Optimal. Leaf size=79 \[ -\frac {3 \cosh (4 a+4 b x)}{1024 b^4}+\frac {3 x \sinh (4 a+4 b x)}{256 b^3}-\frac {3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac {x^3 \sinh (4 a+4 b x)}{32 b}-\frac {x^4}{32} \]
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Rubi [A] time = 0.11, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5448, 3296, 2638} \[ -\frac {3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac {3 x \sinh (4 a+4 b x)}{256 b^3}-\frac {3 \cosh (4 a+4 b x)}{1024 b^4}+\frac {x^3 \sinh (4 a+4 b x)}{32 b}-\frac {x^4}{32} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5448
Rubi steps
\begin {align*} \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {x^3}{8}+\frac {1}{8} x^3 \cosh (4 a+4 b x)\right ) \, dx\\ &=-\frac {x^4}{32}+\frac {1}{8} \int x^3 \cosh (4 a+4 b x) \, dx\\ &=-\frac {x^4}{32}+\frac {x^3 \sinh (4 a+4 b x)}{32 b}-\frac {3 \int x^2 \sinh (4 a+4 b x) \, dx}{32 b}\\ &=-\frac {x^4}{32}-\frac {3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac {x^3 \sinh (4 a+4 b x)}{32 b}+\frac {3 \int x \cosh (4 a+4 b x) \, dx}{64 b^2}\\ &=-\frac {x^4}{32}-\frac {3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac {3 x \sinh (4 a+4 b x)}{256 b^3}+\frac {x^3 \sinh (4 a+4 b x)}{32 b}-\frac {3 \int \sinh (4 a+4 b x) \, dx}{256 b^3}\\ &=-\frac {x^4}{32}-\frac {3 \cosh (4 a+4 b x)}{1024 b^4}-\frac {3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac {3 x \sinh (4 a+4 b x)}{256 b^3}+\frac {x^3 \sinh (4 a+4 b x)}{32 b}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 58, normalized size = 0.73 \[ \frac {4 b x \left (8 b^2 x^2+3\right ) \sinh (4 (a+b x))-3 \left (8 b^2 x^2+1\right ) \cosh (4 (a+b x))-32 b^4 x^4}{1024 b^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 140, normalized size = 1.77 \[ -\frac {32 \, b^{4} x^{4} + 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - 16 \, {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 18 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 16 \, {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4}}{1024 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 78, normalized size = 0.99 \[ -\frac {1}{32} \, x^{4} + \frac {{\left (32 \, b^{3} x^{3} - 24 \, b^{2} x^{2} + 12 \, b x - 3\right )} e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b^{4}} - \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 404, normalized size = 5.11 \[ \frac {\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2} \left (\cosh ^{4}\left (b x +a \right )\right )}{16}+\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{32}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}-\frac {3 \left (b x +a \right )^{2}}{128}-\frac {3 \left (\cosh ^{4}\left (b x +a \right )\right )}{128}+\frac {3 \left (\cosh ^{2}\left (b x +a \right )\right )}{128}+\frac {3 \left (b x +a \right )^{2} \left (\cosh ^{2}\left (b x +a \right )\right )}{16}-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{3}}{24}-\frac {\left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{8}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{32}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}-\frac {b x}{64}-\frac {a}{64}+\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{8}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{2}}{16}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{16}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{16}\right )-a^{3} \left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 91, normalized size = 1.15 \[ -\frac {1}{32} \, x^{4} + \frac {{\left (32 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 24 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 12 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{2048 \, b^{4}} - \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.80, size = 70, normalized size = 0.89 \[ -\frac {\frac {3\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{1024}-\frac {3\,b\,x\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{256}+\frac {3\,b^2\,x^2\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{128}-\frac {b^3\,x^3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}}{b^4}-\frac {x^4}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.36, size = 250, normalized size = 3.16 \[ \begin {cases} - \frac {x^{4} \sinh ^{4}{\left (a + b x \right )}}{32} + \frac {x^{4} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} - \frac {x^{4} \cosh ^{4}{\left (a + b x \right )}}{32} + \frac {x^{3} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {x^{3} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} - \frac {3 x^{2} \sinh ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {9 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 x^{2} \cosh ^{4}{\left (a + b x \right )}}{128 b^{2}} + \frac {3 x \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{3}} + \frac {3 x \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{3}} - \frac {3 \sinh ^{4}{\left (a + b x \right )}}{256 b^{4}} - \frac {3 \cosh ^{4}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{2}{\relax (a )} \cosh ^{2}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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