3.367 \(\int \frac{-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx\)

Optimal. Leaf size=148 \[ \frac{x^2}{2}-\frac{\log \left (x^2-\sqrt{2} \sqrt [4]{7} x+\sqrt{7}\right )}{4 \sqrt{2} 7^{3/4}}+\frac{\log \left (x^2+\sqrt{2} \sqrt [4]{7} x+\sqrt{7}\right )}{4 \sqrt{2} 7^{3/4}}-\frac{1}{2} \tanh ^{-1}\left (x^2\right )-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )}{2 \sqrt{2} 7^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{7}}+1\right )}{2 \sqrt{2} 7^{3/4}} \]

[Out]

x^2/2 - ArcTan[1 - (Sqrt[2]*x)/7^(1/4)]/(2*Sqrt[2]*7^(3/4)) + ArcTan[1 + (Sqrt[2
]*x)/7^(1/4)]/(2*Sqrt[2]*7^(3/4)) - ArcTanh[x^2]/2 - Log[Sqrt[7] - Sqrt[2]*7^(1/
4)*x + x^2]/(4*Sqrt[2]*7^(3/4)) + Log[Sqrt[7] + Sqrt[2]*7^(1/4)*x + x^2]/(4*Sqrt
[2]*7^(3/4))

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Rubi [A]  time = 0.271577, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{x^2}{2}-\frac{\log \left (x^2-\sqrt{2} \sqrt [4]{7} x+\sqrt{7}\right )}{4 \sqrt{2} 7^{3/4}}+\frac{\log \left (x^2+\sqrt{2} \sqrt [4]{7} x+\sqrt{7}\right )}{4 \sqrt{2} 7^{3/4}}-\frac{1}{2} \tanh ^{-1}\left (x^2\right )-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )}{2 \sqrt{2} 7^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{7}}+1\right )}{2 \sqrt{2} 7^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^4 + 7*x^5 + x^9)/(-7 + 6*x^4 + x^8),x]

[Out]

x^2/2 - ArcTan[1 - (Sqrt[2]*x)/7^(1/4)]/(2*Sqrt[2]*7^(3/4)) + ArcTan[1 + (Sqrt[2
]*x)/7^(1/4)]/(2*Sqrt[2]*7^(3/4)) - ArcTanh[x^2]/2 - Log[Sqrt[7] - Sqrt[2]*7^(1/
4)*x + x^2]/(4*Sqrt[2]*7^(3/4)) + Log[Sqrt[7] + Sqrt[2]*7^(1/4)*x + x^2]/(4*Sqrt
[2]*7^(3/4))

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Rubi in Sympy [A]  time = 40.9431, size = 136, normalized size = 0.92 \[ \frac{x^{2}}{2} - \frac{\sqrt{2} \sqrt [4]{7} \log{\left (x^{2} - \sqrt{2} \sqrt [4]{7} x + \sqrt{7} \right )}}{56} + \frac{\sqrt{2} \sqrt [4]{7} \log{\left (x^{2} + \sqrt{2} \sqrt [4]{7} x + \sqrt{7} \right )}}{56} + \frac{\sqrt{2} \sqrt [4]{7} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 7^{\frac{3}{4}} x}{7} - 1 \right )}}{28} + \frac{\sqrt{2} \sqrt [4]{7} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 7^{\frac{3}{4}} x}{7} + 1 \right )}}{28} - \frac{\operatorname{atanh}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**9+7*x**5+x**4-1)/(x**8+6*x**4-7),x)

[Out]

x**2/2 - sqrt(2)*7**(1/4)*log(x**2 - sqrt(2)*7**(1/4)*x + sqrt(7))/56 + sqrt(2)*
7**(1/4)*log(x**2 + sqrt(2)*7**(1/4)*x + sqrt(7))/56 + sqrt(2)*7**(1/4)*atan(sqr
t(2)*7**(3/4)*x/7 - 1)/28 + sqrt(2)*7**(1/4)*atan(sqrt(2)*7**(3/4)*x/7 + 1)/28 -
atanh(x**2)/2

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Mathematica [A]  time = 0.10222, size = 159, normalized size = 1.07 \[ \frac{1}{56} \left (28 x^2-14 \log \left (x^2+1\right )-\sqrt{2} \sqrt [4]{7} \log \left (\sqrt{7} x^2-\sqrt{2} 7^{3/4} x+7\right )+\sqrt{2} \sqrt [4]{7} \log \left (\sqrt{7} x^2+\sqrt{2} 7^{3/4} x+7\right )+14 \log (1-x)+14 \log (x+1)-2 \sqrt{2} \sqrt [4]{7} \tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{7}}\right )+2 \sqrt{2} \sqrt [4]{7} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{7}}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^4 + 7*x^5 + x^9)/(-7 + 6*x^4 + x^8),x]

