3.1512 \(\int \frac{x^{19}}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=41 \[ \frac{3}{32} \sinh ^{-1}\left (x^4\right )+\frac{1}{16} \sqrt{x^8+1} x^{12}-\frac{3}{32} \sqrt{x^8+1} x^4 \]

[Out]

(-3*x^4*Sqrt[1 + x^8])/32 + (x^12*Sqrt[1 + x^8])/16 + (3*ArcSinh[x^4])/32

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Rubi [A]  time = 0.0564664, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3}{32} \sinh ^{-1}\left (x^4\right )+\frac{1}{16} \sqrt{x^8+1} x^{12}-\frac{3}{32} \sqrt{x^8+1} x^4 \]

Antiderivative was successfully verified.

[In]  Int[x^19/Sqrt[1 + x^8],x]

[Out]

(-3*x^4*Sqrt[1 + x^8])/32 + (x^12*Sqrt[1 + x^8])/16 + (3*ArcSinh[x^4])/32

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Rubi in Sympy [A]  time = 6.29661, size = 36, normalized size = 0.88 \[ \frac{x^{12} \sqrt{x^{8} + 1}}{16} - \frac{3 x^{4} \sqrt{x^{8} + 1}}{32} + \frac{3 \operatorname{asinh}{\left (x^{4} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**19/(x**8+1)**(1/2),x)

[Out]

x**12*sqrt(x**8 + 1)/16 - 3*x**4*sqrt(x**8 + 1)/32 + 3*asinh(x**4)/32

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Mathematica [A]  time = 0.0293303, size = 31, normalized size = 0.76 \[ \frac{1}{32} \left (3 \sinh ^{-1}\left (x^4\right )+\sqrt{x^8+1} \left (2 x^8-3\right ) x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^19/Sqrt[1 + x^8],x]

[Out]

(x^4*Sqrt[1 + x^8]*(-3 + 2*x^8) + 3*ArcSinh[x^4])/32

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Maple [A]  time = 0.036, size = 27, normalized size = 0.7 \[{\frac{{x}^{4} \left ( 2\,{x}^{8}-3 \right ) }{32}\sqrt{{x}^{8}+1}}+{\frac{3\,{\it Arcsinh} \left ({x}^{4} \right ) }{32}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19/(x^8+1)^(1/2),x)

[Out]

1/32*x^4*(2*x^8-3)*(x^8+1)^(1/2)+3/32*arcsinh(x^4)

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Maxima [A]  time = 1.44044, size = 116, normalized size = 2.83 \[ -\frac{\frac{5 \, \sqrt{x^{8} + 1}}{x^{4}} - \frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}}}{x^{12}}}{32 \,{\left (\frac{2 \,{\left (x^{8} + 1\right )}}{x^{8}} - \frac{{\left (x^{8} + 1\right )}^{2}}{x^{16}} - 1\right )}} + \frac{3}{64} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} + 1\right ) - \frac{3}{64} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/sqrt(x^8 + 1),x, algorithm="maxima")

[Out]

-1/32*(5*sqrt(x^8 + 1)/x^4 - 3*(x^8 + 1)^(3/2)/x^12)/(2*(x^8 + 1)/x^8 - (x^8 + 1
)^2/x^16 - 1) + 3/64*log(sqrt(x^8 + 1)/x^4 + 1) - 3/64*log(sqrt(x^8 + 1)/x^4 - 1
)

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Fricas [A]  time = 0.22338, size = 170, normalized size = 4.15 \[ -\frac{16 \, x^{32} - 28 \, x^{16} - 12 \, x^{8} + 3 \,{\left (8 \, x^{16} + 8 \, x^{8} - 4 \,{\left (2 \, x^{12} + x^{4}\right )} \sqrt{x^{8} + 1} + 1\right )} \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) -{\left (16 \, x^{28} - 8 \, x^{20} - 22 \, x^{12} - 3 \, x^{4}\right )} \sqrt{x^{8} + 1}}{32 \,{\left (8 \, x^{16} + 8 \, x^{8} - 4 \,{\left (2 \, x^{12} + x^{4}\right )} \sqrt{x^{8} + 1} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/sqrt(x^8 + 1),x, algorithm="fricas")

[Out]

-1/32*(16*x^32 - 28*x^16 - 12*x^8 + 3*(8*x^16 + 8*x^8 - 4*(2*x^12 + x^4)*sqrt(x^
8 + 1) + 1)*log(-x^4 + sqrt(x^8 + 1)) - (16*x^28 - 8*x^20 - 22*x^12 - 3*x^4)*sqr
t(x^8 + 1))/(8*x^16 + 8*x^8 - 4*(2*x^12 + x^4)*sqrt(x^8 + 1) + 1)

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Sympy [A]  time = 25.9716, size = 49, normalized size = 1.2 \[ \frac{x^{20}}{16 \sqrt{x^{8} + 1}} - \frac{x^{12}}{32 \sqrt{x^{8} + 1}} - \frac{3 x^{4}}{32 \sqrt{x^{8} + 1}} + \frac{3 \operatorname{asinh}{\left (x^{4} \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**19/(x**8+1)**(1/2),x)

[Out]

x**20/(16*sqrt(x**8 + 1)) - x**12/(32*sqrt(x**8 + 1)) - 3*x**4/(32*sqrt(x**8 + 1
)) + 3*asinh(x**4)/32

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GIAC/XCAS [A]  time = 0.239932, size = 59, normalized size = 1.44 \[ \frac{1}{32} \,{\left (2 \, x^{8} - 3\right )} \sqrt{x^{8} + 1} x^{4} + \frac{3}{64} \,{\rm ln}\left (\sqrt{\frac{1}{x^{8}} + 1} + 1\right ) - \frac{3}{64} \,{\rm ln}\left (\sqrt{\frac{1}{x^{8}} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/sqrt(x^8 + 1),x, algorithm="giac")

[Out]

1/32*(2*x^8 - 3)*sqrt(x^8 + 1)*x^4 + 3/64*ln(sqrt(1/x^8 + 1) + 1) - 3/64*ln(sqrt
(1/x^8 + 1) - 1)