Optimal. Leaf size=57 \[ \frac {e^{-2 a-2 b x}}{32 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {e^{6 a+6 b x}}{96 b}-\frac {x}{8} \]
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Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 12, 457,
76} \begin {gather*} \frac {e^{-2 a-2 b x}}{32 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {e^{6 a+6 b x}}{96 b}-\frac {x}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 76
Rule 457
Rule 2320
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh ^3(a+b x) \sinh (a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (1+x^2\right )^3}{16 x^3} \, 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^3} \, dx,x,e^{a+b x}\right )}{16 b}\\ &=\frac {\text {Subst}\left (\int \frac {(-1+x) (1+x)^3}{x^2} \, dx,x,e^{2 a+2 b x}\right )}{32 b}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{x^2}-\frac {2}{x}+2 x+x^2\right ) \, dx,x,e^{2 a+2 b x}\right )}{32 b}\\ &=\frac {e^{-2 a-2 b x}}{32 b}+\frac {e^{4 a+4 b x}}{32 b}+\frac {e^{6 a+6 b x}}{96 b}-\frac {x}{8}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 43, normalized size = 0.75 \begin {gather*} \frac {3 e^{-2 (a+b x)}+3 e^{4 (a+b x)}+e^{6 (a+b x)}-12 b x}{96 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.00, size = 89, normalized size = 1.56
method | result | size |
risch | \(\frac {{\mathrm e}^{-2 b x -2 a}}{32 b}+\frac {{\mathrm e}^{4 b x +4 a}}{32 b}+\frac {{\mathrm e}^{6 b x +6 a}}{96 b}-\frac {x}{8}\) | \(47\) |
default | \(-\frac {x}{8}-\frac {\sinh \left (2 b x +2 a \right )}{32 b}+\frac {\sinh \left (4 b x +4 a \right )}{32 b}+\frac {\sinh \left (6 b x +6 a \right )}{96 b}+\frac {\cosh \left (2 b x +2 a \right )}{32 b}+\frac {\cosh \left (4 b x +4 a \right )}{32 b}+\frac {\cosh \left (6 b x +6 a \right )}{96 b}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 52, normalized size = 0.91 \begin {gather*} \frac {{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{96 \, b} - \frac {b x + a}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs.
\(2 (46) = 92\).
time = 0.35, size = 152, normalized size = 2.67 \begin {gather*} \frac {4 \, \cosh \left (b x + a\right )^{4} - 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 4 \, \sinh \left (b x + a\right )^{4} - 3 \, {\left (4 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} - 3 \, {\left (4 \, b x - 8 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, {\left (4 \, \cosh \left (b x + a\right )^{3} - 3 \, {\left (4 \, b x + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, {\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. 235 vs.
\(2 (44) = 88\).
time = 3.22, size = 235, normalized size = 4.12 \begin {gather*} \begin {cases} \frac {x e^{2 a} e^{2 b x} \sinh ^{4}{\left (a + b x \right )}}{8} - \frac {x e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4} + \frac {x e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{4} - \frac {x e^{2 a} e^{2 b x} \cosh ^{4}{\left (a + b x \right )}}{8} - \frac {e^{2 a} e^{2 b x} \sinh ^{4}{\left (a + b x \right )}}{16 b} + \frac {e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac {e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} + \frac {7 e^{2 a} e^{2 b x} \cosh ^{4}{\left (a + b x \right )}}{48 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh {\left (a \right )} \cosh ^{3}{\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 = 46, normalized size = 0.81 \begin {gather*} -\frac {1}{8} \, x + \frac {e^{\left (6 \, b x + 6 \, a\right )}}{96 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{32 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 42, normalized size = 0.74 \begin {gather*} \frac {\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{32}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{32}+\frac {{\mathrm {e}}^{6\,a+6\,b\,x}}{96}}{b}-\frac {x}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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