Integrand size = 18, antiderivative size = 86 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \]
1/2*arctan(b^(1/4)*x/a^(1/4))*(-d*a^(1/2)+c*b^(1/2))/a^(3/4)/b^(3/4)+1/2*a rctanh(b^(1/4)*x/a^(1/4))*(d*a^(1/2)+c*b^(1/2))/a^(3/4)/b^(3/4)
Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {2 \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (\sqrt {b} c+\sqrt {a} d\right ) \left (\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )\right )}{4 a^{3/4} b^{3/4}} \]
(2*(Sqrt[b]*c - Sqrt[a]*d)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c + Sqrt [a]*d)*(Log[a^(1/4) - b^(1/4)*x] - Log[a^(1/4) + b^(1/4)*x]))/(4*a^(3/4)*b ^(3/4))
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1481, 218, 221}
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.
\(\displaystyle \int \frac {c+d x^2}{a-b x^4} \, dx\) |
\(\Big \downarrow \) 1481 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2}dx-\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {1}{-b x^2-\sqrt {a} \sqrt {b}}dx\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2}dx+\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}{2 \sqrt [4]{a} b^{3/4}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}{2 \sqrt [4]{a} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right )}{2 \sqrt [4]{a} b^{3/4}}\) |
(((Sqrt[b]*c)/Sqrt[a] - d)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(1/4)*b^(3/4) ) + (((Sqrt[b]*c)/Sqrt[a] + d)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(1/4)*b^ (3/4))
3.1.3.3.1 Defintions of rubi rules used
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[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ (-a)*c, 2]}, Simp[(e/2 + c*(d/(2*q))) Int[1/(-q + c*x^2), x], x] + Simp[( e/2 - c*(d/(2*q))) Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] & & NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.42
method | result | size |
risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (\textit {\_R}^{2} d +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b}\) | \(36\) |
default | \(\frac {c \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {d \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(104\) |
Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (58) = 116\).
Time = 0.26 (sec) , antiderivative size = 755, normalized size of antiderivative = 8.78 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x + {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x - {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x + {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) + \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x - {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) \]
1/4*sqrt((a*b*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d) /(a*b))*log(-(b^2*c^4 - a^2*d^4)*x + (a^3*b^2*d*sqrt((b^2*c^4 + 2*a*b*c^2* d^2 + a^2*d^4)/(a^3*b^3)) - a*b^2*c^3 - a^2*b*c*d^2)*sqrt((a*b*sqrt((b^2*c ^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))) - 1/4*sqrt((a*b* sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))*log(-( b^2*c^4 - a^2*d^4)*x - (a^3*b^2*d*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4) /(a^3*b^3)) - a*b^2*c^3 - a^2*b*c*d^2)*sqrt((a*b*sqrt((b^2*c^4 + 2*a*b*c^2 *d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))) - 1/4*sqrt(-(a*b*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) - 2*c*d)/(a*b))*log(-(b^2*c^4 - a^2 *d^4)*x + (a^3*b^2*d*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + a*b^2*c^3 + a^2*b*c*d^2)*sqrt(-(a*b*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d ^4)/(a^3*b^3)) - 2*c*d)/(a*b))) + 1/4*sqrt(-(a*b*sqrt((b^2*c^4 + 2*a*b*c^2 *d^2 + a^2*d^4)/(a^3*b^3)) - 2*c*d)/(a*b))*log(-(b^2*c^4 - a^2*d^4)*x - (a ^3*b^2*d*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + a*b^2*c^3 + a^2*b*c*d^2)*sqrt(-(a*b*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3 )) - 2*c*d)/(a*b)))
Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=- \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{3} - 64 t^{2} a^{2} b^{2} c d - a^{2} d^{4} + 2 a b c^{2} d^{2} - b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{3} b^{2} d + 12 t a^{2} b c d^{2} + 4 t a b^{2} c^{3}}{a^{2} d^{4} - b^{2} c^{4}} \right )} \right )\right )} \]
-RootSum(256*_t**4*a**3*b**3 - 64*_t**2*a**2*b**2*c*d - a**2*d**4 + 2*a*b* c**2*d**2 - b**2*c**4, Lambda(_t, _t*log(x + (-64*_t**3*a**3*b**2*d + 12*_ t*a**2*b*c*d**2 + 4*_t*a*b**2*c**3)/(a**2*d**4 - b**2*c**4))))
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.27 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {{\left (\sqrt {b} c - \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (\sqrt {b} c + \sqrt {a} d\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \]
1/2*(sqrt(b)*c - sqrt(a)*d)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt( a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - 1/4*(sqrt(b)*c + sqrt(a)*d)*log((sqrt( b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a )*sqrt(sqrt(a)*sqrt(b))*sqrt(b))
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (58) = 116\).
