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3.15.33 \(\int \frac {(-1+x^2) \sqrt {1+\sqrt {1+x}}}{1+x^2} \, dx\)

Optimal. Leaf size=101 \[ -\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4+1\& ,\frac {\log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^3-2 \text {$\#$1}}\& \right ]+\frac {4}{15} \sqrt {\sqrt {x+1}+1} (3 x+1)+\frac {4}{15} \sqrt {x+1} \sqrt {\sqrt {x+1}+1} \]

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Rubi [F]  time = 0.81, 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^2\right ) \sqrt {1+\sqrt {1+x}}}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^2)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^2),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 + 4*x^4 - 4*
x^6 + x^8), x], x, Sqrt[1 + Sqrt[1 + x]]] - 8*Defer[Subst][Defer[Int][x^4/(1 + 4*x^4 - 4*x^6 + x^8), x], x, Sq
rt[1 + Sqrt[1 + x]]]

Rubi steps

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

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Mathematica [C]  time = 0.17, size = 281, normalized size = 2.78 \begin {gather*} \frac {4}{15} \sqrt {\sqrt {x+1}+1} \left (3 x+\sqrt {x+1}+1\right )-\frac {(1-i)^{3/2} \left ((-1+i)+\sqrt {1-i}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {1-\sqrt {1-i}}}\right )}{\sqrt {1-\sqrt {1-i}}}+\frac {2 \left ((1+i)+i \sqrt {1-i}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {1+\sqrt {1-i}}}\right )}{\sqrt {(1-i)+(1-i)^{3/2}}}-\frac {(1+i)^{3/2} \left ((-1-i)+\sqrt {1+i}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {1-\sqrt {1+i}}}\right )}{\sqrt {1-\sqrt {1+i}}}-\frac {(1+i)^{3/2} \left ((1+i)+\sqrt {1+i}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {1+\sqrt {1+i}}}\right )}{\sqrt {1+\sqrt {1+i}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^2)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^2),x]

[Out]

(4*Sqrt[1 + Sqrt[1 + x]]*(1 + 3*x + Sqrt[1 + x]))/15 - ((1 - I)^(3/2)*((-1 + I) + Sqrt[1 - I])*ArcTanh[Sqrt[1
+ Sqrt[1 + x]]/Sqrt[1 - Sqrt[1 - I]]])/Sqrt[1 - Sqrt[1 - I]] + (2*((1 + I) + I*Sqrt[1 - I])*ArcTanh[Sqrt[1 + S
qrt[1 + x]]/Sqrt[1 + Sqrt[1 - I]]])/Sqrt[(1 - I) + (1 - I)^(3/2)] - ((1 + I)^(3/2)*((-1 - I) + Sqrt[1 + I])*Ar
cTanh[Sqrt[1 + Sqrt[1 + x]]/Sqrt[1 - Sqrt[1 + I]]])/Sqrt[1 - Sqrt[1 + I]] - ((1 + I)^(3/2)*((1 + I) + Sqrt[1 +
 I])*ArcTanh[Sqrt[1 + Sqrt[1 + x]]/Sqrt[1 + Sqrt[1 + I]]])/Sqrt[1 + Sqrt[1 + I]]

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IntegrateAlgebraic [A]  time = 0.00, size = 86, normalized size = 0.85 \begin {gather*} \frac {4}{15} \sqrt {1+\sqrt {1+x}} \left (-2+\sqrt {1+x}+3 (1+x)\right )-\text {RootSum}\left [1+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{-2 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^2)*Sqrt[1 + Sqrt[1 + x]])/(1 + x^2),x]

[Out]

(4*Sqrt[1 + Sqrt[1 + x]]*(-2 + Sqrt[1 + x] + 3*(1 + x)))/15 - RootSum[1 + 4*#1^4 - 4*#1^6 + #1^8 & , Log[Sqrt[
1 + Sqrt[1 + x]] - #1]/(-2*#1 + #1^3) & ]

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fricas [B]  time = 1.05, size = 5173, normalized size = 51.22

