3.19.78 \(\int \frac {(-1+x^4) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx\)

Optimal. Leaf size=129 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^{16}-8 \text {$\#$1}^{14}+24 \text {$\#$1}^{12}-32 \text {$\#$1}^{10}+16 \text {$\#$1}^8+1\& ,\frac {\log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^{11}-6 \text {$\#$1}^9+12 \text {$\#$1}^7-8 \text {$\#$1}^5}\& \right ]+\frac {4}{15} \sqrt {\sqrt {x+1}+1} (3 x+1)+\frac {4}{15} \sqrt {x+1} \sqrt {\sqrt {x+1}+1} \]

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Rubi [F]  time = 0.65, 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}}}{1+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-4*(1 + Sqrt[1 + x])^(3/2))/3 + (4*(1 + Sqrt[1 + x])^(5/2))/5 + 8*Defer[Subst][Defer[Int][x^2/(1 + 16*x^8 - 3
2*x^10 + 24*x^12 - 8*x^14 + x^16), x], x, Sqrt[1 + Sqrt[1 + x]]] - 8*Defer[Subst][Defer[Int][x^4/(1 + 16*x^8 -
32*x^10 + 24*x^12 - 8*x^14 + x^16), x], x, Sqrt[1 + Sqrt[1 + x]]]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {1+x} \left (-1+\left (-1+x^2\right )^4\right )}{1+\left (-1+x^2\right )^4} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^2\right ) \left (-1+x^8 \left (-2+x^2\right )^4\right )}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (-x^2+x^4-\frac {2 x^2 \left (-1+x^2\right )}{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}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}-8 \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}-8 \operatorname {Subst}\left (\int \left (-\frac {x^2}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}+\frac {x^4}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}+8 \operatorname {Subst}\left (\int \frac {x^2}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-8 \operatorname {Subst}\left (\int \frac {x^4}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 112, normalized size = 0.87 \begin {gather*} \frac {4}{15} \sqrt {\sqrt {x+1}+1} \left (3 x+\sqrt {x+1}+1\right )-\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^{16}-8 \text {$\#$1}^{14}+24 \text {$\#$1}^{12}-32 \text {$\#$1}^{10}+16 \text {$\#$1}^8+1\&,\frac {\log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^{11}-6 \text {$\#$1}^9+12 \text {$\#$1}^7-8 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[1 + Sqrt[1 + x]]*(1 + 3*x + Sqrt[1 + x]))/15 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #
1^16 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(-8*#1^5 + 12*#1^7 - 6*#1^9 + #1^11) & ]/2

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IntegrateAlgebraic [A]  time = 0.00, size = 114, normalized size = 0.88 \begin {gather*} \frac {4}{15} \sqrt {1+\sqrt {1+x}} \left (-2+\sqrt {1+x}+3 (1+x)\right )-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{-8 \text {$\#$1}^5+12 \text {$\#$1}^7-6 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[1 + Sqrt[1 + x]]*(-2 + Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 16*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1
^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + x]] - #1]/(-8*#1^5 + 12*#1^7 - 6*#1^9 + #1^11) & ]/2

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fricas [F(-1)]  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)/(x^4+1),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [B]  time = 0.09, size = 105, normalized size = 0.81

method	result	size
derivativedivides	\(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\)	\(105\)
default	\(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(1+(1+x)^(1/2))^(5/2)-4/3*(1+(1+x)^(1/2))^(3/2)-1/2*sum((_R^4-_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*} \int \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x + 1} + 1}}{x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \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)/(x**4+1),x)

[Out]

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

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