3.113 \(\int \frac{\sqrt{2}+x^2}{1+b x^2+x^4} \, dx\)

Optimal. Leaf size=160 \[ \frac{\left (1-\sqrt{2}\right ) \log \left (-\sqrt{2-b} x+x^2+1\right )}{4 \sqrt{2-b}}-\frac{\left (1-\sqrt{2}\right ) \log \left (\sqrt{2-b} x+x^2+1\right )}{4 \sqrt{2-b}}-\frac{\left (1+\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2-b}-2 x}{\sqrt{b+2}}\right )}{2 \sqrt{b+2}}+\frac{\left (1+\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2-b}+2 x}{\sqrt{b+2}}\right )}{2 \sqrt{b+2}} \]

[Out]

-((1 + Sqrt[2])*ArcTan[(Sqrt[2 - b] - 2*x)/Sqrt[2 + b]])/(2*Sqrt[2 + b]) + ((1 +
Sqrt[2])*ArcTan[(Sqrt[2 - b] + 2*x)/Sqrt[2 + b]])/(2*Sqrt[2 + b]) + ((1 - Sqrt[
2])*Log[1 - Sqrt[2 - b]*x + x^2])/(4*Sqrt[2 - b]) - ((1 - Sqrt[2])*Log[1 + Sqrt[
2 - b]*x + x^2])/(4*Sqrt[2 - b])

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Rubi [A]  time = 0.230413, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (1-\sqrt{2}\right ) \log \left (-\sqrt{2-b} x+x^2+1\right )}{4 \sqrt{2-b}}-\frac{\left (1-\sqrt{2}\right ) \log \left (\sqrt{2-b} x+x^2+1\right )}{4 \sqrt{2-b}}-\frac{\left (1+\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2-b}-2 x}{\sqrt{b+2}}\right )}{2 \sqrt{b+2}}+\frac{\left (1+\sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2-b}+2 x}{\sqrt{b+2}}\right )}{2 \sqrt{b+2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[2] + x^2)/(1 + b*x^2 + x^4),x]

[Out]

-((1 + Sqrt[2])*ArcTan[(Sqrt[2 - b] - 2*x)/Sqrt[2 + b]])/(2*Sqrt[2 + b]) + ((1 +
Sqrt[2])*ArcTan[(Sqrt[2 - b] + 2*x)/Sqrt[2 + b]])/(2*Sqrt[2 + b]) + ((1 - Sqrt[
2])*Log[1 - Sqrt[2 - b]*x + x^2])/(4*Sqrt[2 - b]) - ((1 - Sqrt[2])*Log[1 + Sqrt[
2 - b]*x + x^2])/(4*Sqrt[2 - b])

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Rubi in Sympy [A]  time = 34.3218, size = 134, normalized size = 0.84 \[ \frac{\left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- b + 2}}{\sqrt{b + 2}} \right )}}{\sqrt{b + 2}} + \frac{\left (\frac{1}{2} + \frac{\sqrt{2}}{2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- b + 2}}{\sqrt{b + 2}} \right )}}{\sqrt{b + 2}} + \frac{\left (- \frac{\sqrt{2}}{4} + \frac{1}{4}\right ) \log{\left (x^{2} - x \sqrt{- b + 2} + 1 \right )}}{\sqrt{- b + 2}} - \frac{\left (- \frac{\sqrt{2}}{4} + \frac{1}{4}\right ) \log{\left (x^{2} + x \sqrt{- b + 2} + 1 \right )}}{\sqrt{- b + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+2**(1/2))/(x**4+b*x**2+1),x)

[Out]

(1/2 + sqrt(2)/2)*atan((2*x - sqrt(-b + 2))/sqrt(b + 2))/sqrt(b + 2) + (1/2 + sq
rt(2)/2)*atan((2*x + sqrt(-b + 2))/sqrt(b + 2))/sqrt(b + 2) + (-sqrt(2)/4 + 1/4)
*log(x**2 - x*sqrt(-b + 2) + 1)/sqrt(-b + 2) - (-sqrt(2)/4 + 1/4)*log(x**2 + x*s
qrt(-b + 2) + 1)/sqrt(-b + 2)

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Mathematica [A]  time = 0.0935016, size = 136, normalized size = 0.85 \[ \frac{\frac{\left (\sqrt{b^2-4}-b+2 \sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{b-\sqrt{b^2-4}}}\right )}{\sqrt{b-\sqrt{b^2-4}}}+\frac{\left (\sqrt{b^2-4}+b-2 \sqrt{2}\right ) \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{b^2-4}+b}}\right )}{\sqrt{\sqrt{b^2-4}+b}}}{\sqrt{2} \sqrt{b^2-4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[2] + x^2)/(1 + b*x^2 + x^4),x]

