Optimal. Leaf size=148 \[ -\frac {\cosh (a+b x)}{4 b^3}-\frac {\cosh (3 a+3 b x)}{216 b^3}+\frac {\cosh (5 a+5 b x)}{1000 b^3}+\frac {x \sinh (a+b x)}{4 b^2}+\frac {x \sinh (3 a+3 b x)}{72 b^2}-\frac {x \sinh (5 a+5 b x)}{200 b^2}-\frac {x^2 \cosh (a+b x)}{8 b}-\frac {x^2 \cosh (3 a+3 b x)}{48 b}+\frac {x^2 \cosh (5 a+5 b x)}{80 b} \]
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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, 2638} \[ \frac {x \sinh (a+b x)}{4 b^2}+\frac {x \sinh (3 a+3 b x)}{72 b^2}-\frac {x \sinh (5 a+5 b x)}{200 b^2}-\frac {\cosh (a+b x)}{4 b^3}-\frac {\cosh (3 a+3 b x)}{216 b^3}+\frac {\cosh (5 a+5 b x)}{1000 b^3}-\frac {x^2 \cosh (a+b x)}{8 b}-\frac {x^2 \cosh (3 a+3 b x)}{48 b}+\frac {x^2 \cosh (5 a+5 b x)}{80 b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 5448
Rubi steps
\begin {align*} \int x^2 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {1}{8} x^2 \sinh (a+b x)-\frac {1}{16} x^2 \sinh (3 a+3 b x)+\frac {1}{16} x^2 \sinh (5 a+5 b x)\right ) \, dx\\ &=-\left (\frac {1}{16} \int x^2 \sinh (3 a+3 b x) \, dx\right )+\frac {1}{16} \int x^2 \sinh (5 a+5 b x) \, dx-\frac {1}{8} \int x^2 \sinh (a+b x) \, dx\\ &=-\frac {x^2 \cosh (a+b x)}{8 b}-\frac {x^2 \cosh (3 a+3 b x)}{48 b}+\frac {x^2 \cosh (5 a+5 b x)}{80 b}-\frac {\int x \cosh (5 a+5 b x) \, dx}{40 b}+\frac {\int x \cosh (3 a+3 b x) \, dx}{24 b}+\frac {\int x \cosh (a+b x) \, dx}{4 b}\\ &=-\frac {x^2 \cosh (a+b x)}{8 b}-\frac {x^2 \cosh (3 a+3 b x)}{48 b}+\frac {x^2 \cosh (5 a+5 b x)}{80 b}+\frac {x \sinh (a+b x)}{4 b^2}+\frac {x \sinh (3 a+3 b x)}{72 b^2}-\frac {x \sinh (5 a+5 b x)}{200 b^2}+\frac {\int \sinh (5 a+5 b x) \, dx}{200 b^2}-\frac {\int \sinh (3 a+3 b x) \, dx}{72 b^2}-\frac {\int \sinh (a+b x) \, dx}{4 b^2}\\ &=-\frac {\cosh (a+b x)}{4 b^3}-\frac {x^2 \cosh (a+b x)}{8 b}-\frac {\cosh (3 a+3 b x)}{216 b^3}-\frac {x^2 \cosh (3 a+3 b x)}{48 b}+\frac {\cosh (5 a+5 b x)}{1000 b^3}+\frac {x^2 \cosh (5 a+5 b x)}{80 b}+\frac {x \sinh (a+b x)}{4 b^2}+\frac {x \sinh (3 a+3 b x)}{72 b^2}-\frac {x \sinh (5 a+5 b x)}{200 b^2}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 98, normalized size = 0.66 \[ \frac {-6750 \left (b^2 x^2+2\right ) \cosh (a+b x)-125 \left (9 b^2 x^2+2\right ) \cosh (3 (a+b x))+27 \left (25 b^2 x^2+2\right ) \cosh (5 (a+b x))+30 b x (450 \sinh (a+b x)+25 \sinh (3 (a+b x))-9 \sinh (5 (a+b x)))}{54000 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 214, normalized size = 1.45 \[ -\frac {270 \, b x \sinh \left (b x + a\right )^{5} - 27 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{5} - 135 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 125 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} + 150 \, {\left (18 \, b x \cosh \left (b x + a\right )^{2} - 5 \, b x\right )} \sinh \left (b x + a\right )^{3} - 15 \, {\left (18 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - 25 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 6750 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) + 450 \, {\left (3 \, b x \cosh \left (b x + a\right )^{4} - 5 \, b x \cosh \left (b x + a\right )^{2} - 30 \, b x\right )} \sinh \left (b x + a\right )}{54000 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, 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 [A] time = 0.30, size = 246, normalized size = 1.66 \[ \frac {\frac {\left (b x +a \right )^{2} \left (\sinh ^{2}\left (b x +a \right )\right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{5}-\frac {2 \left (b x +a \right )^{2} \left (\cosh ^{3}\left (b x +a \right )\right )}{15}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{25}+\frac {52 \left (b x +a \right ) \sinh \left (b x +a \right )}{225}+\frac {26 \left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{225}+\frac {2 \left (\cosh ^{5}\left (b x +a \right )\right )}{125}-\frac {52 \cosh \left (b x +a \right )}{225}-\frac {26 \left (\cosh ^{3}\left (b x +a \right )\right )}{675}-2 a \left (\frac {\left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{5}-\frac {2 \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{15}-\frac {\sinh \left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{25}+\frac {26 \sinh \left (b x +a \right )}{225}+\frac {13 \left (\cosh ^{2}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{225}\right )+a^{2} \left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{5}-\frac {2 \left (\cosh ^{3}\left (b x +a \right )\right )}{15}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, 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 = 0.23, size = 112, normalized size = 0.76 \[ -\frac {780\,\mathrm {cosh}\left (a+b\,x\right )+130\,{\mathrm {cosh}\left (a+b\,x\right )}^3-54\,{\mathrm {cosh}\left (a+b\,x\right )}^5-780\,b\,x\,\mathrm {sinh}\left (a+b\,x\right )+1125\,b^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3-675\,b^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^5-390\,b\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )+270\,b\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,\mathrm {sinh}\left (a+b\,x\right )}{3375\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.06, size = 182, normalized size = 1.23 \[ \begin {cases} \frac {x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 x^{2} \cosh ^{5}{\left (a + b x \right )}}{15 b} + \frac {52 x \sinh ^{5}{\left (a + b x \right )}}{225 b^{2}} - \frac {26 x \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{45 b^{2}} + \frac {4 x \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{15 b^{2}} - \frac {52 \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{225 b^{3}} + \frac {338 \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{675 b^{3}} - \frac {856 \cosh ^{5}{\left (a + b x \right )}}{3375 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh ^{3}{\relax (a )} \cosh ^{2}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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