\(\int \frac {1}{(d+e x^2) (d^2-e^2 x^4)} \, dx\) [193]

Optimal result

Integrand size = 24, antiderivative size = 72 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {x}{4 d^2 \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \sqrt {e}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2} \sqrt {e}} \]

[Out]

1/4*x/d^2/(e*x^2+d)+1/2*arctan(x*e^(1/2)/d^(1/2))/d^(5/2)/e^(1/2)+1/4*arctanh(x*e^(1/2)/d^(1/2))/d^(5/2)/e^(1/
2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1164, 425, 536, 214, 211} \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \sqrt {e}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2} \sqrt {e}}+\frac {x}{4 d^2 \left (d+e x^2\right )} \]

[In]

Int[1/((d + e*x^2)*(d^2 - e^2*x^4)),x]

[Out]

x/(4*d^2*(d + e*x^2)) + ArcTan[(Sqrt[e]*x)/Sqrt[d]]/(2*d^(5/2)*Sqrt[e]) + ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/(4*d^(5
/2)*Sqrt[e])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

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]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 1164

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p + q)*(a/d + (c/e)
*x^2)^p, x] /; FreeQ[{a, c, d, e, q}, x] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (d-e x^2\right ) \left (d+e x^2\right )^2} \, dx \\ & = \frac {x}{4 d^2 \left (d+e x^2\right )}-\frac {\int \frac {-3 d e+e^2 x^2}{\left (d-e x^2\right ) \left (d+e x^2\right )} \, dx}{4 d^2 e} \\ & = \frac {x}{4 d^2 \left (d+e x^2\right )}+\frac {\int \frac {1}{d-e x^2} \, dx}{4 d^2}+\frac {\int \frac {1}{d+e x^2} \, dx}{2 d^2} \\ & = \frac {x}{4 d^2 \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \sqrt {e}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{4 d^{5/2} \sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {\frac {\sqrt {d} x}{d+e x^2}+\frac {2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}}{4 d^{5/2}} \]

[In]

Integrate[1/((d + e*x^2)*(d^2 - e^2*x^4)),x]

[Out]

((Sqrt[d]*x)/(d + e*x^2) + (2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + ArcTanh[(Sqrt[e]*x)/Sqrt[d]]/Sqrt[e])/(4*
d^(5/2))

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {\operatorname {arctanh}\left (\frac {e x}{\sqrt {e d}}\right )}{4 d^{2} \sqrt {e d}}+\frac {\frac {x}{e \,x^{2}+d}+\frac {2 \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{\sqrt {e d}}}{4 d^{2}}\)	\(54\)
risch	\(\frac {x}{4 d^{2} \left (e \,x^{2}+d \right )}-\frac {\ln \left (-e x -\sqrt {-e d}\right )}{4 \sqrt {-e d}\, d^{2}}+\frac {\ln \left (e x -\sqrt {-e d}\right )}{4 \sqrt {-e d}\, d^{2}}+\frac {\ln \left (e x +\sqrt {e d}\right )}{8 \sqrt {e d}\, d^{2}}-\frac {\ln \left (-e x +\sqrt {e d}\right )}{8 \sqrt {e d}\, d^{2}}\)	\(107\)

[In]

int(1/(e*x^2+d)/(-e^2*x^4+d^2),x,method=_RETURNVERBOSE)

[Out]

1/4/d^2/(e*d)^(1/2)*arctanh(e*x/(e*d)^(1/2))+1/4/d^2*(x/(e*x^2+d)+2/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.62 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\left [\frac {2 \, d e x + 4 \, {\left (e x^{2} + d\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (e x^{2} + d\right )} \sqrt {d e} \log \left (\frac {e x^{2} + 2 \, \sqrt {d e} x + d}{e x^{2} - d}\right )}{8 \, {\left (d^{3} e^{2} x^{2} + d^{4} e\right )}}, \frac {d e x - {\left (e x^{2} + d\right )} \sqrt {-d e} \arctan \left (\frac {\sqrt {-d e} x}{d}\right ) - {\left (e x^{2} + d\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right )}{4 \, {\left (d^{3} e^{2} x^{2} + d^{4} e\right )}}\right ] \]

