Optimal. Leaf size=103 \[ -\frac{\left (1-\sqrt{2}\right ) \log \left (e^{2 c (a+b x)}+3-2 \sqrt{2}\right )}{2 b c}-\frac{\left (1+\sqrt{2}\right ) \log \left (e^{2 c (a+b x)}+3+2 \sqrt{2}\right )}{2 b c}+\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c} \]
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Rubi [A] time = 0.154934, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2194, 5207, 2282, 12, 1247, 632, 31} \[ -\frac{\left (1-\sqrt{2}\right ) \log \left (e^{2 c (a+b x)}+3-2 \sqrt{2}\right )}{2 b c}-\frac{\left (1+\sqrt{2}\right ) \log \left (e^{2 c (a+b x)}+3+2 \sqrt{2}\right )}{2 b c}+\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 5207
Rule 2282
Rule 12
Rule 1247
Rule 632
Rule 31
Rubi steps
\begin{align*} \int e^{c (a+b x)} \tan ^{-1}(\cosh (a c+b c x)) \, dx &=\frac{\operatorname{Subst}\left (\int e^x \tan ^{-1}(\cosh (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c}-\frac{\operatorname{Subst}\left (\int \frac{e^x \sinh (x)}{1+\cosh ^2(x)} \, dx,x,a c+b c x\right )}{b c}\\ &=\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c}-\frac{\operatorname{Subst}\left (\int \frac{2 x \left (-1+x^2\right )}{1+6 x^2+x^4} \, dx,x,e^{a c+b c x}\right )}{b c}\\ &=\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c}-\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-1+x^2\right )}{1+6 x^2+x^4} \, dx,x,e^{a c+b c x}\right )}{b c}\\ &=\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c}-\frac{\operatorname{Subst}\left (\int \frac{-1+x}{1+6 x+x^2} \, dx,x,e^{2 a c+2 b c x}\right )}{b c}\\ &=\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c}-\frac{\left (1-\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{3-2 \sqrt{2}+x} \, dx,x,e^{2 a c+2 b c x}\right )}{2 b c}-\frac{\left (1+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{3+2 \sqrt{2}+x} \, dx,x,e^{2 a c+2 b c x}\right )}{2 b c}\\ &=\frac{e^{a c+b c x} \tan ^{-1}(\cosh (c (a+b x)))}{b c}-\frac{\left (1-\sqrt{2}\right ) \log \left (3-2 \sqrt{2}+e^{2 a c+2 b c x}\right )}{2 b c}-\frac{\left (1+\sqrt{2}\right ) \log \left (3+2 \sqrt{2}+e^{2 a c+2 b c x}\right )}{2 b c}\\ \end{align*}
Mathematica [C] time = 0.144337, size = 146, normalized size = 1.42 \[ \frac{\text{RootSum}\left [\text{$\#$1}^4+6 \text{$\#$1}^2+1\& ,\frac{-7 \text{$\#$1}^2 \log \left (e^{c (a+b x)}-\text{$\#$1}\right )+7 \text{$\#$1}^2 a c+7 \text{$\#$1}^2 b c x-\log \left (e^{c (a+b x)}-\text{$\#$1}\right )+a c+b c x}{3 \text{$\#$1}^2+1}\& \right ]-4 c (a+b x)+2 e^{c (a+b x)} \tan ^{-1}\left (\frac{1}{2} e^{-c (a+b x)} \left (e^{2 c (a+b x)}+1\right )\right )}{2 b c} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.801, size = 1375, normalized size = 13.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53232, size = 177, normalized size = 1.72 \begin{align*} \frac{\arctan \left (\cosh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} - \frac{\sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-2 \, b c x - 2 \, a c\right )} - 3}{2 \, \sqrt{2} + e^{\left (-2 \, b c x - 2 \, a c\right )} + 3}\right )}{2 \, b c} - \frac{2 \,{\left (b c x + a c\right )}}{b c} - \frac{\log \left (6 \, e^{\left (-2 \, b c x - 2 \, a c\right )} + e^{\left (-4 \, b c x - 4 \, a c\right )} + 1\right )}{2 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99728, size = 591, normalized size = 5.74 \begin{align*} \frac{2 \,{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \arctan \left (\cosh \left (b c x + a c\right )\right ) + \sqrt{2} \log \left (-\frac{3 \,{\left (2 \, \sqrt{2} - 3\right )} \cosh \left (b c x + a c\right )^{2} - 4 \,{\left (3 \, \sqrt{2} - 4\right )} \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 3 \,{\left (2 \, \sqrt{2} - 3\right )} \sinh \left (b c x + a c\right )^{2} + 2 \, \sqrt{2} - 3}{\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} + 3}\right ) - \log \left (\frac{2 \,{\left (\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} + 3\right )}}{\cosh \left (b c x + a c\right )^{2} - 2 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )^{2}}\right )}{2 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12482, size = 208, normalized size = 2.02 \begin{align*} \frac{{\left (\sqrt{2} e^{\left (-a c\right )} \log \left (-\frac{2 \, \sqrt{2} e^{\left (2 \, a c\right )} - e^{\left (2 \, b c x + 4 \, a c\right )} - 3 \, e^{\left (2 \, a c\right )}}{2 \, \sqrt{2} e^{\left (2 \, a c\right )} + e^{\left (2 \, b c x + 4 \, a c\right )} + 3 \, e^{\left (2 \, a c\right )}}\right ) + 2 \, \arctan \left (\frac{1}{2} \, e^{\left (b c x + a c\right )} + \frac{1}{2} \, e^{\left (-b c x - a c\right )}\right ) e^{\left (b c x\right )} - e^{\left (-a c\right )} \log \left (e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )\right )} e^{\left (a c\right )}}{2 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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