3.218 \(\int \frac{1}{(d+e x^2) (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\)
Optimal. Leaf size=136 \[ -\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^2}-\frac{(4 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \]
[Out]
-x/(2*d*(2*c*d - b*e)*(d + e*x^2)) - ((4*c*d - b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]*(2*c*d - b
*e)^2) - (c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^2)
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Rubi [A] time = 0.178477, antiderivative size = 136, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used =
{1149, 414, 522, 205, 208} \[ -\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^2}-\frac{(4 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \]
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]
[Out]
-x/(2*d*(2*c*d - b*e)*(d + e*x^2)) - ((4*c*d - b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]*(2*c*d - b
*e)^2) - (c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^2)
Rule 1149
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + 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 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]
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]
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right ) \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx &=\int \frac{1}{\left (d+e x^2\right )^2 \left (\frac{-c d^2+b d e}{d}+c e x^2\right )} \, dx\\ &=-\frac{x}{2 d (2 c d-b e) \left (d+e x^2\right )}+\frac{\int \frac{e (3 c d-b e)-c e^2 x^2}{\left (d+e x^2\right ) \left (\frac{-c d^2+b d e}{d}+c e x^2\right )} \, dx}{2 d e (2 c d-b e)}\\ &=-\frac{x}{2 d (2 c d-b e) \left (d+e x^2\right )}+\frac{c^2 \int \frac{1}{\frac{-c d^2+b d e}{d}+c e x^2} \, dx}{(2 c d-b e)^2}-\frac{(4 c d-b e) \int \frac{1}{d+e x^2} \, dx}{2 d (2 c d-b e)^2}\\ &=-\frac{x}{2 d (2 c d-b e) \left (d+e x^2\right )}-\frac{(4 c d-b e) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^2}\\ \end{align*}
Mathematica [A] time = 0.20775, size = 133, normalized size = 0.98 \[ \frac{c^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{e} x}{\sqrt{b e-c d}}\right )}{\sqrt{e} (b e-2 c d)^2 \sqrt{b e-c d}}+\frac{(b e-4 c d) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} (2 c d-b e)^2}-\frac{x}{2 d \left (d+e x^2\right ) (2 c d-b e)} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x^2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]
[Out]
-x/(2*d*(2*c*d - b*e)*(d + e*x^2)) + ((-4*c*d + b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]*(2*c*d -
b*e)^2) + (c^(3/2)*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*(-2*c*d + b*e)^2*Sqrt[-(c*d) + b*e
])
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Maple [A] time = 0.02, size = 155, normalized size = 1.1 \begin{align*}{\frac{bxe}{2\, \left ( be-2\,cd \right ) ^{2}d \left ( e{x}^{2}+d \right ) }}-{\frac{cx}{ \left ( be-2\,cd \right ) ^{2} \left ( e{x}^{2}+d \right ) }}+{\frac{be}{2\, \left ( be-2\,cd \right ) ^{2}d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-2\,{\frac{c}{ \left ( be-2\,cd \right ) ^{2}\sqrt{de}}\arctan \left ({\frac{ex}{\sqrt{de}}} \right ) }+{\frac{{c}^{2}}{ \left ( be-2\,cd \right ) ^{2}}\arctan \left ({cex{\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) ce}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)
[Out]
1/2/(b*e-2*c*d)^2/d*x/(e*x^2+d)*b*e-1/(b*e-2*c*d)^2*x/(e*x^2+d)*c+1/2/(b*e-2*c*d)^2/d/(d*e)^(1/2)*arctan(x*e/(
d*e)^(1/2))*b*e-2/(b*e-2*c*d)^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*c+c^2/(b*e-2*c*d)^2/((b*e-c*d)*c*e)^(1/2)*
arctan(c*e*x/((b*e-c*d)*c*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)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.54581, size = 1829, normalized size = 13.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")
[Out]
[1/4*(2*(c*d^2*e^2*x^2 + c*d^3*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 - 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b
*e^2)) + c*d - b*e)/(c*e*x^2 - c*d + b*e)) + (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(-d*e)*log((e*x^2 -
2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 2*(2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b^2*d^3*e^3 + (4
*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2), -1/2*((4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(d*e)*ar
ctan(sqrt(d*e)*x/d) - (c*d^2*e^2*x^2 + c*d^3*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 - 2*(c*d*e - b*e^2)*x*sqr
t(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2 - c*d + b*e)) + (2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e
^2 + b^2*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2), 1/4*(4*(c*d^2*e^2*x^2 + c*d^3*e)*sqrt(-
c/(c*d*e - b*e^2))*arctan(e*x*sqrt(-c/(c*d*e - b*e^2))) + (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(-d*e)
*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 2*(2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b^2
*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4)*x^2), 1/2*(2*(c*d^2*e^2*x^2 + c*d^3*e)*sqrt(-c/(c*d*e
- b*e^2))*arctan(e*x*sqrt(-c/(c*d*e - b*e^2))) - (4*c*d^2 - b*d*e + (4*c*d*e - b*e^2)*x^2)*sqrt(d*e)*arctan(s
qrt(d*e)*x/d) - (2*c*d^2*e - b*d*e^2)*x)/(4*c^2*d^5*e - 4*b*c*d^4*e^2 + b^2*d^3*e^3 + (4*c^2*d^4*e^2 - 4*b*c*d
