Optimal. Leaf size=66 \[ \frac {e^{-a-b x}}{8 b}-\frac {e^{a+b x}}{8 b}+\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{40 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2320, 12, 459}
\begin {gather*} \frac {e^{-a-b x}}{8 b}-\frac {e^{a+b x}}{8 b}+\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{40 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 459
Rule 2320
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh ^2(a+b x) \sinh (a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^2}{8 x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^2}{x^2} \, dx,x,e^{a+b x}\right )}{8 b}\\ &=\frac {\text {Subst}\left (\int \left (-1-\frac {1}{x^2}+x^2+x^4\right ) \, dx,x,e^{a+b x}\right )}{8 b}\\ &=\frac {e^{-a-b x}}{8 b}-\frac {e^{a+b x}}{8 b}+\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{40 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 54, normalized size = 0.82 \begin {gather*} \frac {e^{a+b x} \left (-3+e^{2 (a+b x)}\right )}{24 b}+\frac {e^{-a-b x} \left (5+e^{6 (a+b x)}\right )}{40 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.84, size = 69, normalized size = 1.05
method | result | size |
risch | \(\frac {{\mathrm e}^{-b x -a}}{8 b}-\frac {{\mathrm e}^{b x +a}}{8 b}+\frac {{\mathrm e}^{3 b x +3 a}}{24 b}+\frac {{\mathrm e}^{5 b x +5 a}}{40 b}\) | \(55\) |
default | \(-\frac {\sinh \left (b x +a \right )}{4 b}+\frac {\sinh \left (3 b x +3 a \right )}{24 b}+\frac {\sinh \left (5 b x +5 a \right )}{40 b}+\frac {\cosh \left (3 b x +3 a \right )}{24 b}+\frac {\cosh \left (5 b x +5 a \right )}{40 b}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 53, normalized size = 0.80 \begin {gather*} \frac {{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 15 \, e^{\left (-4 \, b x - 4 \, a\right )} + 3\right )} e^{\left (5 \, b x + 5 \, a\right )}}{120 \, b} + \frac {e^{\left (-b x - a\right )}}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 105, normalized size = 1.59 \begin {gather*} \frac {9 \, \cosh \left (b x + a\right )^{3} + 27 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 6 \, \sinh \left (b x + a\right )^{3} - 2 \, {\left (9 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right ) - 5 \, \cosh \left (b x + a\right )}{60 \, {\left (b \cosh \left (b x + a\right )^{2} - 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (48) = 96\).
time = 1.21, size = 128, normalized size = 1.94 \begin {gather*} \begin {cases} - \frac {4 e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )}}{15 b} + \frac {8 e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{15 b} - \frac {2 e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{15 b} + \frac {e^{2 a} e^{2 b x} \cosh ^{3}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh {\left (a \right )} \cosh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 54, normalized size = 0.82 \begin {gather*} \frac {e^{\left (5 \, b x + 5 \, a\right )}}{40 \, b} + \frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} - \frac {e^{\left (b x + a\right )}}{8 \, b} + \frac {e^{\left (-b x - a\right )}}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 47, normalized size = 0.71 \begin {gather*} \frac {15\,{\mathrm {e}}^{-a-b\,x}-15\,{\mathrm {e}}^{a+b\,x}+5\,{\mathrm {e}}^{3\,a+3\,b\,x}+3\,{\mathrm {e}}^{5\,a+5\,b\,x}}{120\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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