(1/8)*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)))*a*d + 2* c**(3/8)*a**(5/8)*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqr t(sqrt(2) + 2) + 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2))...