Optimal. Leaf size=82 \[ -\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cosh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cosh (x)\right )+\cosh (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1690, 1180, 213}
\begin {gather*} \cosh (x)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cosh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cosh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 213
Rule 1180
Rule 1690
Rubi steps
\begin {align*} \int \cosh (x) \tanh (5 x) \, dx &=\text {Subst}\left (\int \frac {1-12 x^2+16 x^4}{5-20 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\text {Subst}\left (\int \left (1-\frac {4 \left (1-2 x^2\right )}{5-20 x^2+16 x^4}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-4 \text {Subst}\left (\int \frac {1-2 x^2}{5-20 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac {1}{5} \left (4 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-10+2 \sqrt {5}+16 x^2} \, dx,x,\cosh (x)\right )+\frac {1}{5} \left (4 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-10-2 \sqrt {5}+16 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \cosh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \cosh (x)\right )+\cosh (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.02, size = 249, normalized size = 3.04 \begin {gather*} \cosh (x)+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4-\text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-x-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )+x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2-x \text {$\#$1}^4-2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^4+x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}+2 \text {$\#$1}^3-3 \text {$\#$1}^5+4 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.91, size = 70, normalized size = 0.85
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\left (\munderset {\textit {\_R} =\RootOf \left (2000 \textit {\_Z}^{4}-100 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-10 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+1\right )\right )\) | \(42\) |
derivativedivides | \(\cosh \left (x \right )-\frac {\sqrt {5}\, \left (\sqrt {5}+1\right ) \arctanh \left (\frac {4 \cosh \left (x \right )}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \cosh \left (x \right )}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\) | \(70\) |
default | \(\cosh \left (x \right )-\frac {\sqrt {5}\, \left (\sqrt {5}+1\right ) \arctanh \left (\frac {4 \cosh \left (x \right )}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \cosh \left (x \right )}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 293 vs.
\(2 (54) = 108\).
time = 0.43, size = 293, normalized size = 3.57 \begin {gather*} -\frac {{\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\sqrt {5} + 5} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\sqrt {5} + 5} + 2\right ) - {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\sqrt {5} + 5} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\sqrt {5} + 5} + 2\right ) + {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {-\sqrt {5} + 5} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {-\sqrt {5} + 5} + 2\right ) - {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {-\sqrt {5} + 5} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {-\sqrt {5} + 5} + 2\right ) - 10 \, \cosh \left (x\right )^{2} - 20 \, \cosh \left (x\right ) \sinh \left (x\right ) - 10 \, \sinh \left (x\right )^{2} - 10}{20 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cosh {\left (x \right )} \tanh {\left (5 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 127 vs.
\(2 (54) = 108\).
time = 0.44, size = 127, normalized size = 1.55 \begin {gather*} -\frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left (-\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left (\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left (-\sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 141, normalized size = 1.72 \begin {gather*} \frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}+\ln \left (4\,{\mathrm {e}}^{2\,x}-40\,{\mathrm {e}}^x\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+4\right )\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}-\ln \left (4\,{\mathrm {e}}^{2\,x}+40\,{\mathrm {e}}^x\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+4\right )\,\sqrt {\frac {1}{40}-\frac {\sqrt {5}}{200}}+\ln \left (4\,{\mathrm {e}}^{2\,x}-40\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+4\right )\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}-\ln \left (4\,{\mathrm {e}}^{2\,x}+40\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}}+4\right )\,\sqrt {\frac {\sqrt {5}}{200}+\frac {1}{40}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________