[In]

integrate(1/(e*x^2+d)/(-e^2*x^4+d^2),x, algorithm="fricas")

[Out]

[1/8*(2*d*e*x + 4*(e*x^2 + d)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + (e*x^2 + d)*sqrt(d*e)*log((e*x^2 + 2*sqrt(d*e)
*x + d)/(e*x^2 - d)))/(d^3*e^2*x^2 + d^4*e), 1/4*(d*e*x - (e*x^2 + d)*sqrt(-d*e)*arctan(sqrt(-d*e)*x/d) - (e*x
^2 + d)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)))/(d^3*e^2*x^2 + d^4*e)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (63) = 126\).

Time = 0.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.14 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {x}{4 d^{3} + 4 d^{2} e x^{2}} - \frac {\sqrt {\frac {1}{d^{5} e}} \log {\left (- \frac {d^{8} e \left (\frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{10} - \frac {9 d^{3} \sqrt {\frac {1}{d^{5} e}}}{10} + x \right )}}{8} + \frac {\sqrt {\frac {1}{d^{5} e}} \log {\left (\frac {d^{8} e \left (\frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{10} + \frac {9 d^{3} \sqrt {\frac {1}{d^{5} e}}}{10} + x \right )}}{8} - \frac {\sqrt {- \frac {1}{d^{5} e}} \log {\left (- \frac {4 d^{8} e \left (- \frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{5} - \frac {9 d^{3} \sqrt {- \frac {1}{d^{5} e}}}{5} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{5} e}} \log {\left (\frac {4 d^{8} e \left (- \frac {1}{d^{5} e}\right )^{\frac {3}{2}}}{5} + \frac {9 d^{3} \sqrt {- \frac {1}{d^{5} e}}}{5} + x \right )}}{4} \]

[In]

integrate(1/(e*x**2+d)/(-e**2*x**4+d**2),x)

[Out]

x/(4*d**3 + 4*d**2*e*x**2) - sqrt(1/(d**5*e))*log(-d**8*e*(1/(d**5*e))**(3/2)/10 - 9*d**3*sqrt(1/(d**5*e))/10
+ x)/8 + sqrt(1/(d**5*e))*log(d**8*e*(1/(d**5*e))**(3/2)/10 + 9*d**3*sqrt(1/(d**5*e))/10 + x)/8 - sqrt(-1/(d**
5*e))*log(-4*d**8*e*(-1/(d**5*e))**(3/2)/5 - 9*d**3*sqrt(-1/(d**5*e))/5 + x)/4 + sqrt(-1/(d**5*e))*log(4*d**8*
e*(-1/(d**5*e))**(3/2)/5 + 9*d**3*sqrt(-1/(d**5*e))/5 + x)/4

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x^2+d)/(-e^2*x^4+d^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d^{2}} - \frac {\arctan \left (\frac {e x}{\sqrt {-d e}}\right )}{4 \, \sqrt {-d e} d^{2}} + \frac {x}{4 \, {\left (e x^{2} + d\right )} d^{2}} \]

[In]

integrate(1/(e*x^2+d)/(-e^2*x^4+d^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx=\frac {x}{4\,d^2\,\left (e\,x^2+d\right )}+\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {d^5\,e}}{d^3}\right )\,\sqrt {d^5\,e}}{4\,d^5\,e}-\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {-d^5\,e}}{d^3}\right )\,\sqrt {-d^5\,e}}{2\,d^5\,e} \]

[In]

int(1/((d^2 - e^2*x^4)*(d + e*x^2)),x)

[Out]

x/(4*d^2*(d + e*x^2)) + (atanh((x*(d^5*e)^(1/2))/d^3)*(d^5*e)^(1/2))/(4*d^5*e) - (atanh((x*(-d^5*e)^(1/2))/d^3
)*(-d^5*e)^(1/2))/(2*d^5*e)