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/80*sqrt(2)*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2
*sqrt(2) + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*
sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(1/4
)*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt
(2) + 4))*sqrt(-2*sqrt(2) + 4) - 20)*arctan(1/6400*sqrt(2*sqrt(2)*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(
2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqr
t(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(
2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(1/4)*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt
(2) + 2)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) - 20)*sqrt(sqrt(x + 1) + 1)
 + 160*sqrt(2)*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 320*sqrt(x + 1) + 320)*sqrt(
8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*s
qrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*sqrt(2)*(sqrt(2) +
2) + 16*sqrt(2) + 32)*(sqrt(5)*(2*2^(3/4)*(sqrt(2) + 1) + 3*2^(1/4)*(sqrt(2) + 2))*sqrt(-2*sqrt(2) + 4) - 2*sq
rt(5)*(sqrt(2)*(sqrt(2) + 3) + 2*sqrt(2) + 3))*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16
)^(3/4) + 1/160*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(
-2*sqrt(2) + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) +
8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3
/4)*(2*sqrt(2)*(sqrt(2) + 3) - (2*2^(3/4)*(sqrt(2) + 1) + 3*2^(1/4)*(sqrt(2) + 2))*sqrt(-2*sqrt(2) + 4) + 4*sq
rt(2) + 6)*sqrt(sqrt(x + 1) + 1) + 1/2*2^(3/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + 1/2*2^(1/4)*sqrt(2^(1/4)*(
sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) + sqrt(2) + 1) - 1/80*sq
rt(2)*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2)
 + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*sqrt(2)*
(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(1/4)*sqrt(2
^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt(2) + 4)
)*sqrt(-2*sqrt(2) + 4) - 20)*arctan(1/6400*sqrt(-2*sqrt(2)*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4)
 + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) +
2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)
*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(1/4)*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2
)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) - 20)*sqrt(sqrt(x + 1) + 1) + 160*
sqrt(2)*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 320*sqrt(x + 1) + 320)*sqrt(8*2^(3/
4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*
(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*sqrt(2)*(sqrt(2) + 2) + 16
*sqrt(2) + 32)*(sqrt(5)*(2*2^(3/4)*(sqrt(2) + 1) + 3*2^(1/4)*(sqrt(2) + 2))*sqrt(-2*sqrt(2) + 4) - 2*sqrt(5)*(
sqrt(2)*(sqrt(2) + 3) + 2*sqrt(2) + 3))*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4)
 + 1/160*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt
(2) + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*sqrt(
2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4)*(2*
sqrt(2)*(sqrt(2) + 3) - (2*2^(3/4)*(sqrt(2) + 1) + 3*2^(1/4)*(sqrt(2) + 2))*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) +
 6)*sqrt(sqrt(x + 1) + 1) - 1/2*2^(3/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) - 1/2*2^(1/4)*sqrt(2^(1/4)*(sqrt(2)
 + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) - sqrt(2) - 1) - 1/160*sqrt(-8*
2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - (2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt
(2) + 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16
*sqrt(2) + 32)*(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4)*((2^(3/4)*(3*sqrt(2) +
2) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) + 20)*arctan(1/6400*sqrt(sqrt(-8*2^(3/4)*(2*sqrt(2) + 3)*sq
rt(-2*sqrt(2) + 4) - (2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(1/4)*
(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(-8*2^(1/4)
*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt(2) +
4))*sqrt(-2*sqrt(2) + 4) + 20)*sqrt(sqrt(x + 1) + 1) + 80*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) +
 16*sqrt(2) + 16) + 320*sqrt(x + 1) + 320)*sqrt(-8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - (2^(3/4)*(2*
sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4
) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(sqrt(5)*(2*2^(3/4)*(sqrt(2) + 1) + 3*2^(1/4
)*(sqrt(2) + 2))*sqrt(-2*sqrt(2) + 4) + 2*sqrt(5)*(sqrt(2)*(sqrt(2) + 3) + 2*sqrt(2) + 3))*(-8*2^(1/4)*(sqrt(2
) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4) - 1/8*2^(3/4)*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(
2) + 4) + 16*sqrt(2) + 16)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) - 1/160*sqrt(-8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*
sqrt(2) + 4) - (2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(1/4)*(sqrt(
2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(-8*2^(1/4)*(sqrt
(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4)*(2*sqrt(2)*(sqrt(2) + 3) + (2*2^(3/4)*(sqrt(2) + 1) + 3
*2^(1/4)*(sqrt(2) + 2))*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2) + 6)*sqrt(sqrt(x + 1) + 1) - 1/2*2^(3/4)*(sqrt(2) + 1
)*sqrt(-2*sqrt(2) + 4) + sqrt(2) + 1) - 1/160*sqrt(-8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - (2^(3/4)*
(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2)
+ 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2)
 + 4) + 16*sqrt(2) + 16)^(3/4)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) + 20)
*arctan(1/6400*sqrt(-sqrt(-8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - (2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*s
qrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) +
 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^
(3/4)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) + 20)*sqrt(sqrt(x + 1) + 1) +
80*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) + 320*sqrt(x + 1) + 320)*sqrt(-8*2^(3
/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - (2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2)
+ 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqr
t(2) + 32)*(sqrt(5)*(2*2^(3/4)*(sqrt(2) + 1) + 3*2^(1/4)*(sqrt(2) + 2))*sqrt(-2*sqrt(2) + 4) + 2*sqrt(5)*(sqrt
(2)*(sqrt(2) + 3) + 2*sqrt(2) + 3))*(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4) +
1/8*2^(3/4)*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)*(sqrt(2) + 1)*sqrt(-2*sqrt(2
) + 4) - 1/160*sqrt(-8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - (2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2)
 + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) + 8*sqr
t(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4)*
(2*sqrt(2)*(sqrt(2) + 3) + (2*2^(3/4)*(sqrt(2) + 1) + 3*2^(1/4)*(sqrt(2) + 2))*sqrt(-2*sqrt(2) + 4) + 4*sqrt(2
) + 6)*sqrt(sqrt(x + 1) + 1) + 1/2*2^(3/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4) - sqrt(2) - 1) - 1/320*(sqrt(2)*
(2^(3/4)*(sqrt(2) + 4) + 2^(1/4)*(sqrt(2) - 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) +
2)*sqrt(-2*sqrt(2) + 4) - 40)*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt
(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*s
qrt(2) + 2) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sq
rt(2) + 16)^(1/4)*log(1/40*sqrt(2)*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2
*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4)
+ 2*sqrt(2) + 2) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) +
16*sqrt(2) + 16)^(1/4)*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*((2^(3/4)*(3*sqrt(2) +
 2) + 2^(1/4)*(3*sqrt(2) + 4))*sqrt(-2*sqrt(2) + 4) - 20)*sqrt(sqrt(x + 1) + 1) + 2*sqrt(2)*sqrt(2^(1/4)*(sqrt
(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 4*sqrt(x + 1) + 4) + 1/320*(sqrt(2)*(2^(3/4)*(sqrt(2) + 4) +
2^(1/4)*(sqrt(2) - 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) -
 40)*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2)
+ 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*sqrt(2)*(
sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(1/4)*log(-1/
40*sqrt(2)*sqrt(8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2)*(2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sq
rt(2) + 4) + 2*sqrt(2)*(sqrt(2) + 2))*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2) + 8*sqr
t(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(1/4)*s
qrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 2*sqrt(2) + 2)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt(2)
 + 4))*sqrt(-2*sqrt(2) + 4) - 20)*sqrt(sqrt(x + 1) + 1) + 2*sqrt(2)*sqrt(2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2)
 + 4) + 2*sqrt(2) + 2) + 4*sqrt(x + 1) + 4) - 1/640*sqrt(-8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - (2^
(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sq
rt(2) + 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*((2^(3/4)*(sqrt(2) + 4) + 2^(1/4)*(
sqrt(2) - 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)*sqrt(-2*sqrt(2) + 4) + 80)
*(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(1/4)*log(1/80*sqrt(-8*2^(3/4)*(2*sqrt(2) +
 3)*sqrt(-2*sqrt(2) + 4) - (2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^
(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(-8*2
^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt
(2) + 4))*sqrt(-2*sqrt(2) + 4) + 20)*sqrt(sqrt(x + 1) + 1) + sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4
) + 16*sqrt(2) + 16) + 4*sqrt(x + 1) + 4) + 1/640*sqrt(-8*2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - (2^(3
/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt
(2) + 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*((2^(3/4)*(sqrt(2) + 4) + 2^(1/4)*(sq
rt(2) - 2))*sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)*sqrt(-2*sqrt(2) + 4) + 80)*(
-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(1/4)*log(-1/80*sqrt(-8*2^(3/4)*(2*sqrt(2) +
3)*sqrt(-2*sqrt(2) + 4) - (2^(3/4)*(2*sqrt(2) + 3)*sqrt(-2*sqrt(2) + 4) - 2*sqrt(2)*(sqrt(2) + 2))*sqrt(-8*2^(
1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16) + 8*sqrt(2)*(sqrt(2) + 2) + 16*sqrt(2) + 32)*(-8*2^
(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4) + 16*sqrt(2) + 16)^(3/4)*((2^(3/4)*(3*sqrt(2) + 2) + 2^(1/4)*(3*sqrt(
2) + 4))*sqrt(-2*sqrt(2) + 4) + 20)*sqrt(sqrt(x + 1) + 1) + sqrt(-8*2^(1/4)*(sqrt(2) + 2)*sqrt(-2*sqrt(2) + 4)
 + 16*sqrt(2) + 16) + 4*sqrt(x + 1) + 4) + 4/15*(3*x + sqrt(x + 1) + 1)*sqrt(sqrt(x + 1) + 1)

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

[Out]

sage0*x

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maple [A]  time = 0.07, size = 85, normalized size = 0.84

method result size
derivativedivides \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) \(85\)
default \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) \(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(1+(1+x)^(1/2))^(5/2)-4/3*(1+(1+x)^(1/2))^(3/2)-sum((_R^4-_R^2)/(_R^7-3*_R^5+2*_R^3)*ln((1+(1+x)^(1/2))^(1
/2)-_R),_R=RootOf(_Z^8-4*_Z^6+4*_Z^4+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {\sqrt {x+1}+1}}{x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((x^2 - 1)*((x + 1)^(1/2) + 1)^(1/2))/(x^2 + 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 ) \sqrt {\sqrt {x + 1} + 1}}{x^{2} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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