3.7 \(\int \frac{d+e x^4}{d^2-b x^4+e^2 x^8} \, dx\)

Optimal. Leaf size=349 \[ -\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}} \]

[Out]

-((Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/
(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])) - (Sqrt[e]*Ar
cTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt
[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]
*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*S
qrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/S
qrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b -
2*d*e] + Sqrt[b + 2*d*e]])

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Rubi [A]  time = 0.924635, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}-\sqrt{b+2 d e}}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}}\right )}{\sqrt{2} \sqrt{b-2 d e} \sqrt{\sqrt{b-2 d e}+\sqrt{b+2 d e}}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]

[Out]

-((Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/
(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]])) - (Sqrt[e]*Ar
cTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt
[b - 2*d*e]*Sqrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]
*Sqrt[e]*x)/Sqrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*S
qrt[Sqrt[b - 2*d*e] - Sqrt[b + 2*d*e]]) - (Sqrt[e]*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/S
qrt[Sqrt[b - 2*d*e] + Sqrt[b + 2*d*e]]])/(Sqrt[2]*Sqrt[b - 2*d*e]*Sqrt[Sqrt[b -
2*d*e] + Sqrt[b + 2*d*e]])

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Rubi in Sympy [A]  time = 76.3425, size = 333, normalized size = 0.95 \[ - \frac{\sqrt{2} \sqrt{e} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b - 2 d e} + \sqrt{b + 2 d e}}} \right )}}{2 \sqrt{b - 2 d e} \sqrt{\sqrt{b - 2 d e} + \sqrt{b + 2 d e}}} - \frac{\sqrt{2} \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b - 2 d e} + \sqrt{b + 2 d e}}} \right )}}{2 \sqrt{b - 2 d e} \sqrt{\sqrt{b - 2 d e} + \sqrt{b + 2 d e}}} - \frac{\sqrt{2} \sqrt{e} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b - 2 d e} - \sqrt{b + 2 d e}}} \right )}}{2 \sqrt{b - 2 d e} \sqrt{\sqrt{b - 2 d e} - \sqrt{b + 2 d e}}} - \frac{\sqrt{2} \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{\sqrt{b - 2 d e} - \sqrt{b + 2 d e}}} \right )}}{2 \sqrt{b - 2 d e} \sqrt{\sqrt{b - 2 d e} - \sqrt{b + 2 d e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**4+d)/(e**2*x**8-b*x**4+d**2),x)

[Out]

-sqrt(2)*sqrt(e)*atan(sqrt(2)*sqrt(e)*x/sqrt(sqrt(b - 2*d*e) + sqrt(b + 2*d*e)))
/(2*sqrt(b - 2*d*e)*sqrt(sqrt(b - 2*d*e) + sqrt(b + 2*d*e))) - sqrt(2)*sqrt(e)*a
tanh(sqrt(2)*sqrt(e)*x/sqrt(sqrt(b - 2*d*e) + sqrt(b + 2*d*e)))/(2*sqrt(b - 2*d*
e)*sqrt(sqrt(b - 2*d*e) + sqrt(b + 2*d*e))) - sqrt(2)*sqrt(e)*atan(sqrt(2)*sqrt(
e)*x/sqrt(sqrt(b - 2*d*e) - sqrt(b + 2*d*e)))/(2*sqrt(b - 2*d*e)*sqrt(sqrt(b - 2
*d*e) - sqrt(b + 2*d*e))) - sqrt(2)*sqrt(e)*atanh(sqrt(2)*sqrt(e)*x/sqrt(sqrt(b
- 2*d*e) - sqrt(b + 2*d*e)))/(2*sqrt(b - 2*d*e)*sqrt(sqrt(b - 2*d*e) - sqrt(b +
2*d*e)))

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Mathematica [C]  time = 0.0612332, size = 69, normalized size = 0.2 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8 e^2-\text{$\#$1}^4 b+d^2\&,\frac{\text{$\#$1}^4 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 e^2-\text{$\#$1}^3 b}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^4)/(d^2 - b*x^4 + e^2*x^8),x]

[Out]

RootSum[d^2 - b*#1^4 + e^2*#1^8 & , (d*Log[x - #1] + e*Log[x - #1]*#1^4)/(-(b*#1
^3) + 2*e^2*#1^7) & ]/4

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Maple [C]  time = 0.041, size = 55, normalized size = 0.2 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({e}^{2}{{\it \_Z}}^{8}-b{{\it \_Z}}^{4}+{d}^{2} \right ) }{\frac{ \left ({{\it \_R}}^{4}e+d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}{e}^{2}-{{\it \_R}}^{3}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^4+d)/(e^2*x^8-b*x^4+d^2),x)

