\(\int \frac {d+e x^2}{d^2-b x^2+e^2 x^4} \, dx\) [28]

Optimal result

Integrand size = 27, antiderivative size = 78 \[ \int \frac {d+e x^2}{d^2-b x^2+e^2 x^4} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b+2 d e}-2 e x}{\sqrt {b-2 d e}}\right )}{\sqrt {b-2 d e}}-\frac {\text {arctanh}\left (\frac {\sqrt {b+2 d e}+2 e x}{\sqrt {b-2 d e}}\right )}{\sqrt {b-2 d e}} \]

[Out]

arctanh((-2*e*x+(2*d*e+b)^(1/2))/(-2*d*e+b)^(1/2))/(-2*d*e+b)^(1/2)-arctanh((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 = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1175, 632, 212} \[ \int \frac {d+e x^2}{d^2-b x^2+e^2 x^4} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b+2 d e}-2 e x}{\sqrt {b-2 d e}}\right )}{\sqrt {b-2 d e}}-\frac {\text {arctanh}\left (\frac {\sqrt {b+2 d e}+2 e x}{\sqrt {b-2 d e}}\right )}{\sqrt {b-2 d e}} \]

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 {\tanh ^{-1}\left (\frac {\sqrt {b+2 d e}-2 e x}{\sqrt {b-2 d e}}\right )}{\sqrt {b-2 d e}}-\frac {\tanh ^{-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. \(189\) vs. \(2(78)=156\).

Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.42 \[ \int \frac {d+e x^2}{d^2-b x^2+e^2 x^4} \, dx=\frac {\frac {\left (b+2 d e+\sqrt {b^2-4 d^2 e^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {-b-\sqrt {b^2-4 d^2 e^2}}}\right )}{\sqrt {-b-\sqrt {b^2-4 d^2 e^2}}}+\frac {\left (-b-2 d e+\sqrt {b^2-4 d^2 e^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {-b+\sqrt {b^2-4 d^2 e^2}}}\right )}{\sqrt {-b+\sqrt {b^2-4 d^2 e^2}}}}{\sqrt {2} \sqrt {b^2-4 d^2 e^2}} \]

[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 = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\arctan \left (\frac {2 e x +\sqrt {2 e d +b}}{\sqrt {2 e d -b}}\right )}{\sqrt {2 e d -b}}-\frac {\arctan \left (\frac {-2 e x +\sqrt {2 e d +b}}{\sqrt {2 e d -b}}\right )}{\sqrt {2 e d -b}}\)	\(75\)
risch	\(\frac {\ln \left (e \,x^{2} \sqrt {-2 e d +b}+\left (2 e d -b \right ) x -d \sqrt {-2 e d +b}\right )}{2 \sqrt {-2 e d +b}}-\frac {\ln \left (e \,x^{2} \sqrt {-2 e d +b}+\left (-2 e d +b \right ) x -d \sqrt {-2 e d +b}\right )}{2 \sqrt {-2 e d +b}}\)	\(92\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.26 \[ \int \frac {d+e x^2}{d^2-b x^2+e^2 x^4} \, dx=\left [-\frac {\sqrt {-2 \, d e + b} \log \left (\frac {e^{2} x^{4} - {\left (4 \, d e - b\right )} x^{2} + d^{2} - 2 \, {\left (e x^{3} - d x\right )} \sqrt {-2 \, d e + b}}{e^{2} x^{4} - b x^{2} + d^{2}}\right )}{2 \, {\left (2 \, d e - b\right )}}, \frac {\sqrt {2 \, d e - b} \arctan \left (\frac {e x}{\sqrt {2 \, d e - b}}\right ) + \sqrt {2 \, d e - b} \arctan \left (\frac {{\left (e^{2} x^{3} + {\left (d e - b\right )} x\right )} \sqrt {2 \, d e - b}}{2 \, d^{2} e - b d}\right )}{2 \, d e - b}\right ] \]

[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 = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.41 \[ \int \frac {d+e x^2}{d^2-b x^2+e^2 x^4} \, dx=\frac {\sqrt {\frac {1}{b - 2 d e}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (- b \sqrt {\frac {1}{b - 2 d e}} + 2 d e \sqrt {\frac {1}{b - 2 d e}}\right )}{e} \right )}}{2} - \frac {\sqrt {\frac {1}{b - 2 d e}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (b \sqrt {\frac {1}{b - 2 d e}} - 2 d e \sqrt {\frac {1}{b - 2 d e}}\right )}{e} \right )}}{2} \]

[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. 195 vs. \(2 (69) = 138\).

Time = 0.71 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.50 \[ \int \frac {d+e x^2}{d^2-b x^2+e^2 x^4} \, dx=\frac {{\left (2 \, d^{2} e^{3} + d e^{4} + b d e^{2}\right )} \sqrt {2 \, d e - b} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {b + \sqrt {-4 \, d^{2} e^{2} + b^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} + 2 \, d^{2} e^{5} - b d e^{4} - b^{2} d e^{2}} + \frac {{\left (2 \, d^{2} e^{3} + d e^{4} + b d e^{2}\right )} \sqrt {2 \, d e - b} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {b - \sqrt {-4 \, d^{2} e^{2} + b^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} + 2 \, d^{2} e^{5} - b d e^{4} - b^{2} d e^{2}} \]

[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 = 14.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {d+e x^2}{d^2-b x^2+e^2 x^4} \, dx=\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {b-2\,d\,e}}{d-e\,x^2}\right )}{\sqrt {b-2\,d\,e}} \]

[In]

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

[Out]

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