Optimal. Leaf size=148 \[ -\frac {\sinh (a+b x)}{4 b^3}+\frac {\sinh (3 a+3 b x)}{216 b^3}+\frac {\sinh (5 a+5 b x)}{1000 b^3}+\frac {x \cosh (a+b x)}{4 b^2}-\frac {x \cosh (3 a+3 b x)}{72 b^2}-\frac {x \cosh (5 a+5 b x)}{200 b^2}-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {x^2 \sinh (5 a+5 b x)}{80 b} \]
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Rubi [A] time = 0.18, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5448, 3296, 2637} \[ -\frac {\sinh (a+b x)}{4 b^3}+\frac {\sinh (3 a+3 b x)}{216 b^3}+\frac {\sinh (5 a+5 b x)}{1000 b^3}+\frac {x \cosh (a+b x)}{4 b^2}-\frac {x \cosh (3 a+3 b x)}{72 b^2}-\frac {x \cosh (5 a+5 b x)}{200 b^2}-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {x^2 \sinh (5 a+5 b x)}{80 b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5448
Rubi steps
\begin {align*} \int x^2 \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {1}{8} x^2 \cosh (a+b x)+\frac {1}{16} x^2 \cosh (3 a+3 b x)+\frac {1}{16} x^2 \cosh (5 a+5 b x)\right ) \, dx\\ &=\frac {1}{16} \int x^2 \cosh (3 a+3 b x) \, dx+\frac {1}{16} \int x^2 \cosh (5 a+5 b x) \, dx-\frac {1}{8} \int x^2 \cosh (a+b x) \, dx\\ &=-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {x^2 \sinh (5 a+5 b x)}{80 b}-\frac {\int x \sinh (5 a+5 b x) \, dx}{40 b}-\frac {\int x \sinh (3 a+3 b x) \, dx}{24 b}+\frac {\int x \sinh (a+b x) \, dx}{4 b}\\ &=\frac {x \cosh (a+b x)}{4 b^2}-\frac {x \cosh (3 a+3 b x)}{72 b^2}-\frac {x \cosh (5 a+5 b x)}{200 b^2}-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {x^2 \sinh (5 a+5 b x)}{80 b}+\frac {\int \cosh (5 a+5 b x) \, dx}{200 b^2}+\frac {\int \cosh (3 a+3 b x) \, dx}{72 b^2}-\frac {\int \cosh (a+b x) \, dx}{4 b^2}\\ &=\frac {x \cosh (a+b x)}{4 b^2}-\frac {x \cosh (3 a+3 b x)}{72 b^2}-\frac {x \cosh (5 a+5 b x)}{200 b^2}-\frac {\sinh (a+b x)}{4 b^3}-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {\sinh (3 a+3 b x)}{216 b^3}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {\sinh (5 a+5 b x)}{1000 b^3}+\frac {x^2 \sinh (5 a+5 b x)}{80 b}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 105, normalized size = 0.71 \[ \frac {-6750 \left (\left (b^2 x^2+2\right ) \sinh (a+b x)-2 b x \cosh (a+b x)\right )+125 \left (\left (9 b^2 x^2+2\right ) \sinh (3 (a+b x))-6 b x \cosh (3 (a+b x))\right )+27 \left (\left (25 b^2 x^2+2\right ) \sinh (5 (a+b x))-10 b x \cosh (5 (a+b x))\right )}{54000 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 209, normalized size = 1.41 \[ -\frac {270 \, b x \cosh \left (b x + a\right )^{5} + 1350 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 27 \, {\left (25 \, b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{5} + 750 \, b x \cosh \left (b x + a\right )^{3} - 5 \, {\left (225 \, b^{2} x^{2} + 54 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 50\right )} \sinh \left (b x + a\right )^{3} - 13500 \, b x \cosh \left (b x + a\right ) + 450 \, {\left (6 \, b x \cosh \left (b x + a\right )^{3} + 5 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 15 \, {\left (9 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{4} - 450 \, b^{2} x^{2} + 25 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - 900\right )} \sinh \left (b x + a\right )}{54000 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 164, normalized size = 1.11 \[ \frac {{\left (25 \, b^{2} x^{2} - 10 \, b x + 2\right )} e^{\left (5 \, b x + 5 \, a\right )}}{4000 \, b^{3}} + \frac {{\left (9 \, b^{2} x^{2} - 6 \, b x + 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{864 \, b^{3}} - \frac {{\left (b^{2} x^{2} - 2 \, b x + 2\right )} e^{\left (b x + a\right )}}{16 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{16 \, b^{3}} - \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{3}} - \frac {{\left (25 \, b^{2} x^{2} + 10 \, b x + 2\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{4000 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 278, normalized size = 1.88 \[ \frac {\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{15}-\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{15}-\frac {2 \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{25}-\frac {856 \sinh \left (b x +a \right )}{3375}+\frac {2 \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{125}+\frac {22 \left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{3375}+\frac {4 \left (b x +a \right ) \cosh \left (b x +a \right )}{15}+\frac {2 \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{45}-2 a \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{15}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{15}-\frac {\left (\cosh ^{5}\left (b x +a \right )\right )}{25}+\frac {2 \cosh \left (b x +a \right )}{15}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{45}\right )+a^{2} \left (\frac {\sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{5}-\frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{5}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 187, normalized size = 1.26 \[ \frac {{\left (25 \, b^{2} x^{2} e^{\left (5 \, a\right )} - 10 \, b x e^{\left (5 \, a\right )} + 2 \, e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )}}{4000 \, b^{3}} + \frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{864 \, b^{3}} - \frac {{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{16 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{16 \, b^{3}} - \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{3}} - \frac {{\left (25 \, b^{2} x^{2} + 10 \, b x + 2\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{4000 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.83, size = 123, normalized size = 0.83 \[ \frac {\frac {x^2\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{48}+\frac {x^2\,\mathrm {sinh}\left (5\,a+5\,b\,x\right )}{80}-\frac {x^2\,\mathrm {sinh}\left (a+b\,x\right )}{8}}{b}-\frac {\mathrm {sinh}\left (a+b\,x\right )}{4\,b^3}-\frac {\frac {x\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{72}-\frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{4}+\frac {x\,\mathrm {cosh}\left (5\,a+5\,b\,x\right )}{200}}{b^2}+\frac {\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{216\,b^3}+\frac {\mathrm {sinh}\left (5\,a+5\,b\,x\right )}{1000\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.02, size = 182, normalized size = 1.23 \[ \begin {cases} - \frac {2 x^{2} \sinh ^{5}{\left (a + b x \right )}}{15 b} + \frac {x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} + \frac {4 x \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{15 b^{2}} - \frac {26 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{45 b^{2}} + \frac {52 x \cosh ^{5}{\left (a + b x \right )}}{225 b^{2}} - \frac {856 \sinh ^{5}{\left (a + b x \right )}}{3375 b^{3}} + \frac {338 \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{675 b^{3}} - \frac {52 \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{225 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh ^{2}{\relax (a )} \cosh ^{3}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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