Optimal. Leaf size=79 \[ \frac {\sin ^3\left (a+b x^2\right )}{18 b^2}+\frac {\sin \left (a+b x^2\right )}{3 b^2}-\frac {x^2 \cos \left (a+b x^2\right )}{3 b}-\frac {x^2 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{6 b} \]
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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 = {3379, 3310, 3296, 2637} \[ \frac {\sin ^3\left (a+b x^2\right )}{18 b^2}+\frac {\sin \left (a+b x^2\right )}{3 b^2}-\frac {x^2 \cos \left (a+b x^2\right )}{3 b}-\frac {x^2 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{6 b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3310
Rule 3379
Rubi steps
\begin {align*} \int x^3 \sin ^3\left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \sin ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac {x^2 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac {\sin ^3\left (a+b x^2\right )}{18 b^2}+\frac {1}{3} \operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,x^2\right )\\ &=-\frac {x^2 \cos \left (a+b x^2\right )}{3 b}-\frac {x^2 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac {\sin ^3\left (a+b x^2\right )}{18 b^2}+\frac {\operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,x^2\right )}{3 b}\\ &=-\frac {x^2 \cos \left (a+b x^2\right )}{3 b}+\frac {\sin \left (a+b x^2\right )}{3 b^2}-\frac {x^2 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac {\sin ^3\left (a+b x^2\right )}{18 b^2}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 58, normalized size = 0.73 \[ -\frac {-27 \sin \left (a+b x^2\right )+\sin \left (3 \left (a+b x^2\right )\right )+27 b x^2 \cos \left (a+b x^2\right )-3 b x^2 \cos \left (3 \left (a+b x^2\right )\right )}{72 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 58, normalized size = 0.73 \[ \frac {3 \, b x^{2} \cos \left (b x^{2} + a\right )^{3} - 9 \, b x^{2} \cos \left (b x^{2} + a\right ) - {\left (\cos \left (b x^{2} + a\right )^{2} - 7\right )} \sin \left (b x^{2} + a\right )}{18 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 60, normalized size = 0.76 \[ \frac {3 \, b x^{2} \cos \left (3 \, b x^{2} + 3 \, a\right ) - 27 \, b x^{2} \cos \left (b x^{2} + a\right ) - \sin \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, \sin \left (b x^{2} + a\right )}{72 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.84 \[ -\frac {3 x^{2} \cos \left (b \,x^{2}+a \right )}{8 b}+\frac {3 \sin \left (b \,x^{2}+a \right )}{8 b^{2}}+\frac {x^{2} \cos \left (3 b \,x^{2}+3 a \right )}{24 b}-\frac {\sin \left (3 b \,x^{2}+3 a \right )}{72 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 60, normalized size = 0.76 \[ \frac {3 \, b x^{2} \cos \left (3 \, b x^{2} + 3 \, a\right ) - 27 \, b x^{2} \cos \left (b x^{2} + a\right ) - \sin \left (3 \, b x^{2} + 3 \, a\right ) + 27 \, \sin \left (b x^{2} + a\right )}{72 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.75, size = 66, normalized size = 0.84 \[ \frac {\frac {7\,\sin \left (b\,x^2+a\right )}{18}-\frac {{\cos \left (b\,x^2+a\right )}^2\,\sin \left (b\,x^2+a\right )}{18}+b\,\left (\frac {x^2\,{\cos \left (b\,x^2+a\right )}^3}{6}-\frac {x^2\,\cos \left (b\,x^2+a\right )}{2}\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.53, size = 92, normalized size = 1.16 \[ \begin {cases} - \frac {x^{2} \sin ^{2}{\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{2 b} - \frac {x^{2} \cos ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac {7 \sin ^{3}{\left (a + b x^{2} \right )}}{18 b^{2}} + \frac {\sin {\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{3 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sin ^{3}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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