Optimal. Leaf size=69 \[ \sinh (x)-\frac {1}{4} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {12, 1279, 1166, 203} \[ \sinh (x)-\frac {1}{4} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 1166
Rule 1279
Rubi steps
\begin {align*} \int \sinh (x) \tanh (4 x) \, dx &=-\operatorname {Subst}\left (\int \frac {4 x^2 \left (-1-2 x^2\right )}{1+8 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {x^2 \left (-1-2 x^2\right )}{1+8 x^2+8 x^4} \, dx,x,\sinh (x)\right )\right )\\ &=\sinh (x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-2-8 x^2}{1+8 x^2+8 x^4} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)+\left (-2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{4-2 \sqrt {2}+8 x^2} \, dx,x,\sinh (x)\right )-\left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{4+2 \sqrt {2}+8 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac {1}{4} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right )+\sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 69, normalized size = 1.00 \[ \sinh (x)-\frac {1}{4} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {2}}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 168, normalized size = 2.43 \[ -\frac {1}{2} \, {\left (\sqrt {\sqrt {2} + 2} \arctan \left (\frac {1}{2} \, {\left (\sqrt {\sqrt {2} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt {\sqrt {2} + 2} {\left (\sqrt {2} - 2\right )} - {\left ({\left (\sqrt {2} - 2\right )} e^{\left (2 \, x\right )} - \sqrt {2} + 2\right )} \sqrt {\sqrt {2} + 2}\right )} e^{\left (-x\right )}\right ) e^{x} - \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {1}{2} \, {\left (\sqrt {-\sqrt {2} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-\sqrt {2} + 2} - {\left ({\left (\sqrt {2} + 2\right )} e^{\left (2 \, x\right )} - \sqrt {2} - 2\right )} \sqrt {-\sqrt {2} + 2}\right )} e^{\left (-x\right )}\right ) e^{x} - e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 71, normalized size = 1.03 \[ -\frac {1}{4} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {e^{\left (-x\right )} - e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {e^{\left (-x\right )} - e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 42, normalized size = 0.61 \[ \frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}+\left (\munderset {\textit {\_R} =\RootOf \left (2048 \textit {\_Z}^{4}+128 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-8 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}-1\right )\right ) \]
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 \[ \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} - \frac {1}{2} \, \int \frac {2 \, {\left (e^{\left (7 \, x\right )} + e^{x}\right )}}{e^{\left (8 \, x\right )} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.77, size = 71, normalized size = 1.03 \[ \frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{\sqrt {\sqrt {2}+2}}\right )\,\sqrt {\sqrt {2}+2}}{4}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{\sqrt {2-\sqrt {2}}}\right )\,\sqrt {2-\sqrt {2}}}{4} \]
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 \[ \int \sinh {\relax (x )} \tanh {\left (4 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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