Optimal. Leaf size=67 \[ -\frac {5 x}{16}+\frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 8}
\begin {gather*} \frac {\sinh ^5(a+b x) \cosh (a+b x)}{6 b}-\frac {5 \sinh ^3(a+b x) \cosh (a+b x)}{24 b}+\frac {5 \sinh (a+b x) \cosh (a+b x)}{16 b}-\frac {5 x}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rubi steps
\begin {align*} \int \sinh ^6(a+b x) \, dx &=\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b}-\frac {5}{6} \int \sinh ^4(a+b x) \, dx\\ &=-\frac {5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b}+\frac {5}{8} \int \sinh ^2(a+b x) \, dx\\ &=\frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b}-\frac {5 \int 1 \, dx}{16}\\ &=-\frac {5 x}{16}+\frac {5 \cosh (a+b x) \sinh (a+b x)}{16 b}-\frac {5 \cosh (a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh (a+b x) \sinh ^5(a+b x)}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.64 \begin {gather*} \frac {-60 a-60 b x+45 \sinh (2 (a+b x))-9 \sinh (4 (a+b x))+\sinh (6 (a+b x))}{192 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.84, size = 47, normalized size = 0.70
method | result | size |
default | \(-\frac {5 x}{16}+\frac {15 \sinh \left (2 b x +2 a \right )}{64 b}-\frac {3 \sinh \left (4 b x +4 a \right )}{64 b}+\frac {\sinh \left (6 b x +6 a \right )}{192 b}\) | \(47\) |
risch | \(-\frac {5 x}{16}+\frac {{\mathrm e}^{6 b x +6 a}}{384 b}-\frac {3 \,{\mathrm e}^{4 b x +4 a}}{128 b}+\frac {15 \,{\mathrm e}^{2 b x +2 a}}{128 b}-\frac {15 \,{\mathrm e}^{-2 b x -2 a}}{128 b}+\frac {3 \,{\mathrm e}^{-4 b x -4 a}}{128 b}-\frac {{\mathrm e}^{-6 b x -6 a}}{384 b}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 86, normalized size = 1.28 \begin {gather*} -\frac {{\left (9 \, e^{\left (-2 \, b x - 2 \, a\right )} - 45 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac {5 \, {\left (b x + a\right )}}{16 \, b} - \frac {45 \, e^{\left (-2 \, b x - 2 \, a\right )} - 9 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 90, normalized size = 1.34 \begin {gather*} \frac {3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 2 \, {\left (5 \, \cosh \left (b x + a\right )^{3} - 9 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{5} - 6 \, \cosh \left (b x + a\right )^{3} + 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{96 \, b} \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.
\(2 (61) = 122\).
time = 0.41, size = 139, normalized size = 2.07 \begin {gather*} \begin {cases} \frac {5 x \sinh ^{6}{\left (a + b x \right )}}{16} - \frac {15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} + \frac {15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{16} - \frac {5 x \cosh ^{6}{\left (a + b x \right )}}{16} + \frac {11 \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16 b} - \frac {5 \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{6 b} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{16 b} & \text {for}\: b \neq 0 \\x \sinh ^{6}{\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 = 0.43, size = 88, normalized size = 1.31 \begin {gather*} -\frac {5}{16} \, x + \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac {3 \, e^{\left (4 \, b x + 4 \, a\right )}}{128 \, b} + \frac {15 \, e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} - \frac {15 \, e^{\left (-2 \, b x - 2 \, a\right )}}{128 \, b} + \frac {3 \, e^{\left (-4 \, b x - 4 \, a\right )}}{128 \, b} - \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 42, normalized size = 0.63 \begin {gather*} \frac {\frac {15\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}-\frac {3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{64}+\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{192}}{b}-\frac {5\,x}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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