[Out]

(((2*Sqrt[2] - b + Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x)/Sqrt[b - Sqrt[-4 + b^2]]])
/Sqrt[b - Sqrt[-4 + b^2]] + ((-2*Sqrt[2] + b + Sqrt[-4 + b^2])*ArcTan[(Sqrt[2]*x
)/Sqrt[b + Sqrt[-4 + b^2]]])/Sqrt[b + Sqrt[-4 + b^2]])/(Sqrt[2]*Sqrt[-4 + b^2])

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Maple [B]  time = 0.026, size = 283, normalized size = 1.8 \[{1\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}}}-{b\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }}}{\frac{1}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}}}+2\,{\frac{\sqrt{2}}{\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ) }+{1\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}}}+{b\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ){\frac{1}{\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }}}{\frac{1}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}}}-2\,{\frac{\sqrt{2}}{\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{ \left ( b-2 \right ) \left ( 2+b \right ) }+2\,b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+2^(1/2))/(x^4+b*x^2+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} + \sqrt{2}}{x^{4} + b x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + sqrt(2))/(x^4 + b*x^2 + 1),x, algorithm="maxima")

[Out]

integrate((x^2 + sqrt(2))/(x^4 + b*x^2 + 1), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + sqrt(2))/(x^4 + b*x^2 + 1),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 6.3439, size = 330, normalized size = 2.06 \[ \operatorname{RootSum}{\left (t^{4} \left (16 b^{4} - 128 b^{2} + 256\right ) + t^{2} \left (12 b^{3} - 16 \sqrt{2} b^{2} - 48 b + 64 \sqrt{2}\right ) + 2 b^{2} - 6 \sqrt{2} b + 9, \left ( t \mapsto t \log{\left (\frac{t^{3} \left (64 b^{12} - 672 \sqrt{2} b^{11} + 5760 b^{10} - 12064 \sqrt{2} b^{9} + 17744 b^{8} + 27480 \sqrt{2} b^{7} - 154608 b^{6} + 141376 \sqrt{2} b^{5} - 69072 b^{4} - 61704 \sqrt{2} b^{3} + 78192 b^{2} + 2592 \sqrt{2} b - 15552\right )}{8 b^{10} - 88 \sqrt{2} b^{9} + 828 b^{8} - 2144 \sqrt{2} b^{7} + 6470 b^{6} - 5310 \sqrt{2} b^{5} + 2781 b^{4} + 2322 \sqrt{2} b^{3} - 3402 b^{2} + 729} + \frac{t \left (16 b^{7} - 116 \sqrt{2} b^{6} + 668 b^{5} - 942 \sqrt{2} b^{4} + 1226 b^{3} - 144 \sqrt{2} b^{2} - 378 b + 108 \sqrt{2}\right )}{4 b^{6} - 28 \sqrt{2} b^{5} + 152 b^{4} - 192 \sqrt{2} b^{3} + 189 b^{2} + 27 \sqrt{2} b - 81} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+2**(1/2))/(x**4+b*x**2+1),x)

[Out]

RootSum(_t**4*(16*b**4 - 128*b**2 + 256) + _t**2*(12*b**3 - 16*sqrt(2)*b**2 - 48
*b + 64*sqrt(2)) + 2*b**2 - 6*sqrt(2)*b + 9, Lambda(_t, _t*log(_t**3*(64*b**12 -
672*sqrt(2)*b**11 + 5760*b**10 - 12064*sqrt(2)*b**9 + 17744*b**8 + 27480*sqrt(2
)*b**7 - 154608*b**6 + 141376*sqrt(2)*b**5 - 69072*b**4 - 61704*sqrt(2)*b**3 + 7
8192*b**2 + 2592*sqrt(2)*b - 15552)/(8*b**10 - 88*sqrt(2)*b**9 + 828*b**8 - 2144
*sqrt(2)*b**7 + 6470*b**6 - 5310*sqrt(2)*b**5 + 2781*b**4 + 2322*sqrt(2)*b**3 -
3402*b**2 + 729) + _t*(16*b**7 - 116*sqrt(2)*b**6 + 668*b**5 - 942*sqrt(2)*b**4
+ 1226*b**3 - 144*sqrt(2)*b**2 - 378*b + 108*sqrt(2))/(4*b**6 - 28*sqrt(2)*b**5
+ 152*b**4 - 192*sqrt(2)*b**3 + 189*b**2 + 27*sqrt(2)*b - 81) + x)))

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GIAC/XCAS [A]  time = 0.317026, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + sqrt(2))/(x^4 + b*x^2 + 1),x, algorithm="giac")

[Out]

Done