Optimal. Leaf size=67 \[ \frac{\cos ^7\left (a+b x^2\right )}{14 b}-\frac{3 \cos ^5\left (a+b x^2\right )}{10 b}+\frac{\cos ^3\left (a+b x^2\right )}{2 b}-\frac{\cos \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.0446329, 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 = {3379, 2633} \[ \frac{\cos ^7\left (a+b x^2\right )}{14 b}-\frac{3 \cos ^5\left (a+b x^2\right )}{10 b}+\frac{\cos ^3\left (a+b x^2\right )}{2 b}-\frac{\cos \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 3379
Rule 2633
Rubi steps
\begin{align*} \int x \sin ^7\left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sin ^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,\cos \left (a+b x^2\right )\right )}{2 b}\\ &=-\frac{\cos \left (a+b x^2\right )}{2 b}+\frac{\cos ^3\left (a+b x^2\right )}{2 b}-\frac{3 \cos ^5\left (a+b x^2\right )}{10 b}+\frac{\cos ^7\left (a+b x^2\right )}{14 b}\\ \end{align*}
Mathematica [A] time = 0.0426408, size = 67, normalized size = 1. \[ -\frac{35 \cos \left (a+b x^2\right )}{128 b}+\frac{7 \cos \left (3 \left (a+b x^2\right )\right )}{128 b}-\frac{7 \cos \left (5 \left (a+b x^2\right )\right )}{640 b}+\frac{\cos \left (7 \left (a+b x^2\right )\right )}{896 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 50, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( b{x}^{2}+a \right ) }{14\,b} \left ({\frac{16}{5}}+ \left ( \sin \left ( b{x}^{2}+a \right ) \right ) ^{6}+{\frac{6\, \left ( \sin \left ( b{x}^{2}+a \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \sin \left ( b{x}^{2}+a \right ) \right ) ^{2}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983536, size = 74, normalized size = 1.1 \begin{align*} \frac{5 \, \cos \left (7 \, b x^{2} + 7 \, a\right ) - 49 \, \cos \left (5 \, b x^{2} + 5 \, a\right ) + 245 \, \cos \left (3 \, b x^{2} + 3 \, a\right ) - 1225 \, \cos \left (b x^{2} + a\right )}{4480 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30148, size = 126, normalized size = 1.88 \begin{align*} \frac{5 \, \cos \left (b x^{2} + a\right )^{7} - 21 \, \cos \left (b x^{2} + a\right )^{5} + 35 \, \cos \left (b x^{2} + a\right )^{3} - 35 \, \cos \left (b x^{2} + a\right )}{70 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.6649, size = 95, normalized size = 1.42 \begin{align*} \begin{cases} - \frac{\sin ^{6}{\left (a + b x^{2} \right )} \cos{\left (a + b x^{2} \right )}}{2 b} - \frac{\sin ^{4}{\left (a + b x^{2} \right )} \cos ^{3}{\left (a + b x^{2} \right )}}{b} - \frac{4 \sin ^{2}{\left (a + b x^{2} \right )} \cos ^{5}{\left (a + b x^{2} \right )}}{5 b} - \frac{8 \cos ^{7}{\left (a + b x^{2} \right )}}{35 b} & \text{for}\: b \neq 0 \\\frac{x^{2} \sin ^{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 = 1.10784, size = 70, normalized size = 1.04 \begin{align*} \frac{5 \, \cos \left (b x^{2} + a\right )^{7} - 21 \, \cos \left (b x^{2} + a\right )^{5} + 35 \, \cos \left (b x^{2} + a\right )^{3} - 35 \, \cos \left (b x^{2} + a\right )}{70 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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