Optimal. Leaf size=65 \[ -\frac {3 x}{32 b}+\frac {x \cosh ^4(a+b x)}{4 b}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{16 b^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5481, 2715, 8}
\begin {gather*} -\frac {\sinh (a+b x) \cosh ^3(a+b x)}{16 b^2}-\frac {3 \sinh (a+b x) \cosh (a+b x)}{32 b^2}+\frac {x \cosh ^4(a+b x)}{4 b}-\frac {3 x}{32 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 5481
Rubi steps
\begin {align*} \int x \cosh ^3(a+b x) \sinh (a+b x) \, dx &=\frac {x \cosh ^4(a+b x)}{4 b}-\frac {\int \cosh ^4(a+b x) \, dx}{4 b}\\ &=\frac {x \cosh ^4(a+b x)}{4 b}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{16 b^2}-\frac {3 \int \cosh ^2(a+b x) \, dx}{16 b}\\ &=\frac {x \cosh ^4(a+b x)}{4 b}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{16 b^2}-\frac {3 \int 1 \, dx}{32 b}\\ &=-\frac {3 x}{32 b}+\frac {x \cosh ^4(a+b x)}{4 b}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{16 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 50, normalized size = 0.77 \begin {gather*} -\frac {-16 b x \cosh (2 (a+b x))-4 b x \cosh (4 (a+b x))+8 \sinh (2 (a+b x))+\sinh (4 (a+b x))}{128 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.02, size = 96, normalized size = 1.48
method | result | size |
risch | \(\frac {\left (4 b x -1\right ) {\mathrm e}^{4 b x +4 a}}{256 b^{2}}+\frac {\left (2 b x -1\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{2}}+\frac {\left (2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{2}}+\frac {\left (4 b x +1\right ) {\mathrm e}^{-4 b x -4 a}}{256 b^{2}}\) | \(82\) |
default | \(\frac {\left (2 b x +2 a \right ) \cosh \left (2 b x +2 a \right )-\sinh \left (2 b x +2 a \right )-2 a \cosh \left (2 b x +2 a \right )}{16 b^{2}}+\frac {\left (4 b x +4 a \right ) \cosh \left (4 b x +4 a \right )-\sinh \left (4 b x +4 a \right )-4 a \cosh \left (4 b x +4 a \right )}{128 b^{2}}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 91, normalized size = 1.40 \begin {gather*} \frac {{\left (4 \, b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{256 \, b^{2}} + \frac {{\left (2 \, b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{32 \, b^{2}} + \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{2}} + \frac {{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 108, normalized size = 1.66 \begin {gather*} \frac {b x \cosh \left (b x + a\right )^{4} + b x \sinh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right )^{2} - \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} + 2 \, b x\right )} \sinh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{32 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 110, normalized size = 1.69 \begin {gather*} \begin {cases} - \frac {3 x \sinh ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16 b} + \frac {5 x \cosh ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{32 b^{2}} - \frac {5 \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{32 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \sinh {\left (a \right )} \cosh ^{3}{\left (a \right )}}{2} & \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 = 81, normalized size = 1.25 \begin {gather*} \frac {{\left (4 \, b x - 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b^{2}} + \frac {{\left (2 \, b x - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{2}} + \frac {{\left (2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{2}} + \frac {{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 57, normalized size = 0.88 \begin {gather*} -\frac {\frac {3\,x}{32}-\frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^4}{4}}{b}-\frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )}{16\,b^2}-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )}{32\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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