Optimal. Leaf size=23 \[ \frac {e^{4 a+4 b x}}{16 b}-\frac {x}{4} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, 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, 14}
\begin {gather*} \frac {e^{4 a+4 b x}}{16 b}-\frac {x}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2320
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh (a+b x) \sinh (a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {-1+x^4}{4 x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {-1+x^4}{x} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{x}+x^3\right ) \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {e^{4 a+4 b x}}{16 b}-\frac {x}{4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{4} \left (\frac {e^{4 a+4 b x}}{4 b}-x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 33, normalized size = 1.43
method | result | size |
risch | \(\frac {{\mathrm e}^{4 b x +4 a}}{16 b}-\frac {x}{4}\) | \(19\) |
default | \(-\frac {x}{4}+\frac {\sinh \left (4 b x +4 a \right )}{16 b}+\frac {\cosh \left (4 b x +4 a \right )}{16 b}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 1.04 \begin {gather*} -\frac {1}{4} \, x - \frac {a}{4 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{16 \, 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. 91 vs.
\(2 (18) = 36\).
time = 0.43, size = 91, normalized size = 3.96 \begin {gather*} -\frac {{\left (4 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (4 \, b x + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (4 \, b x - 1\right )} \sinh \left (b x + a\right )^{2}}{16 \, {\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. 117 vs.
\(2 (15) = 30\).
time = 0.50, size = 117, normalized size = 5.09 \begin {gather*} \begin {cases} - \frac {x e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )}}{4} + \frac {x e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{2} - \frac {x e^{2 a} e^{2 b x} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac {e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh {\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 = 18, normalized size = 0.78 \begin {gather*} -\frac {1}{4} \, x + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{16 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.81, size = 18, normalized size = 0.78 \begin {gather*} \frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{16\,b}-\frac {x}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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