Optimal. Leaf size=40 \[ -\frac{3 \coth (a+b x)}{2 b}+\frac{\cosh ^2(a+b x) \coth (a+b x)}{2 b}+\frac{3 x}{2} \]
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Rubi [A] time = 0.0411313, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2591, 288, 321, 206} \[ -\frac{3 \coth (a+b x)}{2 b}+\frac{\cosh ^2(a+b x) \coth (a+b x)}{2 b}+\frac{3 x}{2} \]
Antiderivative was successfully verified.
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Rule 2591
Rule 288
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \cosh ^2(a+b x) \coth ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac{\cosh ^2(a+b x) \coth (a+b x)}{2 b}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,\coth (a+b x)\right )}{2 b}\\ &=-\frac{3 \coth (a+b x)}{2 b}+\frac{\cosh ^2(a+b x) \coth (a+b x)}{2 b}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\coth (a+b x)\right )}{2 b}\\ &=\frac{3 x}{2}-\frac{3 \coth (a+b x)}{2 b}+\frac{\cosh ^2(a+b x) \coth (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.10684, size = 31, normalized size = 0.78 \[ \frac{6 (a+b x)+\sinh (2 (a+b x))-4 \coth (a+b x)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 39, normalized size = 1. \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{2\,\sinh \left ( bx+a \right ) }}+{\frac{3\,bx}{2}}+{\frac{3\,a}{2}}-{\frac{3\,{\rm coth} \left (bx+a\right )}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12189, size = 89, normalized size = 2.22 \begin{align*} \frac{3 \,{\left (b x + a\right )}}{2 \, b} - \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac{17 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1}{8 \, b{\left (e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72865, size = 166, normalized size = 4.15 \begin{align*} \frac{\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 4 \,{\left (3 \, b x + 2\right )} \sinh \left (b x + a\right ) - 9 \, \cosh \left (b x + a\right )}{8 \, b \sinh \left (b x + a\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 \cosh ^{2}{\left (a + b x \right )} \coth ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25637, size = 90, normalized size = 2.25 \begin{align*} \frac{12 \, b x + \frac{{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 14 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, a\right )}}{e^{\left (2 \, b x\right )} - e^{\left (4 \, b x + 2 \, a\right )}} + e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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