[Out]

(28*x^2 - 2*Sqrt[2]*7^(1/4)*ArcTan[1 - (Sqrt[2]*x)/7^(1/4)] + 2*Sqrt[2]*7^(1/4)*
ArcTan[1 + (Sqrt[2]*x)/7^(1/4)] + 14*Log[1 - x] + 14*Log[1 + x] - 14*Log[1 + x^2
] - Sqrt[2]*7^(1/4)*Log[7 - Sqrt[2]*7^(3/4)*x + Sqrt[7]*x^2] + Sqrt[2]*7^(1/4)*L
og[7 + Sqrt[2]*7^(3/4)*x + Sqrt[7]*x^2])/56

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Maple [A]  time = 0.017, size = 110, normalized size = 0.7 \[{\frac{{x}^{2}}{2}}+{\frac{\ln \left ( -1+x \right ) }{4}}+{\frac{\sqrt [4]{7}\sqrt{2}}{28}\arctan \left ( -1+{\frac{x\sqrt{2}{7}^{{\frac{3}{4}}}}{7}} \right ) }+{\frac{\sqrt [4]{7}\sqrt{2}}{56}\ln \left ({\frac{{x}^{2}+\sqrt [4]{7}x\sqrt{2}+\sqrt{7}}{{x}^{2}-\sqrt [4]{7}x\sqrt{2}+\sqrt{7}}} \right ) }+{\frac{\sqrt [4]{7}\sqrt{2}}{28}\arctan \left ( 1+{\frac{x\sqrt{2}{7}^{{\frac{3}{4}}}}{7}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{4}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^9+7*x^5+x^4-1)/(x^8+6*x^4-7),x)

[Out]

1/2*x^2+1/4*ln(-1+x)+1/28*arctan(-1+1/7*x*2^(1/2)*7^(3/4))*7^(1/4)*2^(1/2)+1/56*
7^(1/4)*2^(1/2)*ln((x^2+7^(1/4)*x*2^(1/2)+7^(1/2))/(x^2-7^(1/4)*x*2^(1/2)+7^(1/2
)))+1/28*arctan(1+1/7*x*2^(1/2)*7^(3/4))*7^(1/4)*2^(1/2)+1/4*ln(1+x)-1/4*ln(x^2+
1)

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Maxima [A]  time = 0.87486, size = 178, normalized size = 1.2 \[ \frac{1}{2} \, x^{2} + \frac{1}{28} \cdot 7^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{14} \cdot 7^{\frac{3}{4}} \sqrt{2}{\left (2 \, x + 7^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{28} \cdot 7^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{14} \cdot 7^{\frac{3}{4}} \sqrt{2}{\left (2 \, x - 7^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{56} \cdot 7^{\frac{1}{4}} \sqrt{2} \log \left (x^{2} + 7^{\frac{1}{4}} \sqrt{2} x + \sqrt{7}\right ) - \frac{1}{56} \cdot 7^{\frac{1}{4}} \sqrt{2} \log \left (x^{2} - 7^{\frac{1}{4}} \sqrt{2} x + \sqrt{7}\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^9 + 7*x^5 + x^4 - 1)/(x^8 + 6*x^4 - 7),x, algorithm="maxima")

[Out]

1/2*x^2 + 1/28*7^(1/4)*sqrt(2)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x + 7^(1/4)*sqrt(2
))) + 1/28*7^(1/4)*sqrt(2)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x - 7^(1/4)*sqrt(2)))
+ 1/56*7^(1/4)*sqrt(2)*log(x^2 + 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/56*7^(1/4)*sqr
t(2)*log(x^2 - 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/4*log(x^2 + 1) + 1/4*log(x + 1)
+ 1/4*log(x - 1)

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Fricas [A]  time = 0.295521, size = 257, normalized size = 1.74 \[ \frac{1}{2744} \cdot 343^{\frac{3}{4}} \sqrt{2}{\left (2 \cdot 343^{\frac{1}{4}} \sqrt{2} x^{2} - 343^{\frac{1}{4}} \sqrt{2} \log \left (x^{2} + 1\right ) + 343^{\frac{1}{4}} \sqrt{2} \log \left (x^{2} - 1\right ) - 4 \, \arctan \left (\frac{7}{343^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{7}} \sqrt{\sqrt{7}{\left (\sqrt{7} x^{2} + 343^{\frac{1}{4}} \sqrt{2} x + 7\right )}} + 343^{\frac{1}{4}} \sqrt{2} x + 7}\right ) - 4 \, \arctan \left (\frac{7}{343^{\frac{1}{4}} \sqrt{2} \sqrt{\frac{1}{7}} \sqrt{\sqrt{7}{\left (\sqrt{7} x^{2} - 343^{\frac{1}{4}} \sqrt{2} x + 7\right )}} + 343^{\frac{1}{4}} \sqrt{2} x - 7}\right ) + \log \left (7 \, \sqrt{7} x^{2} + 7 \cdot 343^{\frac{1}{4}} \sqrt{2} x + 49\right ) - \log \left (7 \, \sqrt{7} x^{2} - 7 \cdot 343^{\frac{1}{4}} \sqrt{2} x + 49\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^9 + 7*x^5 + x^4 - 1)/(x^8 + 6*x^4 - 7),x, algorithm="fricas")

