Integrand size = 18, antiderivative size = 263 \[ \int \frac {d+e x^4}{a-c x^8} \, dx=\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{c}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} c^{5/8}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{a} \sqrt [8]{c} x}{\sqrt [4]{a}+\sqrt [4]{c} x^2}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{c}} \] Output:
1/4*(c^(1/2)*d+a^(1/2)*e)*arctan(c^(1/8)*x/a^(1/8))/a^(7/8)/c^(5/8)+1/8*(d -a^(1/2)*e/c^(1/2))*arctan(-1+2^(1/2)*c^(1/8)*x/a^(1/8))*2^(1/2)/a^(7/8)/c ^(1/8)+1/8*(d-a^(1/2)*e/c^(1/2))*arctan(1+2^(1/2)*c^(1/8)*x/a^(1/8))*2^(1/ 2)/a^(7/8)/c^(1/8)+1/4*(c^(1/2)*d+a^(1/2)*e)*arctanh(c^(1/8)*x/a^(1/8))/a^ (7/8)/c^(5/8)+1/8*(d-a^(1/2)*e/c^(1/2))*arctanh(2^(1/2)*a^(1/8)*c^(1/8)*x/ (a^(1/4)+c^(1/4)*x^2))*2^(1/2)/a^(7/8)/c^(1/8)
Time = 0.20 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.62 \[ \int \frac {d+e x^4}{a-c x^8} \, dx=\frac {\left (\sqrt [8]{a} \sqrt {c} d+a^{5/8} e\right ) \arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{4 a c^{5/8}}-\frac {\left (-\sqrt [8]{a} \sqrt {c} d+a^{5/8} e\right ) \arctan \left (\frac {-\sqrt {2} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2} \sqrt [8]{a}}\right )}{4 \sqrt {2} a c^{5/8}}-\frac {\left (-\sqrt [8]{a} \sqrt {c} d+a^{5/8} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt [8]{a}+2 \sqrt [8]{c} x}{\sqrt {2} \sqrt [8]{a}}\right )}{4 \sqrt {2} a c^{5/8}}-\frac {\left (\sqrt [8]{a} \sqrt {c} d+a^{5/8} e\right ) \log \left (\sqrt [8]{a}-\sqrt [8]{c} x\right )}{8 a c^{5/8}}-\frac {\left (-\sqrt [8]{a} \sqrt {c} d-a^{5/8} e\right ) \log \left (\sqrt [8]{a}+\sqrt [8]{c} x\right )}{8 a c^{5/8}}+\frac {\left (-\sqrt [8]{a} \sqrt {c} d+a^{5/8} e\right ) \log \left (\sqrt [4]{a}-\sqrt {2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2} a c^{5/8}}-\frac {\left (-\sqrt [8]{a} \sqrt {c} d+a^{5/8} e\right ) \log \left (\sqrt [4]{a}+\sqrt {2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{c} x^2\right )}{8 \sqrt {2} a c^{5/8}} \] Input:
Integrate[(d + e*x^4)/(a - c*x^8),x]
Output:
((a^(1/8)*Sqrt[c]*d + a^(5/8)*e)*ArcTan[(c^(1/8)*x)/a^(1/8)])/(4*a*c^(5/8) ) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*ArcTan[(-(Sqrt[2]*a^(1/8)) + 2*c^( 1/8)*x)/(Sqrt[2]*a^(1/8))])/(4*Sqrt[2]*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*ArcTan[(Sqrt[2]*a^(1/8) + 2*c^(1/8)*x)/(Sqrt[2]*a^(1/8))])/( 4*Sqrt[2]*a*c^(5/8)) - ((a^(1/8)*Sqrt[c]*d + a^(5/8)*e)*Log[a^(1/8) - c^(1 /8)*x])/(8*a*c^(5/8)) - ((-(a^(1/8)*Sqrt[c]*d) - a^(5/8)*e)*Log[a^(1/8) + c^(1/8)*x])/(8*a*c^(5/8)) + ((-(a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*Log[a^(1/4 ) - Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a*c^(5/8)) - ((-( a^(1/8)*Sqrt[c]*d) + a^(5/8)*e)*Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2])/(8*Sqrt[2]*a*c^(5/8))
Time = 0.53 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {1747, 755, 27, 756, 218, 221, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {d+e x^4}{a-c x^8} \, dx\) |
| \(\Big \downarrow \) 1747 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \int \frac {1}{a-\sqrt {a} \sqrt {c} x^4}dx+\frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {a} \sqrt {c} x^4+a}dx\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \int \frac {1}{a-\sqrt {a} \sqrt {c} x^4}dx+\frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{c} x^2}{\sqrt {a} \left (\sqrt {c} x^4+\sqrt {a}\right )}dx}{2 \sqrt [4]{a}}+\frac {\int \frac {\sqrt [4]{c} x^2+\sqrt [4]{a}}{\sqrt {a} \left (\sqrt {c} x^4+\sqrt {a}\right )}dx}{2 \sqrt [4]{a}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{c} x^2}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}+\frac {\int \frac {\sqrt [4]{c} x^2+\sqrt [4]{a}}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \int \frac {1}{a-\sqrt {a} \sqrt {c} x^4}dx\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\int \frac {1}{\sqrt [4]{a}-\sqrt [4]{c} x^2}dx}{2 a^{3/4}}+\frac {\int \frac {1}{\sqrt [4]{c} x^2+\sqrt [4]{a}}dx}{2 a^{3/4}}\right )+\frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{c} x^2}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}+\frac {\int \frac {\sqrt [4]{c} x^2+\sqrt [4]{a}}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\int \frac {1}{\sqrt [4]{a}-\sqrt [4]{c} x^2}dx}{2 a^{3/4}}+\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )+\frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{c} x^2}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}+\frac {\int \frac {\sqrt [4]{c} x^2+\sqrt [4]{a}}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{c} x^2}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}+\frac {\int \frac {\sqrt [4]{c} x^2+\sqrt [4]{a}}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [4]{c}}}{2 a^{3/4}}+\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{c} x^2}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{c} x^2}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{c} x^2}{\sqrt {c} x^4+\sqrt {a}}dx}{2 a^{3/4}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} x+\sqrt [8]{a}\right )}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{a}-2 \sqrt [8]{c} x}{\sqrt [8]{c} \left (x^2-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} x+\sqrt [8]{a}\right )}{\sqrt [8]{c} \left (x^2+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}\right )}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{a}-2 \sqrt [8]{c} x}{x^2-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt {2} \sqrt [8]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} x+\sqrt [8]{a}}{x^2+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{c}}+\frac {\sqrt [4]{a}}{\sqrt [4]{c}}}dx}{2 \sqrt [8]{a} \sqrt [4]{c}}}{2 a^{3/4}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}\right )+\frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt {a} e}{\sqrt {c}}+d\right ) \left (\frac {\arctan \left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{2 a^{7/8} \sqrt [8]{c}}\right )+\frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}+1\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} x}{\sqrt [8]{a}}\right )}{\sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{a} \sqrt [8]{c} x+\sqrt [4]{a}+\sqrt [4]{c} x^2\right )}{2 \sqrt {2} \sqrt [8]{a} \sqrt [8]{c}}}{2 a^{3/4}}\right )\) |
Input:
Int[(d + e*x^4)/(a - c*x^8),x]
Output:
((d + (Sqrt[a]*e)/Sqrt[c])*(ArcTan[(c^(1/8)*x)/a^(1/8)]/(2*a^(7/8)*c^(1/8) ) + ArcTanh[(c^(1/8)*x)/a^(1/8)]/(2*a^(7/8)*c^(1/8))))/2 + ((d - (Sqrt[a]* e)/Sqrt[c])*((-(ArcTan[1 - (Sqrt[2]*c^(1/8)*x)/a^(1/8)]/(Sqrt[2]*a^(1/8)*c ^(1/8))) + ArcTan[1 + (Sqrt[2]*c^(1/8)*x)/a^(1/8)]/(Sqrt[2]*a^(1/8)*c^(1/8 )))/(2*a^(3/4)) + (-1/2*Log[a^(1/4) - Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)* x^2]/(Sqrt[2]*a^(1/8)*c^(1/8)) + Log[a^(1/4) + Sqrt[2]*a^(1/8)*c^(1/8)*x + c^(1/4)*x^2]/(2*Sqrt[2]*a^(1/8)*c^(1/8)))/(2*a^(3/4))))/2
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{ q = Rt[-a/c, 2]}, Simp[(d + e*q)/2 Int[1/(a + c*q*x^n), x], x] + Simp[(d - e*q)/2 Int[1/(a - c*q*x^n), x], x]] /; FreeQ[{a, c, d, e, n}, x] && EqQ [n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NegQ[a*c] && IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14
| method | result | size |
| default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c}\) | \(36\) |
| risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}-a \right )}{\sum }\frac {\left (\textit {\_R}^{4} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 c}\) | \(36\) |
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. 2741 vs. \(2 (176) = 352\).
Time = 0.39 (sec) , antiderivative size = 2741, normalized size of antiderivative = 10.42 \[ \int \frac {d+e x^4}{a-c x^8} \, dx=\text {Too large to display} \] 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 + 12*a*c^...
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)
Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (176) = 352\).
