Optimal. Leaf size=37 \[ -\frac {2}{7} (\text {csch}(x)+1)^{7/2}+\frac {4}{5} (\text {csch}(x)+1)^{5/2}-\frac {4}{3} (\text {csch}(x)+1)^{3/2} \]
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Rubi [A] time = 0.11, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4372, 1570, 1469, 697} \[ -\frac {2}{7} (\text {csch}(x)+1)^{7/2}+\frac {4}{5} (\text {csch}(x)+1)^{5/2}-\frac {4}{3} (\text {csch}(x)+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 697
Rule 1469
Rule 1570
Rule 4372
Rubi steps
\begin {align*} \int \coth ^3(x) \text {csch}(x) \sqrt {1+\text {csch}(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}} \left (1+x^2\right )}{x^4} \, dx,x,\sinh (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {\left (1+\frac {1}{x^2}\right ) \sqrt {1+\frac {1}{x}}}{x^2} \, dx,x,\sinh (x)\right )\\ &=-\operatorname {Subst}\left (\int \sqrt {1+x} \left (1+x^2\right ) \, dx,x,\text {csch}(x)\right )\\ &=-\operatorname {Subst}\left (\int \left (2 \sqrt {1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\text {csch}(x)\right )\\ &=-\frac {4}{3} (1+\text {csch}(x))^{3/2}+\frac {4}{5} (1+\text {csch}(x))^{5/2}-\frac {2}{7} (1+\text {csch}(x))^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 34, normalized size = 0.92 \[ -\frac {1}{210} \text {csch}^3(x) \sqrt {\text {csch}(x)+1} (-117 \sinh (x)+43 \sinh (3 x)+62 \cosh (2 x)-2) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 271, normalized size = 7.32 \[ -\frac {2 \, {\left (43 \, \cosh \relax (x)^{6} + 2 \, {\left (129 \, \cosh \relax (x) + 31\right )} \sinh \relax (x)^{5} + 43 \, \sinh \relax (x)^{6} + 62 \, \cosh \relax (x)^{5} + {\left (645 \, \cosh \relax (x)^{2} + 310 \, \cosh \relax (x) - 117\right )} \sinh \relax (x)^{4} - 117 \, \cosh \relax (x)^{4} + 4 \, {\left (215 \, \cosh \relax (x)^{3} + 155 \, \cosh \relax (x)^{2} - 117 \, \cosh \relax (x) - 1\right )} \sinh \relax (x)^{3} - 4 \, \cosh \relax (x)^{3} + {\left (645 \, \cosh \relax (x)^{4} + 620 \, \cosh \relax (x)^{3} - 702 \, \cosh \relax (x)^{2} - 12 \, \cosh \relax (x) + 117\right )} \sinh \relax (x)^{2} + 117 \, \cosh \relax (x)^{2} + 2 \, {\left (129 \, \cosh \relax (x)^{5} + 155 \, \cosh \relax (x)^{4} - 234 \, \cosh \relax (x)^{3} - 6 \, \cosh \relax (x)^{2} + 117 \, \cosh \relax (x) + 31\right )} \sinh \relax (x) + 62 \, \cosh \relax (x) - 43\right )} \sqrt {\frac {\sinh \relax (x) + 1}{\sinh \relax (x)}}}{105 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} - 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} - 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {csch}\relax (x) + 1} \coth \relax (x)^{3} \operatorname {csch}\relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (\coth ^{3}\relax (x )\right ) \mathrm {csch}\relax (x ) \sqrt {1+\mathrm {csch}\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 389, normalized size = 10.51 \[ \frac {124 \, \sqrt {-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-x\right )}}{105 \, \sqrt {e^{\left (-x\right )} + 1} \sqrt {e^{\left (-x\right )} - 1} {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac {78 \, \sqrt {-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-2 \, x\right )}}{35 \, \sqrt {e^{\left (-x\right )} + 1} \sqrt {e^{\left (-x\right )} - 1} {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac {8 \, \sqrt {-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-3 \, x\right )}}{105 \, \sqrt {e^{\left (-x\right )} + 1} \sqrt {e^{\left (-x\right )} - 1} {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {78 \, \sqrt {-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-4 \, x\right )}}{35 \, \sqrt {e^{\left (-x\right )} + 1} \sqrt {e^{\left (-x\right )} - 1} {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {124 \, \sqrt {-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-5 \, x\right )}}{105 \, \sqrt {e^{\left (-x\right )} + 1} \sqrt {e^{\left (-x\right )} - 1} {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac {86 \, \sqrt {-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-6 \, x\right )}}{105 \, \sqrt {e^{\left (-x\right )} + 1} \sqrt {e^{\left (-x\right )} - 1} {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {86 \, \sqrt {-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1}}{105 \, \sqrt {e^{\left (-x\right )} + 1} \sqrt {e^{\left (-x\right )} - 1} {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 207, normalized size = 5.59 \[ -\frac {8\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{35\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {8\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{35\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {86\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{105}-\frac {16\,{\mathrm {e}}^x\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{7\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {16\,{\mathrm {e}}^x\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{7\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {124\,{\mathrm {e}}^x\,\sqrt {1-\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}}}}{105\,\left ({\mathrm {e}}^{2\,x}-1\right )} \]
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 \[ \int \sqrt {\operatorname {csch}{\relax (x )} + 1} \coth ^{3}{\relax (x )} \operatorname {csch}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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