\(\int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx\) [26]

Optimal result

Integrand size = 26, antiderivative size = 82 \[ \int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {-b+2 d e}-2 e x}{\sqrt {b+2 d e}}\right )}{\sqrt {b+2 d e}}+\frac {\arctan \left (\frac {\sqrt {-b+2 d e}+2 e x}{\sqrt {b+2 d e}}\right )}{\sqrt {b+2 d e}} \]

[Out]

-arctan((-2*e*x+(2*d*e-b)^(1/2))/(2*d*e+b)^(1/2))/(2*d*e+b)^(1/2)+arctan((2*e*x+(2*d*e-b)^(1/2))/(2*d*e+b)^(1/
2))/(2*d*e+b)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1175, 632, 210} \[ \int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {2 d e-b}+2 e x}{\sqrt {b+2 d e}}\right )}{\sqrt {b+2 d e}}-\frac {\arctan \left (\frac {\sqrt {2 d e-b}-2 e x}{\sqrt {b+2 d e}}\right )}{\sqrt {b+2 d e}} \]

[In]

Int[(d + e*x^2)/(d^2 + b*x^2 + e^2*x^4),x]

[Out]

-(ArcTan[(Sqrt[-b + 2*d*e] - 2*e*x)/Sqrt[b + 2*d*e]]/Sqrt[b + 2*d*e]) + ArcTan[(Sqrt[-b + 2*d*e] + 2*e*x)/Sqrt
[b + 2*d*e]]/Sqrt[b + 2*d*e]

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1175

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !Lt
Q[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\frac {d}{e}-\frac {\sqrt {-b+2 d e} x}{e}+x^2} \, dx}{2 e}+\frac {\int \frac {1}{\frac {d}{e}+\frac {\sqrt {-b+2 d e} x}{e}+x^2} \, dx}{2 e} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{-\frac {b+2 d e}{e^2}-x^2} \, dx,x,-\frac {\sqrt {-b+2 d e}}{e}+2 x\right )}{e}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {b+2 d e}{e^2}-x^2} \, dx,x,\frac {\sqrt {-b+2 d e}}{e}+2 x\right )}{e} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {-b+2 d e}-2 e x}{\sqrt {b+2 d e}}\right )}{\sqrt {b+2 d e}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-b+2 d e}+2 e x}{\sqrt {b+2 d e}}\right )}{\sqrt {b+2 d e}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(181\) vs. \(2(82)=164\).

Time = 0.08 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.21 \[ \int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx=\frac {\frac {\left (-b+2 d e+\sqrt {b^2-4 d^2 e^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {b-\sqrt {b^2-4 d^2 e^2}}}\right )}{\sqrt {b-\sqrt {b^2-4 d^2 e^2}}}+\frac {\left (b-2 d e+\sqrt {b^2-4 d^2 e^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {b+\sqrt {b^2-4 d^2 e^2}}}\right )}{\sqrt {b+\sqrt {b^2-4 d^2 e^2}}}}{\sqrt {2} \sqrt {b^2-4 d^2 e^2}} \]

[In]

Integrate[(d + e*x^2)/(d^2 + b*x^2 + e^2*x^4),x]

[Out]

(((-b + 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b - Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[b - Sqrt[b^
2 - 4*d^2*e^2]] + ((b - 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b + Sqrt[b^2 - 4*d^2*e^2]]])/
Sqrt[b + Sqrt[b^2 - 4*d^2*e^2]])/(Sqrt[2]*Sqrt[b^2 - 4*d^2*e^2])

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87

method	result	size
default	\(-\frac {\arctan \left (\frac {-2 e x +\sqrt {2 e d -b}}{\sqrt {2 e d +b}}\right )}{\sqrt {2 e d +b}}+\frac {\arctan \left (\frac {2 e x +\sqrt {2 e d -b}}{\sqrt {2 e d +b}}\right )}{\sqrt {2 e d +b}}\)	\(71\)
risch	\(-\frac {\ln \left (-e \,x^{2} \sqrt {-2 e d -b}+\left (2 e d +b \right ) x +d \sqrt {-2 e d -b}\right )}{2 \sqrt {-2 e d -b}}+\frac {\ln \left (-e \,x^{2} \sqrt {-2 e d -b}+\left (-2 e d -b \right ) x +d \sqrt {-2 e d -b}\right )}{2 \sqrt {-2 e d -b}}\)	\(104\)

[In]

int((e*x^2+d)/(e^2*x^4+b*x^2+d^2),x,method=_RETURNVERBOSE)

[Out]

-arctan((-2*e*x+(2*d*e-b)^(1/2))/(2*d*e+b)^(1/2))/(2*d*e+b)^(1/2)+arctan((2*e*x+(2*d*e-b)^(1/2))/(2*d*e+b)^(1/
2))/(2*d*e+b)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.98 \[ \int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx=\left [-\frac {\sqrt {-2 \, d e - b} \log \left (\frac {e^{2} x^{4} - {\left (4 \, d e + b\right )} x^{2} + d^{2} - 2 \, {\left (e x^{3} - d x\right )} \sqrt {-2 \, d e - b}}{e^{2} x^{4} + b x^{2} + d^{2}}\right )}{2 \, {\left (2 \, d e + b\right )}}, \frac {\sqrt {2 \, d e + b} \arctan \left (\frac {e x}{\sqrt {2 \, d e + b}}\right ) + \sqrt {2 \, d e + b} \arctan \left (\frac {{\left (e^{2} x^{3} + {\left (d e + b\right )} x\right )} \sqrt {2 \, d e + b}}{2 \, d^{2} e + b d}\right )}{2 \, d e + b}\right ] \]

