Integrand size = 25, antiderivative size = 74 \[ \int e^{c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \, dx=\frac {e^{2 c (a+b x)} \text {csch}(a c+b c x) \sqrt {\sinh ^2(a c+b c x)}}{4 b c}-\frac {1}{2} x \text {csch}(a c+b c x) \sqrt {\sinh ^2(a c+b c x)} \]
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Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6852, 2320, 12, 14} \[ \int e^{c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \, dx=\frac {e^{2 c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \text {csch}(a c+b c x)}{4 b c}-\frac {1}{2} x \sqrt {\sinh ^2(a c+b c x)} \text {csch}(a c+b c x) \]
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Rule 12
Rule 14
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \left (\text {csch}(a c+b c x) \sqrt {\sinh ^2(a c+b c x)}\right ) \int e^{c (a+b x)} \sinh (a c+b c x) \, dx \\ & = \frac {\left (\text {csch}(a c+b c x) \sqrt {\sinh ^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {-1+x^2}{2 x} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\left (\text {csch}(a c+b c x) \sqrt {\sinh ^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {-1+x^2}{x} \, dx,x,e^{c (a+b x)}\right )}{2 b c} \\ & = \frac {\left (\text {csch}(a c+b c x) \sqrt {\sinh ^2(a c+b c x)}\right ) \text {Subst}\left (\int \left (-\frac {1}{x}+x\right ) \, dx,x,e^{c (a+b x)}\right )}{2 b c} \\ & = \frac {e^{2 c (a+b x)} \text {csch}(a c+b c x) \sqrt {\sinh ^2(a c+b c x)}}{4 b c}-\frac {1}{2} x \text {csch}(a c+b c x) \sqrt {\sinh ^2(a c+b c x)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.65 \[ \int e^{c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \, dx=\frac {\left (e^{2 c (a+b x)}-2 b c x\right ) \text {csch}(c (a+b x)) \sqrt {\sinh ^2(c (a+b x))}}{4 b c} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.59 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\sinh \left (c \left (b x +a \right )\right )\right ) \left (\frac {\cosh \left (b c x +a c \right )^{2}}{2}+\frac {\sinh \left (b c x +a c \right ) \cosh \left (b c x +a c \right )}{2}-\frac {b c x}{2}-\frac {a c}{2}\right )}{c b}\) | \(60\) |
risch | \(-\frac {x \sqrt {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, {\mathrm e}^{c \left (b x +a \right )}}{2 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )}+\frac {\sqrt {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}\, {\mathrm e}^{3 c \left (b x +a \right )}}{4 c b \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )}\) | \(106\) |
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.89 \[ \int e^{c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \, dx=-\frac {{\left (2 \, b c x - 1\right )} \cosh \left (b c x + a c\right ) - {\left (2 \, b c x + 1\right )} \sinh \left (b c x + a c\right )}{4 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (68) = 136\).
Time = 1.57 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.14 \[ \int e^{c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \, dx=\begin {cases} 0 & \text {for}\: a = 0 \wedge b = 0 \wedge c = 0 \\x \sqrt {\sinh ^{2}{\left (a c \right )}} e^{a c} & \text {for}\: b = 0 \\0 & \text {for}\: a = - b x \vee c = 0 \\\frac {x \sqrt {\sinh ^{2}{\left (a c + b c x \right )}} e^{a c} e^{b c x}}{2} - \frac {x \sqrt {\sinh ^{2}{\left (a c + b c x \right )}} e^{a c} e^{b c x} \cosh {\left (a c + b c x \right )}}{2 \sinh {\left (a c + b c x \right )}} + \frac {\sqrt {\sinh ^{2}{\left (a c + b c x \right )}} e^{a c} e^{b c x} \cosh {\left (a c + b c x \right )}}{2 b c \sinh {\left (a c + b c x \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.49 \[ \int e^{c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \, dx=-\frac {b c x + a c}{2 \, b c} + \frac {e^{\left (2 \, b c x + 2 \, a c\right )}}{4 \, b c} \]
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Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int e^{c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \, dx=-\frac {1}{2} \, x \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) + \frac {e^{\left (2 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )}{4 \, b c} \]
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Time = 1.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.04 \[ \int e^{c (a+b x)} \sqrt {\sinh ^2(a c+b c x)} \, dx=-\frac {\left (x\,{\mathrm {e}}^{a\,c+b\,c\,x}-\frac {{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}}{2\,b\,c}\right )\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1} \]
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