Optimal. Leaf size=105 \[ -\frac {3 \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {\cosh (6 a+6 b x)}{3456 b^3}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2} \]
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Rubi [A]
time = 0.11, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5556, 3377,
2718} \begin {gather*} -\frac {3 \cosh (2 a+2 b x)}{128 b^3}+\frac {\cosh (6 a+6 b x)}{3456 b^3}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 5556
Rubi steps
\begin {align*} \int x^2 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{32} x^2 \sinh (2 a+2 b x)+\frac {1}{32} x^2 \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac {1}{32} \int x^2 \sinh (6 a+6 b x) \, dx-\frac {3}{32} \int x^2 \sinh (2 a+2 b x) \, dx\\ &=-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}-\frac {\int x \cosh (6 a+6 b x) \, dx}{96 b}+\frac {3 \int x \cosh (2 a+2 b x) \, dx}{32 b}\\ &=-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}+\frac {\int \sinh (6 a+6 b x) \, dx}{576 b^2}-\frac {3 \int \sinh (2 a+2 b x) \, dx}{64 b^2}\\ &=-\frac {3 \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^2 \cosh (2 a+2 b x)}{64 b}+\frac {\cosh (6 a+6 b x)}{3456 b^3}+\frac {x^2 \cosh (6 a+6 b x)}{192 b}+\frac {3 x \sinh (2 a+2 b x)}{64 b^2}-\frac {x \sinh (6 a+6 b x)}{576 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 72, normalized size = 0.69 \begin {gather*} \frac {-81 \left (1+2 b^2 x^2\right ) \cosh (2 (a+b x))+\left (1+18 b^2 x^2\right ) \cosh (6 (a+b x))+6 b x (27 \sinh (2 (a+b x))-\sinh (6 (a+b x)))}{3456 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs.
\(2(93)=186\).
time = 1.20, size = 208, normalized size = 1.98
method | result | size |
risch | \(\frac {\left (18 b^{2} x^{2}-6 b x +1\right ) {\mathrm e}^{6 b x +6 a}}{6912 b^{3}}-\frac {3 \left (2 b^{2} x^{2}-2 b x +1\right ) {\mathrm e}^{2 b x +2 a}}{256 b^{3}}-\frac {3 \left (2 b^{2} x^{2}+2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{256 b^{3}}+\frac {\left (18 b^{2} x^{2}+6 b x +1\right ) {\mathrm e}^{-6 b x -6 a}}{6912 b^{3}}\) | \(114\) |
default | \(-\frac {3 \left (\left (2 b x +2 a \right )^{2} \cosh \left (2 b x +2 a \right )-2 \left (2 b x +2 a \right ) \sinh \left (2 b x +2 a \right )+2 \cosh \left (2 b x +2 a \right )-4 a \left (\left (2 b x +2 a \right ) \cosh \left (2 b x +2 a \right )-\sinh \left (2 b x +2 a \right )\right )+4 a^{2} \cosh \left (2 b x +2 a \right )\right )}{256 b^{3}}+\frac {\left (6 b x +6 a \right )^{2} \cosh \left (6 b x +6 a \right )-2 \left (6 b x +6 a \right ) \sinh \left (6 b x +6 a \right )+2 \cosh \left (6 b x +6 a \right )-12 a \left (\left (6 b x +6 a \right ) \cosh \left (6 b x +6 a \right )-\sinh \left (6 b x +6 a \right )\right )+36 a^{2} \cosh \left (6 b x +6 a \right )}{6912 b^{3}}\) | \(208\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 127, normalized size = 1.21 \begin {gather*} \frac {{\left (18 \, b^{2} x^{2} e^{\left (6 \, a\right )} - 6 \, b x e^{\left (6 \, a\right )} + e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )}}{6912 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{256 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac {{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \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. 202 vs.
\(2 (93) = 186\).
time = 0.40, size = 202, normalized size = 1.92 \begin {gather*} -\frac {120 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 36 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} - {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{6} - 15 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - {\left (18 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{6} + 81 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 3 \, {\left (5 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - 54 \, b^{2} x^{2} - 27\right )} \sinh \left (b x + a\right )^{2} + 36 \, {\left (b x \cosh \left (b x + a\right )^{5} - 9 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{3456 \, b^{3}} \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. 212 vs.
\(2 (102) = 204\).
time = 1.00, size = 212, normalized size = 2.02 \begin {gather*} \begin {cases} - \frac {x^{2} \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x^{2} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac {x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac {x^{2} \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{12 b^{2}} - \frac {2 x \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {x \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{12 b^{2}} - \frac {7 \sinh ^{6}{\left (a + b x \right )}}{216 b^{3}} + \frac {\sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{18 b^{3}} - \frac {\cosh ^{6}{\left (a + b x \right )}}{72 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 113, normalized size = 1.08 \begin {gather*} \frac {{\left (18 \, b^{2} x^{2} - 6 \, b x + 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{6912 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{256 \, b^{3}} - \frac {3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{256 \, b^{3}} + \frac {{\left (18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{6912 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 89, normalized size = 0.85 \begin {gather*} -\frac {\frac {3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{128}-\frac {\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{3456}+b^2\,\left (\frac {3\,x^2\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{64}-\frac {x^2\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{192}\right )-b\,\left (\frac {3\,x\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}-\frac {x\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{576}\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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