Optimal. Leaf size=58 \[ \frac {1}{4} x^2 \left (2 a^2+b^2\right )-\frac {a b \cos \left (c+d x^2\right )}{d}-\frac {b^2 \sin \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{4 d} \]
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Rubi [A] time = 0.05, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3379, 2644} \[ \frac {1}{4} x^2 \left (2 a^2+b^2\right )-\frac {a b \cos \left (c+d x^2\right )}{d}-\frac {b^2 \sin \left (c+d x^2\right ) \cos \left (c+d x^2\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 2644
Rule 3379
Rubi steps
\begin {align*} \int x \left (a+b \sin \left (c+d x^2\right )\right )^2 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b \sin (c+d x))^2 \, dx,x,x^2\right )\\ &=\frac {1}{4} \left (2 a^2+b^2\right ) x^2-\frac {a b \cos \left (c+d x^2\right )}{d}-\frac {b^2 \cos \left (c+d x^2\right ) \sin \left (c+d x^2\right )}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 52, normalized size = 0.90 \[ -\frac {-2 \left (2 a^2+b^2\right ) \left (c+d x^2\right )+8 a b \cos \left (c+d x^2\right )+b^2 \sin \left (2 \left (c+d x^2\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 53, normalized size = 0.91 \[ \frac {{\left (2 \, a^{2} + b^{2}\right )} d x^{2} - b^{2} \cos \left (d x^{2} + c\right ) \sin \left (d x^{2} + c\right ) - 4 \, a b \cos \left (d x^{2} + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.81, size = 57, normalized size = 0.98 \[ \frac {4 \, {\left (d x^{2} + c\right )} a^{2} + {\left (2 \, d x^{2} + 2 \, c - \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} b^{2} - 8 \, a b \cos \left (d x^{2} + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 62, normalized size = 1.07 \[ \frac {b^{2} \left (-\frac {\cos \left (d \,x^{2}+c \right ) \sin \left (d \,x^{2}+c \right )}{2}+\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-2 a b \cos \left (d \,x^{2}+c \right )+a^{2} \left (d \,x^{2}+c \right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 52, normalized size = 0.90 \[ \frac {1}{2} \, a^{2} x^{2} + \frac {{\left (2 \, d x^{2} - \sin \left (2 \, d x^{2} + 2 \, c\right )\right )} b^{2}}{8 \, d} - \frac {a b \cos \left (d x^{2} + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 51, normalized size = 0.88 \[ \frac {a^2\,x^2}{2}+\frac {b^2\,x^2}{4}-\frac {b^2\,\sin \left (2\,d\,x^2+2\,c\right )}{8\,d}-\frac {a\,b\,\cos \left (d\,x^2+c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 95, normalized size = 1.64 \[ \begin {cases} \frac {a^{2} x^{2}}{2} - \frac {a b \cos {\left (c + d x^{2} \right )}}{d} + \frac {b^{2} x^{2} \sin ^{2}{\left (c + d x^{2} \right )}}{4} + \frac {b^{2} x^{2} \cos ^{2}{\left (c + d x^{2} \right )}}{4} - \frac {b^{2} \sin {\left (c + d x^{2} \right )} \cos {\left (c + d x^{2} \right )}}{4 d} & \text {for}\: d \neq 0 \\\frac {x^{2} \left (a + b \sin {\relax (c )}\right )^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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