Optimal. Leaf size=33 \[ \frac {\sin \left (a+b x^2\right )}{2 b}-\frac {\sin ^3\left (a+b x^2\right )}{6 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 33, 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 = {3461, 2713}
\begin {gather*} \frac {\sin \left (a+b x^2\right )}{2 b}-\frac {\sin ^3\left (a+b x^2\right )}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3461
Rubi steps
\begin {align*} \int x \cos ^3\left (a+b x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \cos ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin \left (a+b x^2\right )\right )}{2 b}\\ &=\frac {\sin \left (a+b x^2\right )}{2 b}-\frac {\sin ^3\left (a+b x^2\right )}{6 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {\sin \left (a+b x^2\right )}{2 b}-\frac {\sin ^3\left (a+b x^2\right )}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 26, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\left (2+\cos ^{2}\left (b \,x^{2}+a \right )\right ) \sin \left (b \,x^{2}+a \right )}{6 b}\) | \(26\) |
default | \(\frac {\left (2+\cos ^{2}\left (b \,x^{2}+a \right )\right ) \sin \left (b \,x^{2}+a \right )}{6 b}\) | \(26\) |
risch | \(\frac {3 \sin \left (b \,x^{2}+a \right )}{8 b}+\frac {\sin \left (3 b \,x^{2}+3 a \right )}{24 b}\) | \(31\) |
norman | \(\frac {\frac {\tan \left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )}{b}+\frac {\tan ^{5}\left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )}{b}+\frac {2 \left (\tan ^{3}\left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )\right )}{3 b}}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {b \,x^{2}}{2}\right )\right )^{3}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 27, normalized size = 0.82 \begin {gather*} \frac {\sin \left (3 \, b x^{2} + 3 \, a\right ) + 9 \, \sin \left (b x^{2} + a\right )}{24 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 25, normalized size = 0.76 \begin {gather*} \frac {{\left (\cos \left (b x^{2} + a\right )^{2} + 2\right )} \sin \left (b x^{2} + a\right )}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 44, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {\sin ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac {\sin {\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \cos ^{3}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 26, normalized size = 0.79 \begin {gather*} -\frac {\sin \left (b x^{2} + a\right )^{3} - 3 \, \sin \left (b x^{2} + a\right )}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 28, normalized size = 0.85 \begin {gather*} \frac {3\,\sin \left (b\,x^2+a\right )-{\sin \left (b\,x^2+a\right )}^3}{6\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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