Optimal. Leaf size=67 \[ -\frac {5}{2} \text {ArcTan}\left (\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )+2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )+\frac {1}{2} \coth (x) \sqrt {-1-\coth ^2(x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 427, 537,
223, 209, 385} \begin {gather*} -\frac {5}{2} \text {ArcTan}\left (\frac {\coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )+2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-\coth ^2(x)-1}}\right )+\frac {1}{2} \coth (x) \sqrt {-\coth ^2(x)-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 385
Rule 427
Rule 537
Rule 3742
Rubi steps
\begin {align*} \int \left (-1-\coth ^2(x)\right )^{3/2} \, dx &=\text {Subst}\left (\int \frac {\left (-1-x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \coth (x) \sqrt {-1-\coth ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {-3-5 x^2}{\sqrt {-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \coth (x) \sqrt {-1-\coth ^2(x)}-\frac {5}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2}} \, dx,x,\coth (x)\right )+4 \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \left (1-x^2\right )} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \coth (x) \sqrt {-1-\coth ^2(x)}-\frac {5}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )+4 \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )\\ &=-\frac {5}{2} \tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \coth (x)}{\sqrt {-1-\coth ^2(x)}}\right )+\frac {1}{2} \coth (x) \sqrt {-1-\coth ^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 118, normalized size = 1.76 \begin {gather*} -\frac {1}{8} \left (-1-\coth ^2(x)\right )^{3/2} \text {sech}^2(2 x) \left (16 \tanh ^{-1}\left (\frac {\cosh (x)}{\sqrt {\cosh (2 x)}}\right ) \sqrt {\cosh (2 x)} \sinh ^3(x)+4 \left (\text {ArcTan}\left (\frac {\cosh (x)}{\sqrt {-\cosh (2 x)}}\right ) \sqrt {-\cosh (2 x)}-4 \sqrt {2} \sqrt {\cosh (2 x)} \log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)}\right )\right ) \sinh ^3(x)+\sinh (4 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(210\) vs.
\(2(53)=106\).
time = 0.82, size = 211, normalized size = 3.15
method | result | size |
derivativedivides | \(\frac {\left (-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}-\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}+\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )-\frac {\left (-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}+\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}-\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )\) | \(211\) |
default | \(\frac {\left (-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )}{4}-\sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}+\sqrt {2}\, \arctan \left (\frac {\left (-2+2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (1+\coth \left (x \right )\right )^{2}+2 \coth \left (x \right )}}\right )-\frac {\left (-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )\right )^{\frac {3}{2}}}{6}+\frac {\coth \left (x \right ) \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}{4}-\frac {5 \arctan \left (\frac {\coth \left (x \right )}{\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )}{4}+\sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}-\sqrt {2}\, \arctan \left (\frac {\left (-2-2 \coth \left (x \right )\right ) \sqrt {2}}{4 \sqrt {-\left (\coth \left (x \right )-1\right )^{2}-2 \coth \left (x \right )}}\right )\) | \(211\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.39, size = 361, normalized size = 5.39 \begin {gather*} \frac {2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} - 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (2 \, {\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + 2 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )}\right ) - 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} - 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (-2 \, {\left (\sqrt {-2} \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} - 2 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-2 \, x\right )}\right ) - 5 \, {\left (i \, e^{\left (4 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + i\right )} \log \left (-4 \, {\left (i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}\right ) - 5 \, {\left (-i \, e^{\left (4 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (-4 \, {\left (-i \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}\right ) - 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} - 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (4 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} + \sqrt {-2} e^{\left (4 \, x\right )} + \sqrt {-2} e^{\left (2 \, x\right )} + 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) + 2 \, {\left (\sqrt {-2} e^{\left (4 \, x\right )} - 2 \, \sqrt {-2} e^{\left (2 \, x\right )} + \sqrt {-2}\right )} \log \left (4 \, {\left (\sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 2\right )} - \sqrt {-2} e^{\left (4 \, x\right )} - \sqrt {-2} e^{\left (2 \, x\right )} - 2 \, \sqrt {-2}\right )} e^{\left (-4 \, x\right )}\right ) + 2 \, \sqrt {-2 \, e^{\left (4 \, x\right )} - 2} {\left (e^{\left (2 \, x\right )} + 1\right )}}{4 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} \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 \left (- \coth ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 285, normalized size = 4.25 \begin {gather*} -\frac {1}{4} \, \sqrt {2} {\left (-5 i \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 2 \right |}}{2 \, {\left (\sqrt {2} + \sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right )}}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) - 4 i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + 4 i \, \log \left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + 4 i \, \log \left (-\sqrt {e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + \frac {4 \, {\left (3 i \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + i \, {\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + {\left (-i \, \sqrt {e^{\left (4 \, x\right )} + 1} + i \, e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + i \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} + 2 \, \sqrt {e^{\left (4 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}}\right )} \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 {\left (-{\mathrm {coth}\left (x\right )}^2-1\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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