Optimal. Leaf size=79 \[ \frac {\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac {\cos \left (a+b x^2\right )}{3 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{3 b}+\frac {x^2 \sin \left (a+b x^2\right ) \cos ^2\left (a+b x^2\right )}{6 b} \]
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Rubi [A] time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3380, 3310, 3296, 2638} \[ \frac {\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac {\cos \left (a+b x^2\right )}{3 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{3 b}+\frac {x^2 \sin \left (a+b x^2\right ) \cos ^2\left (a+b x^2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3310
Rule 3380
Rubi steps
\begin {align*} \int x^3 \cos ^3\left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \cos ^3(a+b x) \, dx,x,x^2\right )\\ &=\frac {\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac {x^2 \cos ^2\left (a+b x^2\right ) \sin \left (a+b x^2\right )}{6 b}+\frac {1}{3} \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,x^2\right )\\ &=\frac {\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{3 b}+\frac {x^2 \cos ^2\left (a+b x^2\right ) \sin \left (a+b x^2\right )}{6 b}-\frac {\operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,x^2\right )}{3 b}\\ &=\frac {\cos \left (a+b x^2\right )}{3 b^2}+\frac {\cos ^3\left (a+b x^2\right )}{18 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{3 b}+\frac {x^2 \cos ^2\left (a+b x^2\right ) \sin \left (a+b x^2\right )}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 55, normalized size = 0.70 \[ \frac {3 b x^2 \left (9 \sin \left (a+b x^2\right )+\sin \left (3 \left (a+b x^2\right )\right )\right )+27 \cos \left (a+b x^2\right )+\cos \left (3 \left (a+b x^2\right )\right )}{72 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 58, normalized size = 0.73 \[ \frac {\cos \left (b x^{2} + a\right )^{3} + 3 \, {\left (b x^{2} \cos \left (b x^{2} + a\right )^{2} + 2 \, b x^{2}\right )} \sin \left (b x^{2} + a\right ) + 6 \, \cos \left (b x^{2} + a\right )}{18 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 58, normalized size = 0.73 \[ \frac {3 \, b x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, \cos \left (b x^{2} + a\right )}{72 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 0.84 \[ \frac {3 x^{2} \sin \left (b \,x^{2}+a \right )}{8 b}+\frac {3 \cos \left (b \,x^{2}+a \right )}{8 b^{2}}+\frac {x^{2} \sin \left (3 b \,x^{2}+3 a \right )}{24 b}+\frac {\cos \left (3 b \,x^{2}+3 a \right )}{72 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 58, normalized size = 0.73 \[ \frac {3 \, b x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, \cos \left (b x^{2} + a\right )}{72 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 66, normalized size = 0.84 \[ \frac {\frac {\cos \left (b\,x^2+a\right )}{3}+\frac {{\cos \left (b\,x^2+a\right )}^3}{18}+b\,\left (\frac {x^2\,\sin \left (b\,x^2+a\right )}{3}+\frac {x^2\,{\cos \left (b\,x^2+a\right )}^2\,\sin \left (b\,x^2+a\right )}{6}\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.77, size = 92, normalized size = 1.16 \[ \begin {cases} \frac {x^{2} \sin ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac {x^{2} \sin {\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{2 b} + \frac {\sin ^{2}{\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{3 b^{2}} + \frac {7 \cos ^{3}{\left (a + b x^{2} \right )}}{18 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cos ^{3}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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