Time = 0.17 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.38 \[ \int \frac {d+e x^4}{a-c x^8} \, dx =\text {Too large to display} \] 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(-s qrt(2) + 2)*(-a/c)^(1/8))*arctan((2*x + sqrt(sqrt(2) + 2)*(-a/c)^(1/8))/(s qrt(-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 - 1/16*(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(-sq rt(2) + 2)*(-a/c)^(1/8))*log(x^2 - x*sqrt(-sqrt(2) + 2)*(-a/c)^(1/8) + (-a /c)^(1/4))/a
Time = 11.46 (sec) , antiderivative size = 2438, normalized size of antiderivative = 9.27 \[ \int \frac {d+e x^4}{a-c x^8} \, dx=\text {Too large to display} \] Input:
int((d + e*x^4)/(a - c*x^8),x)
Output:
(atan((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)^(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*c^2*d^4*e^2*x - c^3*d^6*x - a^3*e^6*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*...
Time = 0.20 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.43 \[ \int \frac {d+e x^4}{a-c x^8} \, dx=\frac {2 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}-2 c^{\frac {1}{4}} x}{c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}}\right ) e -2 \sqrt {c}\, \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}-2 c^{\frac {1}{4}} x}{c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}}\right ) d -2 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}+2 c^{\frac {1}{4}} x}{c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}}\right ) e +2 \sqrt {c}\, \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}+2 c^{\frac {1}{4}} x}{c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}}\right ) d +4 \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} x}{a^{\frac {1}{8}}}\right ) e +4 \sqrt {c}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} x}{a^{\frac {1}{8}}}\right ) d +\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}\, x +a^{\frac {1}{4}}+c^{\frac {1}{4}} x^{2}\right ) e -\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}\, x +a^{\frac {1}{4}}+c^{\frac {1}{4}} x^{2}\right ) e +2 \sqrt {a}\, \mathrm {log}\left (-c^{\frac {1}{8}} a^{\frac {1}{8}}-c^{\frac {1}{4}} x \right ) e -2 \sqrt {a}\, \mathrm {log}\left (c^{\frac {1}{8}} a^{\frac {1}{8}}-c^{\frac {1}{4}} x \right ) e -\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}\, x +a^{\frac {1}{4}}+c^{\frac {1}{4}} x^{2}\right ) d +\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {2}\, x +a^{\frac {1}{4}}+c^{\frac {1}{4}} x^{2}\right ) d +2 \sqrt {c}\, \mathrm {log}\left (-c^{\frac {1}{8}} a^{\frac {1}{8}}-c^{\frac {1}{4}} x \right ) d -2 \sqrt {c}\, \mathrm {log}\left (c^{\frac {1}{8}} a^{\frac {1}{8}}-c^{\frac {1}{4}} x \right ) d}{16 c^{\frac {5}{8}} a^{\frac {7}{8}}} \] Input:
int((e*x^4+d)/(-c*x^8+a),x)
Output:
(c**(3/8)*a**(1/8)*(2*sqrt(a)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(2) - 2* c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(2)))*e - 2*sqrt(c)*sqrt(2)*atan((c**(1 /8)*a**(1/8)*sqrt(2) - 2*c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(2)))*d - 2*sq rt(a)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(2) + 2*c**(1/4)*x)/(c**(1/8)*a* *(1/8)*sqrt(2)))*e + 2*sqrt(c)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(2) + 2 *c**(1/4)*x)/(c**(1/8)*a**(1/8)*sqrt(2)))*d + 4*sqrt(a)*atan((c**(1/4)*x)/ (c**(1/8)*a**(1/8)))*e + 4*sqrt(c)*atan((c**(1/4)*x)/(c**(1/8)*a**(1/8)))* d + sqrt(a)*sqrt(2)*log( - c**(1/8)*a**(1/8)*sqrt(2)*x + a**(1/4) + c**(1/ 4)*x**2)*e - sqrt(a)*sqrt(2)*log(c**(1/8)*a**(1/8)*sqrt(2)*x + a**(1/4) + c**(1/4)*x**2)*e + 2*sqrt(a)*log( - c**(1/8)*a**(1/8) - c**(1/4)*x)*e - 2* sqrt(a)*log(c**(1/8)*a**(1/8) - c**(1/4)*x)*e - sqrt(c)*sqrt(2)*log( - c** (1/8)*a**(1/8)*sqrt(2)*x + a**(1/4) + c**(1/4)*x**2)*d + sqrt(c)*sqrt(2)*l og(c**(1/8)*a**(1/8)*sqrt(2)*x + a**(1/4) + c**(1/4)*x**2)*d + 2*sqrt(c)*l og( - c**(1/8)*a**(1/8) - c**(1/4)*x)*d - 2*sqrt(c)*log(c**(1/8)*a**(1/8) - c**(1/4)*x)*d))/(16*a*c)