Optimal. Leaf size=113 \[ \frac{e^{a+b x}}{b}+\frac{5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac{8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac{3 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0716714, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2282, 390, 1258, 1157, 385, 206} \[ \frac{e^{a+b x}}{b}+\frac{5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac{8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac{3 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 390
Rule 1258
Rule 1157
Rule 385
Rule 206
Rubi steps
\begin{align*} \int e^{a+b x} \coth ^4(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{\left (1-x^2\right )^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{8 x^2 \left (1+x^4\right )}{\left (1-x^2\right )^4}\right ) \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}+\frac{8 \operatorname{Subst}\left (\int \frac{x^2 \left (1+x^4\right )}{\left (1-x^2\right )^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}+\frac{8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}+\frac{4 \operatorname{Subst}\left (\int \frac{-2-6 x^2-6 x^4}{\left (1-x^2\right )^3} \, dx,x,e^{a+b x}\right )}{3 b}\\ &=\frac{e^{a+b x}}{b}+\frac{8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac{14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-6-24 x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{3 b}\\ &=\frac{e^{a+b x}}{b}+\frac{8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac{14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac{5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}+\frac{8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac{14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac{5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac{3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 10.1054, size = 115, normalized size = 1.02 \[ \frac{-24 e^{a+b x}+50 e^{3 (a+b x)}-48 e^{5 (a+b x)}+6 e^{7 (a+b x)}+9 \left (e^{2 (a+b x)}-1\right )^3 \log \left (1-e^{a+b x}\right )-9 \left (e^{2 (a+b x)}-1\right )^3 \log \left (e^{a+b x}+1\right )}{6 b \left (e^{2 (a+b x)}-1\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 123, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}-3\,{\frac{\cosh \left ( bx+a \right ) }{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}+{\frac{3\,{\rm csch} \left (bx+a\right ){\rm coth} \left (bx+a\right )}{2}}-3\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) +{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}{ \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}-{\frac{4\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\, \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}-{\frac{8\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\,\sinh \left ( bx+a \right ) }}+{\frac{8\,\sinh \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1656, size = 149, normalized size = 1.32 \begin{align*} \frac{e^{\left (b x + a\right )}}{b} - \frac{3 \, \log \left (e^{\left (b x + a\right )} + 1\right )}{2 \, b} + \frac{3 \, \log \left (e^{\left (b x + a\right )} - 1\right )}{2 \, b} - \frac{15 \, e^{\left (5 \, b x + 5 \, a\right )} - 16 \, e^{\left (3 \, b x + 3 \, a\right )} + 9 \, e^{\left (b x + a\right )}}{3 \, b{\left (e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11681, size = 2221, normalized size = 19.65 \begin{align*} \text{result too large to display} \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 = 1.29063, size = 112, normalized size = 0.99 \begin{align*} -\frac{\frac{2 \,{\left (15 \, e^{\left (5 \, b x + 5 \, a\right )} - 16 \, e^{\left (3 \, b x + 3 \, a\right )} + 9 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} - 6 \, e^{\left (b x + a\right )} + 9 \, \log \left (e^{\left (b x + a\right )} + 1\right ) - 9 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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