Optimal. Leaf size=87 \[ \sinh (x)-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}-\frac{1}{6} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{6} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{3}}}\right ) \]
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Rubi [A] time = 0.258683, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {12, 6742, 2073, 203, 1166} \[ \sinh (x)-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}-\frac{1}{6} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{6} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{3}}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 6742
Rule 2073
Rule 203
Rule 1166
Rubi steps
\begin{align*} \int \sinh (x) \tanh (6 x) \, dx &=-\operatorname{Subst}\left (\int \frac{2 x^2 \left (-3-16 x^2-16 x^4\right )}{1+18 x^2+48 x^4+32 x^6} \, dx,x,\sinh (x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{x^2 \left (-3-16 x^2-16 x^4\right )}{1+18 x^2+48 x^4+32 x^6} \, dx,x,\sinh (x)\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \left (-\frac{1}{2}+\frac{1+12 x^2+16 x^4}{2 \left (1+18 x^2+48 x^4+32 x^6\right )}\right ) \, dx,x,\sinh (x)\right )\right )\\ &=\sinh (x)-\operatorname{Subst}\left (\int \frac{1+12 x^2+16 x^4}{1+18 x^2+48 x^4+32 x^6} \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\operatorname{Subst}\left (\int \left (\frac{1}{3 \left (1+2 x^2\right )}+\frac{2 \left (1+8 x^2\right )}{3 \left (1+16 x^2+16 x^4\right )}\right ) \, dx,x,\sinh (x)\right )\\ &=\sinh (x)-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\sinh (x)\right )-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1+8 x^2}{1+16 x^2+16 x^4} \, dx,x,\sinh (x)\right )\\ &=-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}+\sinh (x)-\frac{1}{3} \left (4 \left (2-\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{8-4 \sqrt{3}+16 x^2} \, dx,x,\sinh (x)\right )-\frac{1}{3} \left (4 \left (2+\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{8+4 \sqrt{3}+16 x^2} \, dx,x,\sinh (x)\right )\\ &=-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}-\frac{1}{6} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{6} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{3}}}\right )+\sinh (x)\\ \end{align*}
Mathematica [A] time = 0.123729, size = 87, normalized size = 1. \[ \sinh (x)-\frac{\tan ^{-1}\left (\sqrt{2} \sinh (x)\right )}{3 \sqrt{2}}-\frac{1}{6} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{6} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 \sinh (x)}{\sqrt{2+\sqrt{3}}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.105, size = 84, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{x}}}{2}}-{\frac{{{\rm e}^{-x}}}{2}}+{\frac{i}{12}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}-i\sqrt{2}{{\rm e}^{x}}-1 \right ) -{\frac{i}{12}}\sqrt{2}\ln \left ({{\rm e}^{2\,x}}+i\sqrt{2}{{\rm e}^{x}}-1 \right ) +\sum _{{\it \_R}={\it RootOf} \left ( 20736\,{{\it \_Z}}^{4}+576\,{{\it \_Z}}^{2}+1 \right ) }{\it \_R}\,\ln \left ( -12\,{\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}{6} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) - \frac{1}{6} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{1}{2} \, \int \frac{2 \,{\left (2 \, e^{\left (7 \, x\right )} - e^{\left (5 \, x\right )} - e^{\left (3 \, x\right )} + 2 \, e^{x}\right )}}{3 \,{\left (e^{\left (8 \, x\right )} - e^{\left (4 \, x\right )} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37504, size = 633, normalized size = 7.28 \begin{align*} -\frac{1}{6} \,{\left (2 \, \sqrt{\sqrt{3} + 2} \arctan \left ({\left (\sqrt{\sqrt{3} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1} \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )} -{\left ({\left (\sqrt{3} - 2\right )} e^{\left (2 \, x\right )} - \sqrt{3} + 2\right )} \sqrt{\sqrt{3} + 2}\right )} e^{\left (-x\right )}\right ) e^{x} - 2 \, \sqrt{-\sqrt{3} + 2} \arctan \left ({\left (\sqrt{-\sqrt{3} e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} + 1}{\left (\sqrt{3} + 2\right )} \sqrt{-\sqrt{3} + 2} -{\left ({\left (\sqrt{3} + 2\right )} e^{\left (2 \, x\right )} - \sqrt{3} - 2\right )} \sqrt{-\sqrt{3} + 2}\right )} e^{\left (-x\right )}\right ) e^{x} + \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} e^{\left (3 \, x\right )} + \frac{1}{2} \, \sqrt{2} e^{x}\right ) e^{x} + \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} e^{x}\right ) e^{x} - 3 \, e^{\left (2 \, x\right )} + 3\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 (6 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19229, size = 135, normalized size = 1.55 \begin{align*} -\frac{1}{12} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (-\frac{2 \,{\left (e^{\left (-x\right )} - e^{x}\right )}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{12} \,{\left (\sqrt{6} - \sqrt{2}\right )} \arctan \left (-\frac{2 \,{\left (e^{\left (-x\right )} - e^{x}\right )}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{12} \, \sqrt{2}{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\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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