Optimal. Leaf size=51 \[ \frac {\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \sin \left (a+b x^2\right ) \cos \left (a+b x^2\right )}{4 b}+\frac {x^4}{8} \]
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Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3380, 3310, 30} \[ \frac {\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \sin \left (a+b x^2\right ) \cos \left (a+b x^2\right )}{4 b}+\frac {x^4}{8} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3310
Rule 3380
Rubi steps
\begin {align*} \int x^3 \cos ^2\left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \cos ^2(a+b x) \, dx,x,x^2\right )\\ &=\frac {\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b}+\frac {1}{4} \operatorname {Subst}\left (\int x \, dx,x,x^2\right )\\ &=\frac {x^4}{8}+\frac {\cos ^2\left (a+b x^2\right )}{8 b^2}+\frac {x^2 \cos \left (a+b x^2\right ) \sin \left (a+b x^2\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 40, normalized size = 0.78 \[ \frac {2 b x^2 \left (\sin \left (2 \left (a+b x^2\right )\right )+b x^2\right )+\cos \left (2 \left (a+b x^2\right )\right )}{16 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 45, normalized size = 0.88 \[ \frac {b^{2} x^{4} + 2 \, b x^{2} \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )^{2}}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 55, normalized size = 1.08 \[ \frac {2 \, b x^{2} \sin \left (2 \, b x^{2} + 2 \, a\right ) + 2 \, {\left (b x^{2} + a\right )}^{2} - 4 \, {\left (b x^{2} + a\right )} a + \cos \left (2 \, b x^{2} + 2 \, a\right )}{16 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 0.82 \[ \frac {x^{4}}{8}+\frac {x^{2} \sin \left (2 b \,x^{2}+2 a \right )}{8 b}+\frac {\cos \left (2 b \,x^{2}+2 a \right )}{16 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 42, normalized size = 0.82 \[ \frac {2 \, b^{2} x^{4} + 2 \, b x^{2} \sin \left (2 \, b x^{2} + 2 \, a\right ) + \cos \left (2 \, b x^{2} + 2 \, a\right )}{16 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 41, normalized size = 0.80 \[ \frac {\cos \left (2\,b\,x^2+2\,a\right )}{16\,b^2}+\frac {x^4}{8}+\frac {x^2\,\sin \left (2\,b\,x^2+2\,a\right )}{8\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.55, size = 78, normalized size = 1.53 \[ \begin {cases} \frac {x^{4} \sin ^{2}{\left (a + b x^{2} \right )}}{8} + \frac {x^{4} \cos ^{2}{\left (a + b x^{2} \right )}}{8} + \frac {x^{2} \sin {\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{4 b} + \frac {\cos ^{2}{\left (a + b x^{2} \right )}}{8 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cos ^{2}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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