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 = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 206
Rule 288
Rule 321
Rule 2591
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*}
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Mathematica [A] time = 0.11, 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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fricas [A] time = 0.42, size = 60, normalized size = 1.50 \[ \frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 4 \, {\left (3 \, b x + 2\right )} \sinh \left (b x + a\right ) - 9 \, \cosh \left (b x + a\right )}{8 \, b \sinh \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 67, normalized size = 1.68 \[ \frac {12 \, b x + \frac {{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 14 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-2 \, a\right )}}{e^{\left (2 \, b x\right )} - e^{\left (4 \, b x + 2 \, a\right )}} + e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 39, normalized size = 0.98 \[ \frac {\frac {\cosh ^{3}\left (b x +a \right )}{2 \sinh \left (b x +a \right )}+\frac {3 b x}{2}+\frac {3 a}{2}-\frac {3 \coth \left (b x +a \right )}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 66, normalized size = 1.65 \[ \frac {3 \, {\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac {17 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1}{8 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 50, normalized size = 1.25 \[ \frac {3\,x}{2}-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{8\,b}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{8\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh ^{2}{\left (a + b x \right )} \coth ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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