3.19.85 \(\int \frac {(-1+x^4) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} (1+x^4)} \, dx\) [1885]

Optimal. Leaf size=130 \[ \frac {4}{3} \sqrt {1+\sqrt {1+x}}+\frac {4}{3} \sqrt {1+x} \sqrt {1+\sqrt {1+x}}-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\& ,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{8 \text {$\#$1}^5-20 \text {$\#$1}^7+18 \text {$\#$1}^9-7 \text {$\#$1}^{11}+\text {$\#$1}^{13}}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.50, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(Sqrt[1 + x]*(1 + x^4)),x]

[Out]

(4*(1 + Sqrt[1 + x])^(3/2))/3 - 8*Defer[Subst][Defer[Int][x^2/(1 + x^8*(-2 + x^2)^4), x], x, Sqrt[1 + Sqrt[1 +
x]]]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx &=\int \frac {\sqrt {1+x} \left (-1+x-x^2+x^3\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx\\ &=2 \text {Subst}\left (\int \frac {x^2 \sqrt {1+x} \left (-4+6 x^2-4 x^4+x^6\right )}{1+\left (-1+x^2\right )^4} \, dx,x,\sqrt {1+x}\right )\\ &=4 \text {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2 \left (-4+6 \left (-1+x^2\right )^2-4 \left (-1+x^2\right )^4+\left (-1+x^2\right )^6\right )}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \text {Subst}\left (\int \left (x^2-\frac {2 x^2}{1+x^8 \left (-2+x^2\right )^4}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-8 \text {Subst}\left (\int \frac {x^2}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 106, normalized size = 0.82 \begin {gather*} \frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{8 \text {$\#$1}^5-20 \text {$\#$1}^7+18 \text {$\#$1}^9-7 \text {$\#$1}^{11}+\text {$\#$1}^{13}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^4)*Sqrt[1 + Sqrt[1 + x]])/(Sqrt[1 + x]*(1 + x^4)),x]

[Out]

(4*(1 + Sqrt[1 + x])^(3/2))/3 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + S
qrt[1 + x]] - #1]/(8*#1^5 - 20*#1^7 + 18*#1^9 - 7*#1^11 + #1^13) & ]/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.09, size = 88, normalized size = 0.68

method	result	size
derivativedivides	\(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\)	\(88\)
default	\(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(1+x)^(1/2)/(x^4+1),x,method=_RETURNVERBOSE)

[Out]

4/3*(1+(1+x)^(1/2))^(3/2)-1/2*sum(_R^2/(_R^15-7*_R^13+18*_R^11-20*_R^9+8*_R^7)*ln((1+(1+x)^(1/2))^(1/2)-_R),_R
=RootOf(_Z^16-8*_Z^14+24*_Z^12-32*_Z^10+16*_Z^8+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(1+x)^(1/2)/(x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(1+x)^(1/2)/(x^4+1),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \sqrt {x + 1} \left (x^{2} + 1\right ) \sqrt {\sqrt {x + 1} + 1}}{x^{4} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)*(1+(1+x)**(1/2))**(1/2)/(1+x)**(1/2)/(x**4+1),x)

[Out]

Integral((x - 1)*sqrt(x + 1)*(x**2 + 1)*sqrt(sqrt(x + 1) + 1)/(x**4 + 1), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(1+(1+x)^(1/2))^(1/2)/(1+x)^(1/2)/(x^4+1),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^4-1\right )\,\sqrt {\sqrt {x+1}+1}}{\left (x^4+1\right )\,\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - 1)*((x + 1)^(1/2) + 1)^(1/2))/((x^4 + 1)*(x + 1)^(1/2)),x)

[Out]

int(((x^4 - 1)*((x + 1)^(1/2) + 1)^(1/2))/((x^4 + 1)*(x + 1)^(1/2)), x)

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