Integrand size = 16, antiderivative size = 57 \[ \int e^{a+b x} \cosh ^3(a+b x) \, dx=-\frac {e^{-2 a-2 b x}}{16 b}+\frac {3 e^{2 a+2 b x}}{16 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {3 x}{8} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 12, 272, 45} \[ \int e^{a+b x} \cosh ^3(a+b x) \, dx=-\frac {e^{-2 a-2 b x}}{16 b}+\frac {3 e^{2 a+2 b x}}{16 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {3 x}{8} \]
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Rule 12
Rule 45
Rule 272
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{8 x^3} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^3} \, dx,x,e^{a+b x}\right )}{8 b} \\ & = \frac {\text {Subst}\left (\int \frac {(1+x)^3}{x^2} \, dx,x,e^{2 a+2 b x}\right )}{16 b} \\ & = \frac {\text {Subst}\left (\int \left (3+\frac {1}{x^2}+\frac {3}{x}+x\right ) \, dx,x,e^{2 a+2 b x}\right )}{16 b} \\ & = -\frac {e^{-2 a-2 b x}}{16 b}+\frac {3 e^{2 a+2 b x}}{16 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {3 x}{8} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int e^{a+b x} \cosh ^3(a+b x) \, dx=\frac {-\frac {1}{16} e^{-2 a-2 b x}+\frac {3}{16} e^{2 a+2 b x}+\frac {1}{32} e^{4 a+4 b x}+\frac {3 b x}{8}}{b} \]
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Time = 0.41 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {{\mathrm e}^{-2 b x -2 a}}{16 b}+\frac {3 \,{\mathrm e}^{2 b x +2 a}}{16 b}+\frac {{\mathrm e}^{4 b x +4 a}}{32 b}+\frac {3 x}{8}\) | \(47\) |
derivativedivides | \(\frac {\frac {\cosh \left (b x +a \right )^{4}}{4}+\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}}{b}\) | \(49\) |
default | \(\frac {\frac {\cosh \left (b x +a \right )^{4}}{4}+\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}}{b}\) | \(49\) |
parallelrisch | \(\frac {{\mathrm e}^{b x +a} \left (12 b x \cosh \left (b x +a \right )-12 b x \sinh \left (b x +a \right )+\cosh \left (b x +a \right )+11 \sinh \left (b x +a \right )-\cosh \left (3 b x +3 a \right )+3 \sinh \left (3 b x +3 a \right )\right )}{32 b}\) | \(69\) |
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (46) = 92\).
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.67 \[ \int e^{a+b x} \cosh ^3(a+b x) \, dx=-\frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 3 \, \sinh \left (b x + a\right )^{3} - 6 \, {\left (2 \, b x + 1\right )} \cosh \left (b x + a\right ) + 3 \, {\left (4 \, b x - 3 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right )}{32 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (48) = 96\).
Time = 0.86 (sec) , antiderivative size = 182, normalized size of antiderivative = 3.19 \[ \int e^{a+b x} \cosh ^3(a+b x) \, dx=\begin {cases} \frac {3 x e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )}}{8} - \frac {3 x e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8} - \frac {3 x e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} + \frac {3 x e^{a} e^{b x} \cosh ^{3}{\left (a + b x \right )}}{8} - \frac {3 e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )}}{8 b} + \frac {3 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac {e^{a} e^{b x} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x e^{a} \cosh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int e^{a+b x} \cosh ^3(a+b x) \, dx=\frac {3 \, {\left (b x + a\right )}}{8 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{32 \, b} + \frac {3 \, e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b} \]
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Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int e^{a+b x} \cosh ^3(a+b x) \, dx=\frac {12 \, b x - 2 \, {\left (3 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )} + 12 \, a + e^{\left (4 \, b x + 4 \, a\right )} + 6 \, e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b} \]
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Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74 \[ \int e^{a+b x} \cosh ^3(a+b x) \, dx=\frac {3\,x}{8}+\frac {\frac {3\,{\mathrm {e}}^{2\,a+2\,b\,x}}{16}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{16}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{32}}{b} \]
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