Optimal. Leaf size=101 \[ -\frac {3 x^2}{32 b}+\frac {3 \cosh ^2(a+b x)}{32 b^3}+\frac {\cosh ^4(a+b x)}{32 b^3}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {3 x \cosh (a+b x) \sinh (a+b x)}{16 b^2}-\frac {x \cosh ^3(a+b x) \sinh (a+b x)}{8 b^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5481, 3391, 30}
\begin {gather*} \frac {\cosh ^4(a+b x)}{32 b^3}+\frac {3 \cosh ^2(a+b x)}{32 b^3}-\frac {x \sinh (a+b x) \cosh ^3(a+b x)}{8 b^2}-\frac {3 x \sinh (a+b x) \cosh (a+b x)}{16 b^2}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {3 x^2}{32 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3391
Rule 5481
Rubi steps
\begin {align*} \int x^2 \cosh ^3(a+b x) \sinh (a+b x) \, dx &=\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {\int x \cosh ^4(a+b x) \, dx}{2 b}\\ &=\frac {\cosh ^4(a+b x)}{32 b^3}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {x \cosh ^3(a+b x) \sinh (a+b x)}{8 b^2}-\frac {3 \int x \cosh ^2(a+b x) \, dx}{8 b}\\ &=\frac {3 \cosh ^2(a+b x)}{32 b^3}+\frac {\cosh ^4(a+b x)}{32 b^3}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {3 x \cosh (a+b x) \sinh (a+b x)}{16 b^2}-\frac {x \cosh ^3(a+b x) \sinh (a+b x)}{8 b^2}-\frac {3 \int x \, dx}{16 b}\\ &=-\frac {3 x^2}{32 b}+\frac {3 \cosh ^2(a+b x)}{32 b^3}+\frac {\cosh ^4(a+b x)}{32 b^3}+\frac {x^2 \cosh ^4(a+b x)}{4 b}-\frac {3 x \cosh (a+b x) \sinh (a+b x)}{16 b^2}-\frac {x \cosh ^3(a+b x) \sinh (a+b x)}{8 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 70, normalized size = 0.69 \begin {gather*} \frac {16 \left (1+2 b^2 x^2\right ) \cosh (2 (a+b x))+\left (1+8 b^2 x^2\right ) \cosh (4 (a+b x))-4 b x (8 \sinh (2 (a+b x))+\sinh (4 (a+b x)))}{256 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.
\(2(89)=178\).
time = 1.00, size = 208, normalized size = 2.06
method | result | size |
risch | \(\frac {\left (8 b^{2} x^{2}-4 b x +1\right ) {\mathrm e}^{4 b x +4 a}}{512 b^{3}}+\frac {\left (2 b^{2} x^{2}-2 b x +1\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{3}}+\frac {\left (2 b^{2} x^{2}+2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{3}}+\frac {\left (8 b^{2} x^{2}+4 b x +1\right ) {\mathrm e}^{-4 b x -4 a}}{512 b^{3}}\) | \(114\) |
default | \(\frac {\left (2 b x +2 a \right )^{2} \cosh \left (2 b x +2 a \right )-2 \left (2 b x +2 a \right ) \sinh \left (2 b x +2 a \right )+2 \cosh \left (2 b x +2 a \right )-4 a \left (\left (2 b x +2 a \right ) \cosh \left (2 b x +2 a \right )-\sinh \left (2 b x +2 a \right )\right )+4 a^{2} \cosh \left (2 b x +2 a \right )}{32 b^{3}}+\frac {\left (4 b x +4 a \right )^{2} \cosh \left (4 b x +4 a \right )-2 \left (4 b x +4 a \right ) \sinh \left (4 b x +4 a \right )+2 \cosh \left (4 b x +4 a \right )-8 a \left (\left (4 b x +4 a \right ) \cosh \left (4 b x +4 a \right )-\sinh \left (4 b x +4 a \right )\right )+16 a^{2} \cosh \left (4 b x +4 a \right )}{512 b^{3}}\) | \(208\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 127, normalized size = 1.26 \begin {gather*} \frac {{\left (8 \, b^{2} x^{2} e^{\left (4 \, a\right )} - 4 \, b x e^{\left (4 \, a\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{512 \, b^{3}} + \frac {{\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 )}}{32 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{3}} + \frac {{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 154, normalized size = 1.52 \begin {gather*} -\frac {16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - {\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4} - 16 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (16 \, b^{2} x^{2} + 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} + 8\right )} \sinh \left (b x + a\right )^{2} + 16 \, {\left (b x \cosh \left (b x + a\right )^{3} + 4 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{256 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 150, normalized size = 1.49 \begin {gather*} \begin {cases} - \frac {3 x^{2} \sinh ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16 b} + \frac {5 x^{2} \cosh ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 x \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{16 b^{2}} - \frac {5 x \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{16 b^{2}} - \frac {3 \sinh ^{4}{\left (a + b x \right )}}{64 b^{3}} + \frac {5 \cosh ^{4}{\left (a + b x \right )}}{64 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh {\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 = 1.12 \begin {gather*} \frac {{\left (8 \, b^{2} x^{2} - 4 \, b x + 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{512 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{32 \, b^{3}} + \frac {{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{32 \, b^{3}} + \frac {{\left (8 \, b^{2} x^{2} + 4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 89, normalized size = 0.88 \begin {gather*} \frac {3\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{32\,b^3}-\frac {\frac {3\,x^2}{32}-\frac {x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^4}{4}}{b}-\frac {\frac {x\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{8}+\frac {3\,x\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right )}{16}}{b^2}+\frac {{\mathrm {cosh}\left (a+b\,x\right )}^4}{32\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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