Optimal. Leaf size=67 \[ \frac{\cosh ^7\left (a+b x^2\right )}{14 b}-\frac{3 \cosh ^5\left (a+b x^2\right )}{10 b}+\frac{\cosh ^3\left (a+b x^2\right )}{2 b}-\frac{\cosh \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.0473215, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5320, 2633} \[ \frac{\cosh ^7\left (a+b x^2\right )}{14 b}-\frac{3 \cosh ^5\left (a+b x^2\right )}{10 b}+\frac{\cosh ^3\left (a+b x^2\right )}{2 b}-\frac{\cosh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 5320
Rule 2633
Rubi steps
\begin{align*} \int x \sinh ^7\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sinh ^7(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cosh \left (a+b x^2\right )\right )}{2 b}\\ &=-\frac{\cosh \left (a+b x^2\right )}{2 b}+\frac{\cosh ^3\left (a+b x^2\right )}{2 b}-\frac{3 \cosh ^5\left (a+b x^2\right )}{10 b}+\frac{\cosh ^7\left (a+b x^2\right )}{14 b}\\ \end{align*}
Mathematica [A] time = 0.0248622, size = 67, normalized size = 1. \[ -\frac{35 \cosh \left (a+b x^2\right )}{128 b}+\frac{7 \cosh \left (3 \left (a+b x^2\right )\right )}{128 b}-\frac{7 \cosh \left (5 \left (a+b x^2\right )\right )}{640 b}+\frac{\cosh \left (7 \left (a+b x^2\right )\right )}{896 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 52, normalized size = 0.8 \begin{align*}{\frac{\cosh \left ( b{x}^{2}+a \right ) }{2\,b} \left ( -{\frac{16}{35}}+{\frac{ \left ( \sinh \left ( b{x}^{2}+a \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sinh \left ( b{x}^{2}+a \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( b{x}^{2}+a \right ) \right ) ^{2}}{35}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02966, size = 170, normalized size = 2.54 \begin{align*} \frac{e^{\left (7 \, b x^{2} + 7 \, a\right )}}{1792 \, b} - \frac{7 \, e^{\left (5 \, b x^{2} + 5 \, a\right )}}{1280 \, b} + \frac{7 \, e^{\left (3 \, b x^{2} + 3 \, a\right )}}{256 \, b} - \frac{35 \, e^{\left (b x^{2} + a\right )}}{256 \, b} - \frac{35 \, e^{\left (-b x^{2} - a\right )}}{256 \, b} + \frac{7 \, e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{256 \, b} - \frac{7 \, e^{\left (-5 \, b x^{2} - 5 \, a\right )}}{1280 \, b} + \frac{e^{\left (-7 \, b x^{2} - 7 \, a\right )}}{1792 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68794, size = 398, normalized size = 5.94 \begin{align*} \frac{5 \, \cosh \left (b x^{2} + a\right )^{7} + 35 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{6} - 49 \, \cosh \left (b x^{2} + a\right )^{5} + 35 \,{\left (5 \, \cosh \left (b x^{2} + a\right )^{3} - 7 \, \cosh \left (b x^{2} + a\right )\right )} \sinh \left (b x^{2} + a\right )^{4} + 245 \, \cosh \left (b x^{2} + a\right )^{3} + 35 \,{\left (3 \, \cosh \left (b x^{2} + a\right )^{5} - 14 \, \cosh \left (b x^{2} + a\right )^{3} + 21 \, \cosh \left (b x^{2} + a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} - 1225 \, \cosh \left (b x^{2} + a\right )}{4480 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.3207, size = 94, normalized size = 1.4 \begin{align*} \begin{cases} \frac{\sinh ^{6}{\left (a + b x^{2} \right )} \cosh{\left (a + b x^{2} \right )}}{2 b} - \frac{\sinh ^{4}{\left (a + b x^{2} \right )} \cosh ^{3}{\left (a + b x^{2} \right )}}{b} + \frac{4 \sinh ^{2}{\left (a + b x^{2} \right )} \cosh ^{5}{\left (a + b x^{2} \right )}}{5 b} - \frac{8 \cosh ^{7}{\left (a + b x^{2} \right )}}{35 b} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh ^{7}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3114, size = 146, normalized size = 2.18 \begin{align*} -\frac{{\left (1225 \, e^{\left (6 \, b x^{2} + 6 \, a\right )} - 245 \, e^{\left (4 \, b x^{2} + 4 \, a\right )} + 49 \, e^{\left (2 \, b x^{2} + 2 \, a\right )} - 5\right )} e^{\left (-7 \, b x^{2} - 7 \, a\right )} - 5 \, e^{\left (7 \, b x^{2} + 7 \, a\right )} + 49 \, e^{\left (5 \, b x^{2} + 5 \, a\right )} - 245 \, e^{\left (3 \, b x^{2} + 3 \, a\right )} + 1225 \, e^{\left (b x^{2} + a\right )}}{8960 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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