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 = 0.0419785, antiderivative size = 57, normalized size of antiderivative = 1., 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 3478
Rule 3480
Rule 206
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*}
Mathematica [C] time = 0.258514, 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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Maple [A] time = 0.016, size = 43, normalized size = 0.8 \begin{align*} -{\frac{4}{3} \left ( 1+{\rm coth} \left (x\right ) \right ) ^{{\frac{3}{2}}}}-{\frac{2}{5} \left ( 1+{\rm coth} \left (x\right ) \right ) ^{{\frac{5}{2}}}}+8\,{\it Artanh} \left ( 1/2\,\sqrt{1+{\rm coth} \left (x\right )}\sqrt{2} \right ) \sqrt{2}-8\,\sqrt{1+{\rm coth} \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\coth \left (x\right ) + 1\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.41786, size = 1469, normalized size = 25.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16966, size = 216, normalized size = 3.79 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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