[In]

integrate((e*x^2+d)/(e^2*x^4+b*x^2+d^2),x, algorithm="fricas")

[Out]

[-1/2*sqrt(-2*d*e - b)*log((e^2*x^4 - (4*d*e + b)*x^2 + d^2 - 2*(e*x^3 - d*x)*sqrt(-2*d*e - b))/(e^2*x^4 + b*x
^2 + d^2))/(2*d*e + b), (sqrt(2*d*e + b)*arctan(e*x/sqrt(2*d*e + b)) + sqrt(2*d*e + b)*arctan((e^2*x^3 + (d*e
+ b)*x)*sqrt(2*d*e + b)/(2*d^2*e + b*d)))/(2*d*e + b)]

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.49 \[ \int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx=- \frac {\sqrt {- \frac {1}{b + 2 d e}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (- b \sqrt {- \frac {1}{b + 2 d e}} - 2 d e \sqrt {- \frac {1}{b + 2 d e}}\right )}{e} \right )}}{2} + \frac {\sqrt {- \frac {1}{b + 2 d e}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (b \sqrt {- \frac {1}{b + 2 d e}} + 2 d e \sqrt {- \frac {1}{b + 2 d e}}\right )}{e} \right )}}{2} \]

[In]

integrate((e*x**2+d)/(e**2*x**4+b*x**2+d**2),x)

[Out]

-sqrt(-1/(b + 2*d*e))*log(-d/e + x**2 + x*(-b*sqrt(-1/(b + 2*d*e)) - 2*d*e*sqrt(-1/(b + 2*d*e)))/e)/2 + sqrt(-
1/(b + 2*d*e))*log(-d/e + x**2 + x*(b*sqrt(-1/(b + 2*d*e)) + 2*d*e*sqrt(-1/(b + 2*d*e)))/e)/2

Maxima [F]

\[ \int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx=\int { \frac {e x^{2} + d}{e^{2} x^{4} + b x^{2} + d^{2}} \,d x } \]

[In]

integrate((e*x^2+d)/(e^2*x^4+b*x^2+d^2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)/(e^2*x^4 + b*x^2 + d^2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (73) = 146\).

Time = 0.70 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.30 \[ \int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx=\frac {{\left (2 \, d^{2} e^{3} + d e^{4} - b d e^{2}\right )} \sqrt {2 \, d e + b} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {b + \sqrt {-4 \, d^{2} e^{2} + b^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} + 2 \, d^{2} e^{5} + b d e^{4} - b^{2} d e^{2}} + \frac {{\left (2 \, d^{2} e^{3} + d e^{4} - b d e^{2}\right )} \sqrt {2 \, d e + b} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {b - \sqrt {-4 \, d^{2} e^{2} + b^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} + 2 \, d^{2} e^{5} + b d e^{4} - b^{2} d e^{2}} \]

[In]

integrate((e*x^2+d)/(e^2*x^4+b*x^2+d^2),x, algorithm="giac")

[Out]

(2*d^2*e^3 + d*e^4 - b*d*e^2)*sqrt(2*d*e + b)*arctan(2*sqrt(1/2)*x/sqrt((b + sqrt(-4*d^2*e^2 + b^2))/e^2))/(4*
d^3*e^4 + 2*d^2*e^5 + b*d*e^4 - b^2*d*e^2) + (2*d^2*e^3 + d*e^4 - b*d*e^2)*sqrt(2*d*e + b)*arctan(2*sqrt(1/2)*
x/sqrt((b - sqrt(-4*d^2*e^2 + b^2))/e^2))/(4*d^3*e^4 + 2*d^2*e^5 + b*d*e^4 - b^2*d*e^2)

Mupad [B] (verification not implemented)

Time = 14.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15 \[ \int \frac {d+e x^2}{d^2+b x^2+e^2 x^4} \, dx=\frac {\mathrm {atan}\left (\frac {e\,x}{\sqrt {b+2\,d\,e}}\right )+\mathrm {atan}\left (\frac {b^2\,x-\frac {x\,{\left (b+2\,d\,e\right )}^2}{2}+\frac {b\,x\,\left (b+2\,d\,e\right )}{2}+2\,b\,e^2\,x^3-e^2\,x^3\,\left (b+2\,d\,e\right )}{\left (b\,d-2\,d^2\,e\right )\,\sqrt {b+2\,d\,e}}\right )}{\sqrt {b+2\,d\,e}} \]

[In]

int((d + e*x^2)/(b*x^2 + d^2 + e^2*x^4),x)

[Out]

(atan((e*x)/(b + 2*d*e)^(1/2)) + atan((b^2*x - (x*(b + 2*d*e)^2)/2 + (b*x*(b + 2*d*e))/2 + 2*b*e^2*x^3 - e^2*x
^3*(b + 2*d*e))/((b*d - 2*d^2*e)*(b + 2*d*e)^(1/2))))/(b + 2*d*e)^(1/2)