Time = 0.27 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.67 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=-\frac {\sqrt {2} {\left (b^{2} c + \sqrt {-a b} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} \]
-1/4*sqrt(2)*(b^2*c + sqrt(-a*b)*b*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(- a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^3)^(3/4) - 1/4*sqrt(2)*(b^2*c - sqrt(-a*b) *b*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^ 3)^(3/4) - 1/8*sqrt(2)*(b^2*c - sqrt(-a*b)*b*d)*log(x^2 + sqrt(2)*x*(-a/b) ^(1/4) + sqrt(-a/b))/(-a*b^3)^(3/4) + 1/8*sqrt(2)*(b^2*c - sqrt(-a*b)*b*d) *log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(-a*b^3)^(3/4)
Time = 0.33 (sec) , antiderivative size = 579, normalized size of antiderivative = 6.73 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=2\,\mathrm {atanh}\left (\frac {8\,b^3\,c^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}-\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3-\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}-\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}+\frac {8\,a\,b^2\,d^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}-\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3-\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}-\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}\right )\,\sqrt {-\frac {a\,d^2\,\sqrt {a^3\,b^3}+b\,c^2\,\sqrt {a^3\,b^3}-2\,a^2\,b^2\,c\,d}{16\,a^3\,b^3}}+2\,\mathrm {atanh}\left (\frac {8\,b^3\,c^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}+\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}+\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3+\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}+\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}+\frac {8\,a\,b^2\,d^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}+\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}+\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3+\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}+\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}\right )\,\sqrt {\frac {a\,d^2\,\sqrt {a^3\,b^3}+b\,c^2\,\sqrt {a^3\,b^3}+2\,a^2\,b^2\,c\,d}{16\,a^3\,b^3}} \]
2*atanh((8*b^3*c^2*x*((c*d)/(8*a*b) - (c^2*(a^3*b^3)^(1/2))/(16*a^3*b^2) - (d^2*(a^3*b^3)^(1/2))/(16*a^2*b^3))^(1/2))/(2*b^2*c^2*d + 2*a*b*d^3 - (2* b*c^3*(a^3*b^3)^(1/2))/a^2 - (2*c*d^2*(a^3*b^3)^(1/2))/a) + (8*a*b^2*d^2*x *((c*d)/(8*a*b) - (c^2*(a^3*b^3)^(1/2))/(16*a^3*b^2) - (d^2*(a^3*b^3)^(1/2 ))/(16*a^2*b^3))^(1/2))/(2*b^2*c^2*d + 2*a*b*d^3 - (2*b*c^3*(a^3*b^3)^(1/2 ))/a^2 - (2*c*d^2*(a^3*b^3)^(1/2))/a))*(-(a*d^2*(a^3*b^3)^(1/2) + b*c^2*(a ^3*b^3)^(1/2) - 2*a^2*b^2*c*d)/(16*a^3*b^3))^(1/2) + 2*atanh((8*b^3*c^2*x* ((c*d)/(8*a*b) + (c^2*(a^3*b^3)^(1/2))/(16*a^3*b^2) + (d^2*(a^3*b^3)^(1/2) )/(16*a^2*b^3))^(1/2))/(2*b^2*c^2*d + 2*a*b*d^3 + (2*b*c^3*(a^3*b^3)^(1/2) )/a^2 + (2*c*d^2*(a^3*b^3)^(1/2))/a) + (8*a*b^2*d^2*x*((c*d)/(8*a*b) + (c^ 2*(a^3*b^3)^(1/2))/(16*a^3*b^2) + (d^2*(a^3*b^3)^(1/2))/(16*a^2*b^3))^(1/2 ))/(2*b^2*c^2*d + 2*a*b*d^3 + (2*b*c^3*(a^3*b^3)^(1/2))/a^2 + (2*c*d^2*(a^ 3*b^3)^(1/2))/a))*((a*d^2*(a^3*b^3)^(1/2) + b*c^2*(a^3*b^3)^(1/2) + 2*a^2* b^2*c*d)/(16*a^3*b^3))^(1/2)