Optimal. Leaf size=69 \[ \frac {e^{-3 a-3 b x}}{48 b}-\frac {e^{-a-b x}}{8 b}-\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{80 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2320, 12, 459}
\begin {gather*} \frac {e^{-3 a-3 b x}}{48 b}-\frac {e^{-a-b x}}{8 b}-\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{80 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 459
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-1-x^2\right ) \left (1-x^2\right )^3}{16 x^4} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\left (-1-x^2\right ) \left (1-x^2\right )^3}{x^4} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {2}{x^2}-2 x^2+x^4\right ) \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac {e^{-3 a-3 b x}}{48 b}-\frac {e^{-a-b x}}{8 b}-\frac {e^{3 a+3 b x}}{24 b}+\frac {e^{5 a+5 b x}}{80 b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 51, normalized size = 0.74 \begin {gather*} \frac {e^{-3 (a+b x)} \left (5-30 e^{2 (a+b x)}-10 e^{6 (a+b x)}+3 e^{8 (a+b x)}\right )}{240 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 44, normalized size = 0.64
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sinh ^{5}\left (b x +a \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{5}-\frac {2 \left (\cosh ^{3}\left (b x +a \right )\right )}{15}}{b}\) | \(44\) |
default | \(\frac {\frac {\left (\sinh ^{5}\left (b x +a \right )\right )}{5}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{5}-\frac {2 \left (\cosh ^{3}\left (b x +a \right )\right )}{15}}{b}\) | \(44\) |
risch | \(\frac {{\mathrm e}^{-3 b x -3 a}}{48 b}-\frac {{\mathrm e}^{-b x -a}}{8 b}-\frac {{\mathrm e}^{3 b x +3 a}}{24 b}+\frac {{\mathrm e}^{5 b x +5 a}}{80 b}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 56, normalized size = 0.81 \begin {gather*} -\frac {{\left (6 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, b} + \frac {3 \, e^{\left (5 \, b x + 5 \, a\right )} - 10 \, e^{\left (3 \, b x + 3 \, a\right )}}{240 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 111, normalized size = 1.61 \begin {gather*} \frac {\cosh \left (b x + a\right )^{4} - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + {\left (6 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right )^{2} - 5 \, \cosh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{3} - 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{30 \, {\left (b \cosh \left (b x + a\right ) - b \sinh \left (b x + a\right )\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. 139 vs.
\(2 (53) = 106\).
time = 2.97, size = 139, normalized size = 2.01 \begin {gather*} \begin {cases} \frac {e^{a} e^{b x} \sinh ^{4}{\left (a + b x \right )}}{5 b} - \frac {e^{a} e^{b x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{5 b} + \frac {e^{a} e^{b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{5 b} + \frac {2 e^{a} e^{b x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{15 b} - \frac {2 e^{a} e^{b x} \cosh ^{4}{\left (a + b x \right )}}{15 b} & \text {for}\: b \neq 0 \\x e^{a} \sinh ^{3}{\left (a \right )} \cosh {\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 = 0.41, size = 52, normalized size = 0.75 \begin {gather*} -\frac {5 \, {\left (6 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, e^{\left (5 \, b x + 5 \, a\right )} + 10 \, e^{\left (3 \, b x + 3 \, a\right )}}{240 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 50, normalized size = 0.72 \begin {gather*} -\frac {30\,{\mathrm {e}}^{-a-b\,x}-5\,{\mathrm {e}}^{-3\,a-3\,b\,x}+10\,{\mathrm {e}}^{3\,a+3\,b\,x}-3\,{\mathrm {e}}^{5\,a+5\,b\,x}}{240\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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