Optimal. Leaf size=105 \[ \frac {-8 (-15 x-128)+\sqrt {x+1} \left (8-24 \sqrt {\sqrt {x+1}+1}\right )+64 \sqrt {\sqrt {x+1}+1}}{105 \sqrt {\sqrt {\sqrt {x+1}+1}+1}}+4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.66, antiderivative size = 124, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1629, 63, 206} \begin {gather*} \frac {8}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}-\frac {24}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}+\frac {16}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}+\frac {8}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}+4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 1629
Rubi steps
\begin {align*} \int \frac {-1+x}{(1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {-2+x^2}{x \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x \left (-2+\left (-1+x^2\right )^2\right )}{\sqrt {1+x} \left (-1+x^2\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (-\frac {1}{(1+x)^{3/2}}+\frac {1}{(1-x) \sqrt {1+x}}+2 \sqrt {1+x}-3 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+4 \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+8 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=\frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}}+\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}+4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 106, normalized size = 1.01 \begin {gather*} \frac {8 \left (15 x+\sqrt {x+1}-3 \sqrt {x+1} \sqrt {\sqrt {x+1}+1}+8 \sqrt {\sqrt {x+1}+1}+128\right )}{105 \sqrt {\sqrt {\sqrt {x+1}+1}+1}}+4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 102, normalized size = 0.97 \begin {gather*} \frac {-8 \sqrt {\sqrt {x+1}+1} \left (3 \sqrt {x+1}-8\right )-8 \left (-15 (x+1)-\sqrt {x+1}-113\right )}{105 \sqrt {\sqrt {\sqrt {x+1}+1}+1}}+4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {x+1}+1}+1}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 150, normalized size = 1.43 \begin {gather*} \frac {2 \, {\left (105 \, \sqrt {2} {\left (x + 1\right )} \log \left (\frac {2 \, {\left (\sqrt {2} \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1} + \sqrt {2} \sqrt {x + 1}\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1} + x + 4 \, \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1} + 4 \, \sqrt {x + 1} + 1}{x + 1}\right ) - 4 \, {\left (3 \, {\left (6 \, x + 41\right )} \sqrt {x + 1} - {\left (15 \, {\left (x + 8\right )} \sqrt {x + 1} + 4 \, x + 4\right )} \sqrt {\sqrt {x + 1} + 1} - 4 \, x - 4\right )} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}\right )}}{105 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 86, normalized size = 0.82 \begin {gather*} \frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+4 \sqrt {2}\, \arctanh \left (\frac {\sqrt {1+\sqrt {1+\sqrt {1+x}}}\, \sqrt {2}}{2}\right )+\frac {8}{\sqrt {1+\sqrt {1+\sqrt {1+x}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 107, normalized size = 1.02 \begin {gather*} \frac {8}{7} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {7}{2}} - \frac {24}{5} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {5}{2}} + \frac {16}{3} \, {\left (\sqrt {\sqrt {x + 1} + 1} + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}\right ) + \frac {8}{\sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \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 {x-1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}\,\left (x+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 - 1}{\left (x + 1\right ) \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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