[Out]

1/4*sum((_R^4*e+d)/(2*_R^7*e^2-_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*e^2-_Z^4*b+d^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{4} + d}{e^{2} x^{8} - b x^{4} + d^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^4 + d)/(e^2*x^8 - b*x^4 + d^2),x, algorithm="maxima")

[Out]

integrate((e*x^4 + d)/(e^2*x^8 - b*x^4 + d^2), x)

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Fricas [A]  time = 0.328652, size = 3079, normalized size = 8.82 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^4 + d)/(e^2*x^8 - b*x^4 + d^2),x, algorithm="fricas")

[Out]

-sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7
*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*
d^2)))*arctan(1/2*(2*d*e + (4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(
8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)*sqrt(((4*
d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b
^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))/(e*x + sqrt(1/2)*e
*sqrt((2*e^2*x^2 - sqrt(1/2)*(2*b*d*e - b^2 - (8*d^5*e^3 - 12*b*d^4*e^2 + 6*b^2*
d^3*e - b^3*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3
*d^4)))*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 1
2*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))/e
^2))) + sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b
)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*
e + b^2*d^2)))*arctan(-1/2*(2*d*e - (4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d
*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)*
sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6
*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))/(e*x + s
qrt(1/2)*e*sqrt((2*e^2*x^2 - sqrt(1/2)*(2*b*d*e - b^2 + (8*d^5*e^3 - 12*b*d^4*e^
2 + 6*b^2*d^3*e - b^3*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d
^5*e - b^3*d^4)))*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*
d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b
^2*d^2)))/e^2))) + 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sq
rt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*
e^2 - 4*b*d^3*e + b^2*d^2)))*log(e*x + 1/2*(2*d*e + (4*d^4*e^2 - 4*b*d^3*e + b^2
*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)
*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7
*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*
d^2)))) - 1/4*sqrt(sqrt(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*
e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b
*d^3*e + b^2*d^2)))*log(e*x - 1/2*(2*d*e + (4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqr
t(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqr
t(1/2)*sqrt(((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12
*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) + b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))) +
1/4*sqrt(sqrt(1/2)*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(
8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e +
b^2*d^2)))*log(e*x + 1/2*(2*d*e - (4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*
e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)*s
qrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*
e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)))) - 1/4*sq
rt(sqrt(1/2)*sqrt(-((4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e
^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^
2)))*log(e*x - 1/2*(2*d*e - (4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/
(8*d^7*e^3 - 12*b*d^6*e^2 + 6*b^2*d^5*e - b^3*d^4)) - b)*sqrt(sqrt(1/2)*sqrt(-((
4*d^4*e^2 - 4*b*d^3*e + b^2*d^2)*sqrt(-(2*d*e + b)/(8*d^7*e^3 - 12*b*d^6*e^2 + 6
*b^2*d^5*e - b^3*d^4)) - b)/(4*d^4*e^2 - 4*b*d^3*e + b^2*d^2))))

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Sympy [A]  time = 20.6786, size = 136, normalized size = 0.39 \[ \operatorname{RootSum}{\left (t^{8} \left (65536 b^{4} d^{2} - 524288 b^{3} d^{3} e + 1572864 b^{2} d^{4} e^{2} - 2097152 b d^{5} e^{3} + 1048576 d^{6} e^{4}\right ) + t^{4} \left (- 256 b^{3} + 1024 b^{2} d e - 1024 b d^{2} e^{2}\right ) + e^{2}, \left ( t \mapsto t \log{\left (x + \frac{1024 t^{5} b^{2} d^{2} - 4096 t^{5} b d^{3} e + 4096 t^{5} d^{4} e^{2} - 4 t b + 4 t d e}{e} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**4+d)/(e**2*x**8-b*x**4+d**2),x)

[Out]

RootSum(_t**8*(65536*b**4*d**2 - 524288*b**3*d**3*e + 1572864*b**2*d**4*e**2 - 2
097152*b*d**5*e**3 + 1048576*d**6*e**4) + _t**4*(-256*b**3 + 1024*b**2*d*e - 102
4*b*d**2*e**2) + e**2, Lambda(_t, _t*log(x + (1024*_t**5*b**2*d**2 - 4096*_t**5*
b*d**3*e + 4096*_t**5*d**4*e**2 - 4*_t*b + 4*_t*d*e)/e)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{4} + d}{e^{2} x^{8} - b x^{4} + d^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^4 + d)/(e^2*x^8 - b*x^4 + d^2),x, algorithm="giac")

[Out]

integrate((e*x^4 + d)/(e^2*x^8 - b*x^4 + d^2), x)