Optimal. Leaf size=98 \[ \frac {\text {ArcTan}\left (\frac {1+\sqrt [3]{6} \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cosh (x)+3^{2/3} \cosh ^2(x)\right )}{12 \sqrt [3]{6}} \]
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Rubi [A]
time = 0.08, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3302, 206, 31,
648, 631, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [3]{6} \cosh (x)+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}+\frac {\log \left (3^{2/3} \cosh ^2(x)+2^{2/3} \sqrt [3]{3} \cosh (x)+2 \sqrt [3]{2}\right )}{12 \sqrt [3]{6}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 3302
Rubi steps
\begin {align*} \int \frac {\sinh (x)}{4-3 \cosh ^3(x)} \, dx &=\text {Subst}\left (\int \frac {1}{4-3 x^3} \, dx,x,\cosh (x)\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{3} x} \, dx,x,\cosh (x)\right )}{6 \sqrt [3]{2}}+\frac {\text {Subst}\left (\int \frac {2\ 2^{2/3}+\sqrt [3]{3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{6 \sqrt [3]{2}}\\ &=-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{2\ 2^{2/3}}+\frac {\text {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{3}+2\ 3^{2/3} x}{2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\cosh (x)\right )}{12 \sqrt [3]{6}}\\ &=-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cosh (x)+3^{2/3} \cosh ^2(x)\right )}{12 \sqrt [3]{6}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{6} \cosh (x)\right )}{2 \sqrt [3]{6}}\\ &=\frac {\tan ^{-1}\left (\frac {1+\sqrt [3]{6} \cosh (x)}{\sqrt {3}}\right )}{2 \sqrt [3]{2} 3^{5/6}}-\frac {\log \left (2^{2/3}-\sqrt [3]{3} \cosh (x)\right )}{6 \sqrt [3]{6}}+\frac {\log \left (2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{3} \cosh (x)+3^{2/3} \cosh ^2(x)\right )}{12 \sqrt [3]{6}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 77, normalized size = 0.79 \begin {gather*} \frac {1}{72} \left (6\ 2^{2/3} \sqrt [6]{3} \text {ArcTan}\left (\frac {1+\sqrt [3]{6} \cosh (x)}{\sqrt {3}}\right )+6^{2/3} \left (-2 \log \left (2-\sqrt [3]{6} \cosh (x)\right )+\log \left (4+2 \sqrt [3]{6} \cosh (x)+6^{2/3} \cosh ^2(x)\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 80, normalized size = 0.82
method | result | size |
risch | \(\munderset {\textit {\_R} =\RootOf \left (1296 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (24 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+1\right )\) | \(26\) |
derivativedivides | \(-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh \left (x \right )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh ^{2}\left (x \right )+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cosh \left (x \right )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}+\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cosh \left (x \right )}{2}+1\right )}{3}\right )}{12}\) | \(80\) |
default | \(-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh \left (x \right )-\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{36}+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (\cosh ^{2}\left (x \right )+\frac {4^{\frac {1}{3}} 3^{\frac {2}{3}} \cosh \left (x \right )}{3}+\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{72}+\frac {4^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {4^{\frac {2}{3}} 3^{\frac {1}{3}} \cosh \left (x \right )}{2}+1\right )}{3}\right )}{12}\) | \(80\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 305 vs.
\(2 (69) = 138\).
time = 0.39, size = 305, normalized size = 3.11 \begin {gather*} \frac {1}{12} \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{12} \cdot 6^{\frac {1}{6}} {\left (6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \cosh \left (x\right )^{3} + 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \sinh \left (x\right )^{3} + {\left (3 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \cosh \left (x\right ) + 4 \cdot 6^{\frac {1}{3}} \sqrt {2}\right )} \sinh \left (x\right )^{2} + 4 \cdot 6^{\frac {1}{3}} \sqrt {2} \cosh \left (x\right )^{2} + {\left (6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} - 16 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}}\right )} \cosh \left (x\right ) + {\left (3 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \cosh \left (x\right )^{2} + 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} + 8 \cdot 6^{\frac {1}{3}} \sqrt {2} \cosh \left (x\right ) - 16 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}}\right )} \sinh \left (x\right ) + 2 \cdot 6^{\frac {1}{3}} \sqrt {2}\right )}\right ) - \frac {1}{12} \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{12} \cdot 6^{\frac {1}{6}} {\left (6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \cosh \left (x\right ) + 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} \sinh \left (x\right ) + 2 \cdot 6^{\frac {1}{3}} \sqrt {2}\right )}\right ) - \frac {1}{72} \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2 \, {\left (2 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \cosh \left (x\right ) - 3 \, \cosh \left (x\right )^{2} - 3 \, \sinh \left (x\right )^{2} - 4 \cdot 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} - 3\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \frac {1}{36} \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {2 \, {\left (6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \, \cosh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.53, size = 85, normalized size = 0.87 \begin {gather*} - \frac {6^{\frac {2}{3}} \log {\left (\cosh {\left (x \right )} - \frac {6^{\frac {2}{3}}}{3} \right )}}{36} + \frac {6^{\frac {2}{3}} \log {\left (36 \cosh ^{2}{\left (x \right )} + 12 \cdot 6^{\frac {2}{3}} \cosh {\left (x \right )} + 24 \cdot \sqrt [3]{6} \right )}}{72} + \frac {2^{\frac {2}{3}} \cdot \sqrt [6]{3} \operatorname {atan}{\left (\frac {\sqrt [3]{2} \cdot 3^{\frac {5}{6}} \cosh {\left (x \right )}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 80, normalized size = 0.82 \begin {gather*} \frac {1}{12} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{4} \, \sqrt {3} \left (\frac {4}{3}\right )^{\frac {2}{3}} {\left (\left (\frac {4}{3}\right )^{\frac {1}{3}} + e^{\left (-x\right )} + e^{x}\right )}\right ) + \frac {1}{72} \cdot 36^{\frac {1}{3}} \log \left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 2 \, \left (\frac {4}{3}\right )^{\frac {1}{3}} {\left (e^{\left (-x\right )} + e^{x}\right )} + 4 \, \left (\frac {4}{3}\right )^{\frac {2}{3}}\right ) - \frac {1}{12} \, \left (\frac {4}{3}\right )^{\frac {1}{3}} \log \left ({\left | -2 \, \left (\frac {4}{3}\right )^{\frac {1}{3}} + e^{\left (-x\right )} + e^{x} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.51, size = 205, normalized size = 2.09 \begin {gather*} -\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )}{36}-\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}+\frac {6^{2/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{36}+\frac {6^{2/3}\,\ln \left (\frac {256\,{\mathrm {e}}^{2\,x}}{81}-\frac {128\,{\mathrm {e}}^x}{27}-\frac {6^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {256\,{\mathrm {e}}^{2\,x}}{9}-\frac {2048\,{\mathrm {e}}^x}{27}-\frac {6^{2/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (256\,{\mathrm {e}}^{2\,x}-\frac {2048\,{\mathrm {e}}^x}{3}+256\right )}{36}+\frac {256}{9}\right )}{36}+\frac {256}{81}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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