[Out]

1/2744*343^(3/4)*sqrt(2)*(2*343^(1/4)*sqrt(2)*x^2 - 343^(1/4)*sqrt(2)*log(x^2 +
1) + 343^(1/4)*sqrt(2)*log(x^2 - 1) - 4*arctan(7/(343^(1/4)*sqrt(2)*sqrt(1/7)*sq
rt(sqrt(7)*(sqrt(7)*x^2 + 343^(1/4)*sqrt(2)*x + 7)) + 343^(1/4)*sqrt(2)*x + 7))
- 4*arctan(7/(343^(1/4)*sqrt(2)*sqrt(1/7)*sqrt(sqrt(7)*(sqrt(7)*x^2 - 343^(1/4)*
sqrt(2)*x + 7)) + 343^(1/4)*sqrt(2)*x - 7)) + log(7*sqrt(7)*x^2 + 7*343^(1/4)*sq
rt(2)*x + 49) - log(7*sqrt(7)*x^2 - 7*343^(1/4)*sqrt(2)*x + 49))

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Sympy [A]  time = 1.7953, size = 146, normalized size = 0.99 \[ \frac{x^{2}}{2} + \frac{\log{\left (x^{2} - 1 \right )}}{4} - \frac{\log{\left (x^{2} + 1 \right )}}{4} - \frac{\sqrt{2} \sqrt [4]{7} \log{\left (x^{2} - \sqrt{2} \sqrt [4]{7} x + \sqrt{7} \right )}}{56} + \frac{\sqrt{2} \sqrt [4]{7} \log{\left (x^{2} + \sqrt{2} \sqrt [4]{7} x + \sqrt{7} \right )}}{56} + \frac{\sqrt{2} \sqrt [4]{7} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 7^{\frac{3}{4}} x}{7} - 1 \right )}}{28} + \frac{\sqrt{2} \sqrt [4]{7} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 7^{\frac{3}{4}} x}{7} + 1 \right )}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**9+7*x**5+x**4-1)/(x**8+6*x**4-7),x)

[Out]

x**2/2 + log(x**2 - 1)/4 - log(x**2 + 1)/4 - sqrt(2)*7**(1/4)*log(x**2 - sqrt(2)
*7**(1/4)*x + sqrt(7))/56 + sqrt(2)*7**(1/4)*log(x**2 + sqrt(2)*7**(1/4)*x + sqr
t(7))/56 + sqrt(2)*7**(1/4)*atan(sqrt(2)*7**(3/4)*x/7 - 1)/28 + sqrt(2)*7**(1/4)
*atan(sqrt(2)*7**(3/4)*x/7 + 1)/28

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GIAC/XCAS [A]  time = 0.273128, size = 165, normalized size = 1.11 \[ \frac{1}{2} \, x^{2} + \frac{1}{28} \cdot 28^{\frac{1}{4}} \arctan \left (\frac{1}{14} \cdot 7^{\frac{3}{4}} \sqrt{2}{\left (2 \, x + 7^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{28} \cdot 28^{\frac{1}{4}} \arctan \left (\frac{1}{14} \cdot 7^{\frac{3}{4}} \sqrt{2}{\left (2 \, x - 7^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{56} \cdot 28^{\frac{1}{4}}{\rm ln}\left (x^{2} + 7^{\frac{1}{4}} \sqrt{2} x + \sqrt{7}\right ) - \frac{1}{56} \cdot 28^{\frac{1}{4}}{\rm ln}\left (x^{2} - 7^{\frac{1}{4}} \sqrt{2} x + \sqrt{7}\right ) - \frac{1}{4} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^9 + 7*x^5 + x^4 - 1)/(x^8 + 6*x^4 - 7),x, algorithm="giac")

[Out]

1/2*x^2 + 1/28*28^(1/4)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x + 7^(1/4)*sqrt(2))) + 1
/28*28^(1/4)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x - 7^(1/4)*sqrt(2))) + 1/56*28^(1/4
)*ln(x^2 + 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/56*28^(1/4)*ln(x^2 - 7^(1/4)*sqrt(2)
*x + sqrt(7)) - 1/4*ln(x^2 + 1) + 1/4*ln(abs(x + 1)) + 1/4*ln(abs(x - 1))