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

Optimal. Leaf size=72 \[ \frac{x}{4 d^2 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2} \sqrt{e}} \]

[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])

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Rubi [A]  time = 0.0565035, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1150, 414, 522, 208, 205} \[ \frac{x}{4 d^2 \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 d^{5/2} \sqrt{e}} \]

Antiderivative was successfully verified.

[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 1150

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

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

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 208

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx &=\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]  time = 0.0344483, size = 65, normalized size = 0.9 \[ \frac{\frac{\sqrt{d} x}{d+e x^2}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}}{4 d^{5/2}} \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 0.016, size = 55, normalized size = 0.8 \begin{align*}{\frac{x}{4\,{d}^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{1}{2\,{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{1}{4\,{d}^{2}}{\it Artanh} \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85425, size = 421, normalized size = 5.85 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [B]  time = 0.653694, size = 226, normalized size = 3.14 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError