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 = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 2633
Rule 5320
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*}
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Mathematica [A] time = 0.02, size = 67, normalized size = 1.00 \[ -\frac {35 \cosh \left (a+b x^2\right )}{128 b}+\frac {7 \cosh \left (3 \left (a+b x^2\right )\right )}{128 b}-\frac {7 \cosh \left (5 \left (a+b x^2\right )\right )}{640 b}+\frac {\cosh \left (7 \left (a+b x^2\right )\right )}{896 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 154, normalized size = 2.30 \[ \frac {5 \, \cosh \left (b x^{2} + a\right )^{7} + 35 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{6} - 49 \, \cosh \left (b x^{2} + a\right )^{5} + 35 \, {\left (5 \, \cosh \left (b x^{2} + a\right )^{3} - 7 \, \cosh \left (b x^{2} + a\right )\right )} \sinh \left (b x^{2} + a\right )^{4} + 245 \, \cosh \left (b x^{2} + a\right )^{3} + 35 \, {\left (3 \, \cosh \left (b x^{2} + a\right )^{5} - 14 \, \cosh \left (b x^{2} + a\right )^{3} + 21 \, \cosh \left (b x^{2} + a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} - 1225 \, \cosh \left (b x^{2} + a\right )}{4480 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 108, normalized size = 1.61 \[ -\frac {{\left (1225 \, e^{\left (6 \, b x^{2} + 6 \, a\right )} - 245 \, e^{\left (4 \, b x^{2} + 4 \, a\right )} + 49 \, e^{\left (2 \, b x^{2} + 2 \, a\right )} - 5\right )} e^{\left (-7 \, b x^{2} - 7 \, a\right )} - 5 \, e^{\left (7 \, b x^{2} + 7 \, a\right )} + 49 \, e^{\left (5 \, b x^{2} + 5 \, a\right )} - 245 \, e^{\left (3 \, b x^{2} + 3 \, a\right )} + 1225 \, e^{\left (b x^{2} + a\right )}}{8960 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 52, normalized size = 0.78 \[ \frac {\left (-\frac {16}{35}+\frac {\left (\sinh ^{6}\left (b \,x^{2}+a \right )\right )}{7}-\frac {6 \left (\sinh ^{4}\left (b \,x^{2}+a \right )\right )}{35}+\frac {8 \left (\sinh ^{2}\left (b \,x^{2}+a \right )\right )}{35}\right ) \cosh \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 126, normalized size = 1.88 \[ \frac {e^{\left (7 \, b x^{2} + 7 \, a\right )}}{1792 \, b} - \frac {7 \, e^{\left (5 \, b x^{2} + 5 \, a\right )}}{1280 \, b} + \frac {7 \, e^{\left (3 \, b x^{2} + 3 \, a\right )}}{256 \, b} - \frac {35 \, e^{\left (b x^{2} + a\right )}}{256 \, b} - \frac {35 \, e^{\left (-b x^{2} - a\right )}}{256 \, b} + \frac {7 \, e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{256 \, b} - \frac {7 \, e^{\left (-5 \, b x^{2} - 5 \, a\right )}}{1280 \, b} + \frac {e^{\left (-7 \, b x^{2} - 7 \, a\right )}}{1792 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 52, normalized size = 0.78 \[ -\frac {-5\,{\mathrm {cosh}\left (b\,x^2+a\right )}^7+21\,{\mathrm {cosh}\left (b\,x^2+a\right )}^5-35\,{\mathrm {cosh}\left (b\,x^2+a\right )}^3+35\,\mathrm {cosh}\left (b\,x^2+a\right )}{70\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.69, size = 94, normalized size = 1.40 \[ \begin {cases} \frac {\sinh ^{6}{\left (a + b x^{2} \right )} \cosh {\left (a + b x^{2} \right )}}{2 b} - \frac {\sinh ^{4}{\left (a + b x^{2} \right )} \cosh ^{3}{\left (a + b x^{2} \right )}}{b} + \frac {4 \sinh ^{2}{\left (a + b x^{2} \right )} \cosh ^{5}{\left (a + b x^{2} \right )}}{5 b} - \frac {8 \cosh ^{7}{\left (a + b x^{2} \right )}}{35 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \sinh ^{7}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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