Optimal. Leaf size=87 \[ \sinh (x)-\frac {1}{5} \tan ^{-1}(\sinh (x))-\frac {1}{5} \sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (2 \sqrt {\frac {2}{3+\sqrt {5}}} \sinh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {2 \left (3+\sqrt {5}\right )} \sinh (x)\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6742, 2073, 203, 1166} \[ \sinh (x)-\frac {1}{5} \tan ^{-1}(\sinh (x))-\frac {1}{5} \sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (2 \sqrt {\frac {2}{3+\sqrt {5}}} \sinh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {2 \left (3+\sqrt {5}\right )} \sinh (x)\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 1166
Rule 2073
Rule 6742
Rubi steps
\begin {align*} \int \sinh (x) \tanh (5 x) \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \left (-5-20 x^2-16 x^4\right )}{1+13 x^2+28 x^4+16 x^6} \, dx,x,\sinh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-1+\frac {1+8 x^2+8 x^4}{1+13 x^2+28 x^4+16 x^6}\right ) \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\operatorname {Subst}\left (\int \frac {1+8 x^2+8 x^4}{1+13 x^2+28 x^4+16 x^6} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\operatorname {Subst}\left (\int \left (\frac {1}{5 \left (1+x^2\right )}+\frac {4 \left (1+6 x^2\right )}{5 \left (1+12 x^2+16 x^4\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right )-\frac {4}{5} \operatorname {Subst}\left (\int \frac {1+6 x^2}{1+12 x^2+16 x^4} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{5} \tan ^{-1}(\sinh (x))+\sinh (x)-\frac {1}{5} \left (4 \left (3-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{6-2 \sqrt {5}+16 x^2} \, dx,x,\sinh (x)\right )-\frac {1}{5} \left (4 \left (3+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{6+2 \sqrt {5}+16 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{5} \tan ^{-1}(\sinh (x))-\frac {1}{5} \sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (2 \sqrt {\frac {2}{3+\sqrt {5}}} \sinh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {2 \left (3+\sqrt {5}\right )} \sinh (x)\right )+\sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.20, size = 81, normalized size = 0.93 \[ \frac {1}{10} \left (10 \sinh (x)-2 \tan ^{-1}(\sinh (x))-\sqrt {2 \left (3+\sqrt {5}\right )} \tan ^{-1}\left (2 \sqrt {\frac {2}{3+\sqrt {5}}} \sinh (x)\right )-\sqrt {6-2 \sqrt {5}} \tan ^{-1}\left (\sqrt {2 \left (3+\sqrt {5}\right )} \sinh (x)\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 237, normalized size = 2.72 \[ -\frac {1}{10} \, {\left (2 \, \sqrt {2} \sqrt {\sqrt {5} + 3} \arctan \left (\frac {1}{8} \, {\left (\sqrt {2 \, {\left (\sqrt {5} - 1\right )} e^{\left (2 \, x\right )} + 4 \, e^{\left (4 \, x\right )} + 4} {\left (\sqrt {5} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {5} + 3} - 2 \, {\left ({\left (\sqrt {5} \sqrt {2} - 3 \, \sqrt {2}\right )} e^{\left (2 \, x\right )} - \sqrt {5} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {\sqrt {5} + 3}\right )} e^{\left (-x\right )}\right ) e^{x} - 2 \, \sqrt {2} \sqrt {-\sqrt {5} + 3} \arctan \left (\frac {1}{8} \, {\left (\sqrt {-2 \, {\left (\sqrt {5} + 1\right )} e^{\left (2 \, x\right )} + 4 \, e^{\left (4 \, x\right )} + 4} {\left (\sqrt {5} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {-\sqrt {5} + 3} - 2 \, {\left ({\left (\sqrt {5} \sqrt {2} + 3 \, \sqrt {2}\right )} e^{\left (2 \, x\right )} - \sqrt {5} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {-\sqrt {5} + 3}\right )} e^{\left (-x\right )}\right ) e^{x} + 4 \, \arctan \left (e^{x}\right ) e^{x} - 5 \, e^{\left (2 \, x\right )} + 5\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 81, normalized size = 0.93 \[ -\frac {1}{10} \, \pi - \frac {1}{10} \, {\left (\sqrt {5} + 1\right )} \arctan \left (-\frac {2 \, {\left (e^{\left (-x\right )} - e^{x}\right )}}{\sqrt {5} + 1}\right ) - \frac {1}{10} \, {\left (\sqrt {5} - 1\right )} \arctan \left (-\frac {2 \, {\left (e^{\left (-x\right )} - e^{x}\right )}}{\sqrt {5} - 1}\right ) - \frac {1}{5} \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 60, normalized size = 0.69 \[ \frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{5}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{5}+\left (\munderset {\textit {\_R} =\RootOf \left (10000 \textit {\_Z}^{4}+300 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-10 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}-1\right )\right ) \]
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 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} - \frac {2}{5} \, \arctan \left (e^{x}\right ) - \frac {1}{2} \, \int \frac {2 \, {\left (3 \, e^{\left (7 \, x\right )} - e^{\left (5 \, x\right )} - e^{\left (3 \, x\right )} + 3 \, e^{x}\right )}}{5 \, {\left (e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.66, size = 82, normalized size = 0.94 \[ \frac {{\mathrm {e}}^x}{2}-\frac {2\,\mathrm {atan}\left ({\mathrm {e}}^x\right )}{5}-\frac {{\mathrm {e}}^{-x}}{2}-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{10\,\sqrt {\frac {3}{200}-\frac {\sqrt {5}}{200}}}\right )\,\sqrt {\frac {3}{200}-\frac {\sqrt {5}}{200}}-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{10\,\sqrt {\frac {\sqrt {5}}{200}+\frac {3}{200}}}\right )\,\sqrt {\frac {\sqrt {5}}{200}+\frac {3}{200}} \]
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 \[ \int \sinh {\relax (x )} \tanh {\left (5 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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