^3*e^3 + b^2*d^2*e^4)*x^2)]
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Sympy [B] time = 22.4644, size = 2664, normalized size = 19.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x**2+d)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
x/(2*b*d**2*e - 4*c*d**3 + x**2*(2*b*d*e**2 - 4*c*d**2*e)) - sqrt(-1/(d**3*e))*(b*e - 4*c*d)*log(x + (-b**7*d*
*3*e**8*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(2*(b*e - 2*c*d)**6) + 7*b**6*c*d**4*e**7*(-1/(d**3*e))**(3/2)*(
b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 79*b**5*c**2*d**5*e**6*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(2*(b*e - 2*c*
d)**6) - b**5*e**5*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(2*(b*e - 2*c*d)**2) + 117*b**4*c**3*d**6*e**5*(-1/(d**3*e)
)**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 + 7*b**4*c*d*e**4*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2
- 196*b**3*c**4*d**7*e**4*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 73*b**3*c**2*d**2*e**3*sqrt
(-1/(d**3*e))*(b*e - 4*c*d)/(2*(b*e - 2*c*d)**2) + 184*b**2*c**5*d**8*e**3*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)*
*3/(b*e - 2*c*d)**6 + 86*b**2*c**3*d**3*e**2*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2 - 88*b*c**6*d**9
*e**2*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 88*b*c**4*d**4*e*sqrt(-1/(d**3*e))*(b*e - 4*c*d
)/(b*e - 2*c*d)**2 + 16*c**7*d**10*e*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 + 36*c**5*d**5*sqr
t(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2)/(b**2*c**2*e**2 - 9*b*c**3*d*e + 20*c**4*d**2))/(4*(b*e - 2*c*d
)**2) + sqrt(-1/(d**3*e))*(b*e - 4*c*d)*log(x + (b**7*d**3*e**8*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(2*(b*e
- 2*c*d)**6) - 7*b**6*c*d**4*e**7*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 + 79*b**5*c**2*d**5*e
**6*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(2*(b*e - 2*c*d)**6) + b**5*e**5*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(2*
(b*e - 2*c*d)**2) - 117*b**4*c**3*d**6*e**5*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 7*b**4*c*
d*e**4*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2 + 196*b**3*c**4*d**7*e**4*(-1/(d**3*e))**(3/2)*(b*e -
4*c*d)**3/(b*e - 2*c*d)**6 + 73*b**3*c**2*d**2*e**3*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(2*(b*e - 2*c*d)**2) - 184
*b**2*c**5*d**8*e**3*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 86*b**2*c**3*d**3*e**2*sqrt(-1/(
d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2 + 88*b*c**6*d**9*e**2*(-1/(d**3*e))**(3/2)*(b*e - 4*c*d)**3/(b*e - 2*c
*d)**6 + 88*b*c**4*d**4*e*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2 - 16*c**7*d**10*e*(-1/(d**3*e))**(3
/2)*(b*e - 4*c*d)**3/(b*e - 2*c*d)**6 - 36*c**5*d**5*sqrt(-1/(d**3*e))*(b*e - 4*c*d)/(b*e - 2*c*d)**2)/(b**2*c
**2*e**2 - 9*b*c**3*d*e + 20*c**4*d**2))/(4*(b*e - 2*c*d)**2) - sqrt(-c**3/(e*(b*e - c*d)))*log(x + (-4*b**7*d
**3*e**8*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 + 56*b**6*c*d**4*e**7*(-c**3/(e*(b*e - c*d)))**(3/2)/
(b*e - 2*c*d)**6 - 316*b**5*c**2*d**5*e**6*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - b**5*e**5*sqrt(-c
**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 + 936*b**4*c**3*d**6*e**5*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**
6 + 14*b**4*c*d*e**4*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 - 1568*b**3*c**4*d**7*e**4*(-c**3/(e*(b*e -
c*d)))**(3/2)/(b*e - 2*c*d)**6 - 73*b**3*c**2*d**2*e**3*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 + 1472*b*
*2*c**5*d**8*e**3*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 + 172*b**2*c**3*d**3*e**2*sqrt(-c**3/(e*(b*e
- c*d)))/(b*e - 2*c*d)**2 - 704*b*c**6*d**9*e**2*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - 176*b*c**4
*d**4*e*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 + 128*c**7*d**10*e*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e -
2*c*d)**6 + 72*c**5*d**5*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2)/(b**2*c**2*e**2 - 9*b*c**3*d*e + 20*c**
4*d**2))/(2*(b*e - 2*c*d)**2) + sqrt(-c**3/(e*(b*e - c*d)))*log(x + (4*b**7*d**3*e**8*(-c**3/(e*(b*e - c*d)))*
*(3/2)/(b*e - 2*c*d)**6 - 56*b**6*c*d**4*e**7*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 + 316*b**5*c**2*
d**5*e**6*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 + b**5*e**5*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d
)**2 - 936*b**4*c**3*d**6*e**5*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - 14*b**4*c*d*e**4*sqrt(-c**3/(
e*(b*e - c*d)))/(b*e - 2*c*d)**2 + 1568*b**3*c**4*d**7*e**4*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 +
73*b**3*c**2*d**2*e**3*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 - 1472*b**2*c**5*d**8*e**3*(-c**3/(e*(b*e
- c*d)))**(3/2)/(b*e - 2*c*d)**6 - 172*b**2*c**3*d**3*e**2*sqrt(-c**3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2 + 704*
b*c**6*d**9*e**2*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 + 176*b*c**4*d**4*e*sqrt(-c**3/(e*(b*e - c*d)
))/(b*e - 2*c*d)**2 - 128*c**7*d**10*e*(-c**3/(e*(b*e - c*d)))**(3/2)/(b*e - 2*c*d)**6 - 72*c**5*d**5*sqrt(-c*
*3/(e*(b*e - c*d)))/(b*e - 2*c*d)**2)/(b**2*c**2*e**2 - 9*b*c**3*d*e + 20*c**4*d**2))/(2*(b*e - 2*c*d)**2)
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")
[Out]
Timed out