Optimal. Leaf size=90 \[ \frac {3 x}{128}+\frac {3 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b} \]
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Rubi [A]
time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2648, 2715, 8}
\begin {gather*} \frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{16 b}+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{64 b}+\frac {3 \sinh (a+b x) \cosh (a+b x)}{128 b}+\frac {3 x}{128} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2715
Rubi steps
\begin {align*} \int \cosh ^4(a+b x) \sinh ^4(a+b x) \, dx &=\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}-\frac {3}{8} \int \cosh ^4(a+b x) \sinh ^2(a+b x) \, dx\\ &=-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac {1}{16} \int \cosh ^4(a+b x) \, dx\\ &=\frac {\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac {3}{64} \int \cosh ^2(a+b x) \, dx\\ &=\frac {3 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}+\frac {3 \int 1 \, dx}{128}\\ &=\frac {3 x}{128}+\frac {3 \cosh (a+b x) \sinh (a+b x)}{128 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{8 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 33, normalized size = 0.37 \begin {gather*} \frac {24 (a+b x)-8 \sinh (4 (a+b x))+\sinh (8 (a+b x))}{1024 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.74, size = 33, normalized size = 0.37
method | result | size |
default | \(\frac {3 x}{128}-\frac {\sinh \left (4 b x +4 a \right )}{128 b}+\frac {\sinh \left (8 b x +8 a \right )}{1024 b}\) | \(33\) |
risch | \(\frac {3 x}{128}+\frac {{\mathrm e}^{8 b x +8 a}}{2048 b}-\frac {{\mathrm e}^{4 b x +4 a}}{256 b}+\frac {{\mathrm e}^{-4 b x -4 a}}{256 b}-\frac {{\mathrm e}^{-8 b x -8 a}}{2048 b}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 66, normalized size = 0.73 \begin {gather*} -\frac {{\left (8 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} + \frac {3 \, {\left (b x + a\right )}}{128 \, b} + \frac {8 \, e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 97, normalized size = 1.08 \begin {gather*} \frac {7 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{5} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + {\left (7 \, \cosh \left (b x + a\right )^{5} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b x + {\left (\cosh \left (b x + a\right )^{7} - 4 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )}{128 \, 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. 189 vs.
\(2 (80) = 160\).
time = 0.92, size = 189, normalized size = 2.10 \begin {gather*} \begin {cases} \frac {3 x \sinh ^{8}{\left (a + b x \right )}}{128} - \frac {3 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} + \frac {9 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} - \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} + \frac {3 x \cosh ^{8}{\left (a + b x \right )}}{128} - \frac {3 \sinh ^{7}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{128 b} + \frac {11 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac {11 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{128 b} - \frac {3 \sinh {\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sinh ^{4}{\left (a \right )} \cosh ^{4}{\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 = 60, normalized size = 0.67 \begin {gather*} \frac {3}{128} \, x + \frac {e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} - \frac {e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b} + \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b} - \frac {e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 32, normalized size = 0.36 \begin {gather*} \frac {3\,x}{128}-\frac {\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{128}-\frac {\mathrm {sinh}\left (8\,a+8\,b\,x\right )}{1024}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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