Optimal. Leaf size=67 \[ \frac {\sinh ^7\left (a+b x^2\right )}{14 b}+\frac {3 \sinh ^5\left (a+b x^2\right )}{10 b}+\frac {\sinh ^3\left (a+b x^2\right )}{2 b}+\frac {\sinh \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.04, 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 = {5321, 2633} \[ \frac {\sinh ^7\left (a+b x^2\right )}{14 b}+\frac {3 \sinh ^5\left (a+b x^2\right )}{10 b}+\frac {\sinh ^3\left (a+b x^2\right )}{2 b}+\frac {\sinh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 5321
Rubi steps
\begin {align*} \int x \cosh ^7\left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \cosh ^7(a+b x) \, dx,x,x^2\right )\\ &=\frac {i \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-i \sinh \left (a+b x^2\right )\right )}{2 b}\\ &=\frac {\sinh \left (a+b x^2\right )}{2 b}+\frac {\sinh ^3\left (a+b x^2\right )}{2 b}+\frac {3 \sinh ^5\left (a+b x^2\right )}{10 b}+\frac {\sinh ^7\left (a+b x^2\right )}{14 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 1.00 \[ \frac {\sinh ^7\left (a+b x^2\right )}{14 b}+\frac {3 \sinh ^5\left (a+b x^2\right )}{10 b}+\frac {\sinh ^3\left (a+b x^2\right )}{2 b}+\frac {\sinh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 128, normalized size = 1.91 \[ \frac {5 \, \sinh \left (b x^{2} + a\right )^{7} + 7 \, {\left (15 \, \cosh \left (b x^{2} + a\right )^{2} + 7\right )} \sinh \left (b x^{2} + a\right )^{5} + 35 \, {\left (5 \, \cosh \left (b x^{2} + a\right )^{4} + 14 \, \cosh \left (b x^{2} + a\right )^{2} + 7\right )} \sinh \left (b x^{2} + a\right )^{3} + 35 \, {\left (\cosh \left (b x^{2} + a\right )^{6} + 7 \, \cosh \left (b x^{2} + a\right )^{4} + 21 \, \cosh \left (b x^{2} + a\right )^{2} + 35\right )} \sinh \left (b x^{2} + a\right )}{4480 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, 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 = 0.17, size = 52, normalized size = 0.78 \[ \frac {\left (\frac {16}{35}+\frac {\left (\cosh ^{6}\left (b \,x^{2}+a \right )\right )}{7}+\frac {6 \left (\cosh ^{4}\left (b \,x^{2}+a \right )\right )}{35}+\frac {8 \left (\cosh ^{2}\left (b \,x^{2}+a \right )\right )}{35}\right ) \sinh \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, 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 = 0.99, size = 52, normalized size = 0.78 \[ \frac {5\,{\mathrm {sinh}\left (b\,x^2+a\right )}^7+21\,{\mathrm {sinh}\left (b\,x^2+a\right )}^5+35\,{\mathrm {sinh}\left (b\,x^2+a\right )}^3+35\,\mathrm {sinh}\left (b\,x^2+a\right )}{70\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.16, size = 94, normalized size = 1.40 \[ \begin {cases} - \frac {8 \sinh ^{7}{\left (a + b x^{2} \right )}}{35 b} + \frac {4 \sinh ^{5}{\left (a + b x^{2} \right )} \cosh ^{2}{\left (a + b x^{2} \right )}}{5 b} - \frac {\sinh ^{3}{\left (a + b x^{2} \right )} \cosh ^{4}{\left (a + b x^{2} \right )}}{b} + \frac {\sinh {\left (a + b x^{2} \right )} \cosh ^{6}{\left (a + b x^{2} \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \cosh ^{7}{\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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