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 = 0.0962809, antiderivative size = 69, normalized size of antiderivative = 1., 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 1279
Rule 1166
Rule 203
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*}
Mathematica [A] time = 0.134683, size = 69, normalized size = 1. \[ \sinh (x)-\frac{1}{4} \sqrt{2-\sqrt{2}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{2}}}\right )-\frac{1}{4} \sqrt{2+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{2}}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.083, size = 42, normalized size = 0.6 \begin{align*}{\frac{{{\rm e}^{x}}}{2}}-{\frac{{{\rm e}^{-x}}}{2}}+\sum _{{\it \_R}={\it RootOf} \left ( 2048\,{{\it \_Z}}^{4}+128\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ( -8\,{\it \_R}\,{{\rm e}^{x}}+{{\rm e}^{2\,x}}-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35069, size = 504, normalized size = 7.3 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (x \right )} \tanh{\left (4 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25248, size = 96, normalized size = 1.39 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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