Optimal. Leaf size=31 \[ \frac {\sinh ^4(a+b x)}{4 b}+\frac {\sinh ^6(a+b x)}{6 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2644, 14}
\begin {gather*} \frac {\sinh ^6(a+b x)}{6 b}+\frac {\sinh ^4(a+b x)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2644
Rubi steps
\begin {align*} \int \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,i \sinh (a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,i \sinh (a+b x)\right )}{b}\\ &=\frac {\sinh ^4(a+b x)}{4 b}+\frac {\sinh ^6(a+b x)}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.13 \begin {gather*} \frac {1}{8} \left (-\frac {3 \cosh (2 (a+b x))}{8 b}+\frac {\cosh (6 (a+b x))}{24 b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 34, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {\left (\cosh ^{4}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{12}}{b}\) | \(34\) |
default | \(\frac {\frac {\left (\cosh ^{4}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{12}}{b}\) | \(34\) |
risch | \(\frac {{\mathrm e}^{6 b x +6 a}}{384 b}-\frac {3 \,{\mathrm e}^{2 b x +2 a}}{128 b}-\frac {3 \,{\mathrm e}^{-2 b x -2 a}}{128 b}+\frac {{\mathrm e}^{-6 b x -6 a}}{384 b}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (27) = 54\).
time = 0.26, size = 56, normalized size = 1.81 \begin {gather*} -\frac {{\left (9 \, e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac {9 \, e^{\left (-2 \, b x - 2 \, 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs.
\(2 (27) = 54\).
time = 0.37, size = 72, normalized size = 2.32 \begin {gather*} \frac {\cosh \left (b x + a\right )^{6} + 15 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{4} - 3\right )} \sinh \left (b x + a\right )^{2} - 9 \, \cosh \left (b x + a\right )^{2}}{192 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 42, normalized size = 1.35 \begin {gather*} \begin {cases} - \frac {\sinh ^{6}{\left (a + b x \right )}}{12 b} + \frac {\sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b} & \text {for}\: b \neq 0 \\x \sinh ^{3}{\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (27) = 54\).
time = 0.43, size = 57, normalized size = 1.84 \begin {gather*} \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} - \frac {3 \, e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} - \frac {3 \, e^{\left (-2 \, b x - 2 \, 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 = 1.52, size = 26, normalized size = 0.84 \begin {gather*} \frac {2\,{\mathrm {sinh}\left (a+b\,x\right )}^6+3\,{\mathrm {sinh}\left (a+b\,x\right )}^4}{12\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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