Optimal. Leaf size=147 \[ -16 \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3-18 \text {$\#$1}^2+20 \text {$\#$1}+25\& ,\frac {\text {$\#$1} \log \left (\text {$\#$1}+2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2-9 \text {$\#$1}+5}\& \right ]+\sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}-\frac {7}{4} \log \left (2 \sqrt {x+1}-2 \sqrt {x+\sqrt {x+1}}+1\right ) \]
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Rubi [A] time = 0.57, antiderivative size = 217, normalized size of antiderivative = 1.48, number of steps used = 13, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1586, 6725, 742, 640, 621, 206, 984, 724, 204} \begin {gather*} \sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}+\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\left (4-\sqrt {2}\right ) \sqrt {x+1}+2 \left (1+\sqrt {2}\right )}{2 \sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x+\sqrt {x+1}}}\right )+\frac {7}{4} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )-\sqrt {2 \left (\sqrt {2}-1\right )} \tanh ^{-1}\left (\frac {\left (4+\sqrt {2}\right ) \sqrt {x+1}+2 \left (1-\sqrt {2}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 621
Rule 640
Rule 724
Rule 742
Rule 984
Rule 1586
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x} \left (1+x^2\right )}{\left (-1+x^2\right ) \sqrt {x+\sqrt {1+x}}} \, dx &=\int \frac {1+x^2}{(-1+x) \sqrt {1+x} \sqrt {x+\sqrt {1+x}}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1+\left (-1+x^2\right )^2}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt {-1+x+x^2}}+\frac {2}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+4 \operatorname {Subst}\left (\int \frac {1}{\left (-2+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+2 \operatorname {Subst}\left (\int \frac {1}{\left (-2-\sqrt {2} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {1}{\left (-2+\sqrt {2} x\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {1-\frac {3 x}{2}}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\frac {7}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )-4 \operatorname {Subst}\left (\int \frac {1}{8-8 \sqrt {2}-x^2} \, dx,x,\frac {2+2 \sqrt {2}-\left (-4+\sqrt {2}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )-4 \operatorname {Subst}\left (\int \frac {1}{8+8 \sqrt {2}-x^2} \, dx,x,\frac {2-2 \sqrt {2}+\left (4+\sqrt {2}\right ) \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {2 \left (1+\sqrt {2}\right )+\left (4-\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )-\sqrt {2 \left (-1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {2 \left (1-\sqrt {2}\right )+\left (4+\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )+\frac {7}{2} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=-\frac {3}{2} \sqrt {x+\sqrt {1+x}}+\sqrt {1+x} \sqrt {x+\sqrt {1+x}}+\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {2 \left (1+\sqrt {2}\right )+\left (4-\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )+\frac {7}{4} \tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )-\sqrt {2 \left (-1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {2 \left (1-\sqrt {2}\right )+\left (4+\sqrt {2}\right ) \sqrt {1+x}}{2 \sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 209, normalized size = 1.42 \begin {gather*} \sqrt {x+1} \sqrt {x+\sqrt {x+1}}-\frac {3}{2} \sqrt {x+\sqrt {x+1}}+\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\left (2 \sqrt {2}-1\right ) \sqrt {x+1}+\sqrt {2}+2}{2 \sqrt {\sqrt {2}-1} \sqrt {x+\sqrt {x+1}}}\right )+\frac {7}{4} \tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )+\sqrt {2 \left (\sqrt {2}-1\right )} \tanh ^{-1}\left (\frac {-\left (\left (1+2 \sqrt {2}\right ) \sqrt {x+1}\right )-\sqrt {2}+2}{2 \sqrt {1+\sqrt {2}} \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 157, normalized size = 1.07 \begin {gather*} \frac {1}{2} \sqrt {x+\sqrt {1+x}} \left (-3+2 \sqrt {1+x}\right )-\frac {7}{4} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )-2 \text {RootSum}\left [-1+8 \text {$\#$1}-6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right )+2 \log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}}{2-3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 4.51, size = 378, normalized size = 2.57 \begin {gather*} -2 \, \sqrt {2} \sqrt {\sqrt {2} + 1} \arctan \left (-\frac {\sqrt {5} \sqrt {2} {\left (28 \, x^{2} + \sqrt {2} {\left (7 \, x^{2} + 2 \, x - 9\right )} + 2 \, {\left (\sqrt {2} {\left (x - 13\right )} - 10 \, x + 10\right )} \sqrt {x + 1} - 28 \, x - 48\right )} \sqrt {\sqrt {2} + 1} \sqrt {\sqrt {2} - 1} + 2 \, \sqrt {2} {\left (8 \, \sqrt {2} {\left (x + 2\right )} - {\left (3 \, \sqrt {2} {\left (7 \, x - 11\right )} - 14 \, x + 62\right )} \sqrt {x + 1} + 18 \, x + 6\right )} \sqrt {x + \sqrt {x + 1}} \sqrt {\sqrt {2} + 1}}{2 \, {\left (49 \, x^{2} - 74 \, x - 119\right )}}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (-\frac {2 \, {\left (\sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 3\right )} + 2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} + 4\right )} + 6 \, x + 6\right )} \sqrt {\sqrt {2} - 1} + 4 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + 2\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x - 1}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (\frac {2 \, {\left (\sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 3\right )} + 2 \, \sqrt {x + 1} {\left (3 \, \sqrt {2} + 4\right )} + 6 \, x + 6\right )} \sqrt {\sqrt {2} - 1} - 4 \, {\left (\sqrt {x + 1} {\left (\sqrt {2} + 1\right )} + \sqrt {2} + 2\right )} \sqrt {x + \sqrt {x + 1}}\right )}}{x - 1}\right ) + \frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + \frac {7}{8} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x + 1} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.77, size = 208, normalized size = 1.41 \begin {gather*} \frac {1}{2} \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} - 3\right )} + 4 \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \arctan \left (\frac {\sqrt {2} - \sqrt {x + \sqrt {x + 1}} + \sqrt {x + 1}}{\sqrt {\sqrt {2} - 1}}\right ) - 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left ({\left | 10 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + 20 \, \sqrt {2} + 20 \, \sqrt {x + \sqrt {x + 1}} - 20 \, \sqrt {x + 1} + 20 \, \sqrt {2 \, \sqrt {2} - 2} \right |}\right ) + 2 \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \log \left ({\left | -2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + 4 \, \sqrt {2} + 4 \, \sqrt {x + \sqrt {x + 1}} - 4 \, \sqrt {x + 1} - 4 \, \sqrt {2 \, \sqrt {2} - 2} \right |}\right ) - \frac {7}{4} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 217, normalized size = 1.48
method | result | size |
derivativedivides | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {2-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}\, \sqrt {\left (\sqrt {1+x}+\sqrt {2}\right )^{2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )+1-\sqrt {2}}}\right )}{\sqrt {\sqrt {2}-1}}-\frac {\sqrt {2}\, \arctanh \left (\frac {2+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )}{2 \sqrt {1+\sqrt {2}}\, \sqrt {\left (\sqrt {1+x}-\sqrt {2}\right )^{2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )+1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\) | \(217\) |
default | \(\sqrt {1+x}\, \sqrt {x +\sqrt {1+x}}-\frac {3 \sqrt {x +\sqrt {1+x}}}{2}+\frac {7 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {2-2 \sqrt {2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )}{2 \sqrt {\sqrt {2}-1}\, \sqrt {\left (\sqrt {1+x}+\sqrt {2}\right )^{2}+\left (1-2 \sqrt {2}\right ) \left (\sqrt {1+x}+\sqrt {2}\right )+1-\sqrt {2}}}\right )}{\sqrt {\sqrt {2}-1}}-\frac {\sqrt {2}\, \arctanh \left (\frac {2+2 \sqrt {2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )}{2 \sqrt {1+\sqrt {2}}\, \sqrt {\left (\sqrt {1+x}-\sqrt {2}\right )^{2}+\left (1+2 \sqrt {2}\right ) \left (\sqrt {1+x}-\sqrt {2}\right )+1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\) | \(217\) |
Verification of antiderivative is not currently implemented for this CAS.
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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 {x + 1}}{{\left (x^{2} - 1\right )} \sqrt {x + \sqrt {x + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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 {x+1}}{\sqrt {x+\sqrt {x+1}}\,\left (x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{\left (x - 1\right ) \sqrt {x + 1} \sqrt {x + \sqrt {x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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