Optimal. Leaf size=57 \[ -\frac {2}{5} (\coth (x)+1)^{5/2}-\frac {4}{3} (\coth (x)+1)^{3/2}-8 \sqrt {\coth (x)+1}+8 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3478, 3480, 206} \[ -\frac {2}{5} (\coth (x)+1)^{5/2}-\frac {4}{3} (\coth (x)+1)^{3/2}-8 \sqrt {\coth (x)+1}+8 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 3478
Rule 3480
Rubi steps
\begin {align*} \int (1+\coth (x))^{7/2} \, dx &=-\frac {2}{5} (1+\coth (x))^{5/2}+2 \int (1+\coth (x))^{5/2} \, dx\\ &=-\frac {4}{3} (1+\coth (x))^{3/2}-\frac {2}{5} (1+\coth (x))^{5/2}+4 \int (1+\coth (x))^{3/2} \, dx\\ &=-8 \sqrt {1+\coth (x)}-\frac {4}{3} (1+\coth (x))^{3/2}-\frac {2}{5} (1+\coth (x))^{5/2}+8 \int \sqrt {1+\coth (x)} \, dx\\ &=-8 \sqrt {1+\coth (x)}-\frac {4}{3} (1+\coth (x))^{3/2}-\frac {2}{5} (1+\coth (x))^{5/2}+16 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=8 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-8 \sqrt {1+\coth (x)}-\frac {4}{3} (1+\coth (x))^{3/2}-\frac {2}{5} (1+\coth (x))^{5/2}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 101, normalized size = 1.77 \[ -\frac {2 (\coth (x)+1)^{7/2} \left ((8 \sinh (2 x)+3) \sinh (x) \sqrt {i (\coth (x)+1)}+4 \sinh ^3(x) \left (19 \sqrt {i (\coth (x)+1)}-(15-15 i) \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )\right )\right )}{15 \sqrt {i (\coth (x)+1)} (\sinh (x)+\cosh (x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 438, normalized size = 7.68 \[ -\frac {4 \, {\left (2 \, \sqrt {2} {\left (23 \, \sqrt {2} \cosh \relax (x)^{5} + 115 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{4} + 23 \, \sqrt {2} \sinh \relax (x)^{5} + 5 \, {\left (46 \, \sqrt {2} \cosh \relax (x)^{2} - 7 \, \sqrt {2}\right )} \sinh \relax (x)^{3} - 35 \, \sqrt {2} \cosh \relax (x)^{3} + 5 \, {\left (46 \, \sqrt {2} \cosh \relax (x)^{3} - 21 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 5 \, {\left (23 \, \sqrt {2} \cosh \relax (x)^{4} - 21 \, \sqrt {2} \cosh \relax (x)^{2} + 3 \, \sqrt {2}\right )} \sinh \relax (x) + 15 \, \sqrt {2} \cosh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 15 \, {\left (\sqrt {2} \cosh \relax (x)^{6} + 6 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{5} + \sqrt {2} \sinh \relax (x)^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x)^{4} - 3 \, \sqrt {2} \cosh \relax (x)^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{3} - 3 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{4} - 6 \, \sqrt {2} \cosh \relax (x)^{2} + \sqrt {2}\right )} \sinh \relax (x)^{2} + 3 \, \sqrt {2} \cosh \relax (x)^{2} + 6 \, {\left (\sqrt {2} \cosh \relax (x)^{5} - 2 \, \sqrt {2} \cosh \relax (x)^{3} + \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) - \sqrt {2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )\right )}}{15 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} - 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} - 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 160, normalized size = 2.81 \[ -\frac {4}{15} \, \sqrt {2} {\left (\frac {2 \, {\left (45 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} + 135 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 170 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 100 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 100 \, e^{\left (2 \, x\right )} + 23\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{5}} + 15 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 43, normalized size = 0.75 \[ -\frac {4 \left (1+\coth \relax (x )\right )^{\frac {3}{2}}}{3}-\frac {2 \left (1+\coth \relax (x )\right )^{\frac {5}{2}}}{5}+8 \arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}-8 \sqrt {1+\coth \relax (x )} \]
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 \[ \int {\left (\coth \relax (x) + 1\right )}^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 44, normalized size = 0.77 \[ -8\,\sqrt {\mathrm {coth}\relax (x)+1}-\frac {4\,{\left (\mathrm {coth}\relax (x)+1\right )}^{3/2}}{3}-\frac {2\,{\left (\mathrm {coth}\relax (x)+1\right )}^{5/2}}{5}-\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}\,1{}\mathrm {i}}{2}\right )\,8{}\mathrm {i} \]
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 \[ \int \left (\coth {\relax (x )} + 1\right )^{\frac {7}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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