Optimal. Leaf size=155 \[ -\frac {3 \sinh ^3(a+b x) \cosh (a+b x)}{128 b^4}+\frac {45 \sinh (a+b x) \cosh (a+b x)}{256 b^4}+\frac {3 x \sinh ^4(a+b x)}{32 b^3}-\frac {9 x \sinh ^2(a+b x)}{32 b^3}-\frac {3 x^2 \sinh ^3(a+b x) \cosh (a+b x)}{16 b^2}+\frac {9 x^2 \sinh (a+b x) \cosh (a+b x)}{32 b^2}+\frac {x^3 \sinh ^4(a+b x)}{4 b}-\frac {45 x}{256 b^3}-\frac {3 x^3}{32 b} \]
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Rubi [A] time = 0.14, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5372, 3311, 30, 2635, 8} \[ -\frac {3 x^2 \sinh ^3(a+b x) \cosh (a+b x)}{16 b^2}+\frac {9 x^2 \sinh (a+b x) \cosh (a+b x)}{32 b^2}+\frac {3 x \sinh ^4(a+b x)}{32 b^3}-\frac {9 x \sinh ^2(a+b x)}{32 b^3}-\frac {3 \sinh ^3(a+b x) \cosh (a+b x)}{128 b^4}+\frac {45 \sinh (a+b x) \cosh (a+b x)}{256 b^4}+\frac {x^3 \sinh ^4(a+b x)}{4 b}-\frac {45 x}{256 b^3}-\frac {3 x^3}{32 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2635
Rule 3311
Rule 5372
Rubi steps
\begin {align*} \int x^3 \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\frac {x^3 \sinh ^4(a+b x)}{4 b}-\frac {3 \int x^2 \sinh ^4(a+b x) \, dx}{4 b}\\ &=-\frac {3 x^2 \cosh (a+b x) \sinh ^3(a+b x)}{16 b^2}+\frac {3 x \sinh ^4(a+b x)}{32 b^3}+\frac {x^3 \sinh ^4(a+b x)}{4 b}-\frac {3 \int \sinh ^4(a+b x) \, dx}{32 b^3}+\frac {9 \int x^2 \sinh ^2(a+b x) \, dx}{16 b}\\ &=\frac {9 x^2 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac {9 x \sinh ^2(a+b x)}{32 b^3}-\frac {3 \cosh (a+b x) \sinh ^3(a+b x)}{128 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh ^3(a+b x)}{16 b^2}+\frac {3 x \sinh ^4(a+b x)}{32 b^3}+\frac {x^3 \sinh ^4(a+b x)}{4 b}+\frac {9 \int \sinh ^2(a+b x) \, dx}{128 b^3}+\frac {9 \int \sinh ^2(a+b x) \, dx}{32 b^3}-\frac {9 \int x^2 \, dx}{32 b}\\ &=-\frac {3 x^3}{32 b}+\frac {45 \cosh (a+b x) \sinh (a+b x)}{256 b^4}+\frac {9 x^2 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac {9 x \sinh ^2(a+b x)}{32 b^3}-\frac {3 \cosh (a+b x) \sinh ^3(a+b x)}{128 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh ^3(a+b x)}{16 b^2}+\frac {3 x \sinh ^4(a+b x)}{32 b^3}+\frac {x^3 \sinh ^4(a+b x)}{4 b}-\frac {9 \int 1 \, dx}{256 b^3}-\frac {9 \int 1 \, dx}{64 b^3}\\ &=-\frac {45 x}{256 b^3}-\frac {3 x^3}{32 b}+\frac {45 \cosh (a+b x) \sinh (a+b x)}{256 b^4}+\frac {9 x^2 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac {9 x \sinh ^2(a+b x)}{32 b^3}-\frac {3 \cosh (a+b x) \sinh ^3(a+b x)}{128 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh ^3(a+b x)}{16 b^2}+\frac {3 x \sinh ^4(a+b x)}{32 b^3}+\frac {x^3 \sinh ^4(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 95, normalized size = 0.61 \[ \frac {48 \left (2 b^2 x^2+1\right ) \sinh (2 (a+b x))+2 b x \left (8 b^2 x^2+3\right ) \cosh (4 (a+b x))-\left (3 \left (8 b^2 x^2+1\right ) \sinh (2 (a+b x))+32 b x \left (2 b^2 x^2+3\right )\right ) \cosh (2 (a+b x))}{512 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 191, normalized size = 1.23 \[ \frac {{\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{4} - 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \sinh \left (b x + a\right )^{4} - 16 \, {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (16 \, b^{3} x^{3} - 3 \, {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{2} + 24 \, b x\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left ({\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} - 16 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{256 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 145, normalized size = 0.94 \[ \frac {{\left (32 \, b^{3} x^{3} - 24 \, b^{2} x^{2} + 12 \, b x - 3\right )} e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b^{4}} - \frac {{\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} e^{\left (2 \, b x + 2 \, a\right )}}{64 \, b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{64 \, b^{4}} + \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 304, normalized size = 1.96 \[ \frac {\frac {\left (b x +a \right )^{3} \left (\sinh ^{4}\left (b x +a \right )\right )}{4}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{16}+\frac {9 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{32}-\frac {3 \left (b x +a \right )^{3}}{32}+\frac {3 \left (b x +a \right ) \left (\sinh ^{4}\left (b x +a \right )\right )}{32}-\frac {3 \cosh \left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{128}+\frac {45 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{256}+\frac {27 b x}{256}+\frac {27 a}{256}-\frac {9 \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{32}-3 a \left (\frac {\left (b x +a \right )^{2} \left (\sinh ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{8}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{16}-\frac {3 \left (b x +a \right )^{2}}{32}+\frac {\left (\sinh ^{4}\left (b x +a \right )\right )}{32}-\frac {3 \left (\cosh ^{2}\left (b x +a \right )\right )}{32}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \left (\sinh ^{4}\left (b x +a \right )\right )}{4}-\frac {\cosh \left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{16}+\frac {3 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{32}-\frac {3 b x}{32}-\frac {3 a}{32}\right )-\frac {a^{3} \left (\sinh ^{4}\left (b x +a \right )\right )}{4}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 171, normalized size = 1.10 \[ \frac {{\left (32 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 24 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 12 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{2048 \, b^{4}} - \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{64 \, b^{4}} - \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{64 \, b^{4}} + \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 126, normalized size = 0.81 \[ \frac {\frac {3\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{16}-\frac {3\,x^2\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{128}}{b^2}-\frac {\frac {x^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{8}-\frac {x^3\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{32}}{b}-\frac {\frac {3\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{16}-\frac {3\,x\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{256}}{b^3}+\frac {3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{32\,b^4}-\frac {3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{1024\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.25, size = 226, normalized size = 1.46 \[ \begin {cases} \frac {5 x^{3} \sinh ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16 b} - \frac {3 x^{3} \cosh ^{4}{\left (a + b x \right )}}{32 b} - \frac {15 x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{32 b^{2}} + \frac {9 x^{2} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{32 b^{2}} + \frac {51 x \sinh ^{4}{\left (a + b x \right )}}{256 b^{3}} + \frac {9 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{128 b^{3}} - \frac {45 x \cosh ^{4}{\left (a + b x \right )}}{256 b^{3}} - \frac {51 \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{256 b^{4}} + \frac {45 \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{3}{\relax (a )} \cosh {\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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