Optimal. Leaf size=60 \[ \frac {1}{6} \left (2 a^2+b^2\right ) x^3-\frac {2 a b \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 \cos \left (c+d x^3\right ) \sin \left (c+d x^3\right )}{6 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3460, 2723}
\begin {gather*} \frac {1}{6} x^3 \left (2 a^2+b^2\right )-\frac {2 a b \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 \sin \left (c+d x^3\right ) \cos \left (c+d x^3\right )}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2723
Rule 3460
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\frac {1}{3} \text {Subst}\left (\int (a+b \sin (c+d x))^2 \, dx,x,x^3\right )\\ &=\frac {1}{6} \left (2 a^2+b^2\right ) x^3-\frac {2 a b \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 \cos \left (c+d x^3\right ) \sin \left (c+d x^3\right )}{6 d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 52, normalized size = 0.87 \begin {gather*} -\frac {-2 \left (2 a^2+b^2\right ) \left (c+d x^3\right )+8 a b \cos \left (c+d x^3\right )+b^2 \sin \left (2 \left (c+d x^3\right )\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 62, normalized size = 1.03
method | result | size |
risch | \(\frac {x^{3} a^{2}}{3}+\frac {x^{3} b^{2}}{6}-\frac {2 a b \cos \left (d \,x^{3}+c \right )}{3 d}-\frac {b^{2} \sin \left (2 d \,x^{3}+2 c \right )}{12 d}\) | \(52\) |
derivativedivides | \(\frac {b^{2} \left (-\frac {\cos \left (d \,x^{3}+c \right ) \sin \left (d \,x^{3}+c \right )}{2}+\frac {d \,x^{3}}{2}+\frac {c}{2}\right )-2 a b \cos \left (d \,x^{3}+c \right )+a^{2} \left (d \,x^{3}+c \right )}{3 d}\) | \(62\) |
default | \(\frac {b^{2} \left (-\frac {\cos \left (d \,x^{3}+c \right ) \sin \left (d \,x^{3}+c \right )}{2}+\frac {d \,x^{3}}{2}+\frac {c}{2}\right )-2 a b \cos \left (d \,x^{3}+c \right )+a^{2} \left (d \,x^{3}+c \right )}{3 d}\) | \(62\) |
norman | \(\frac {\left (\frac {a^{2}}{3}+\frac {b^{2}}{6}\right ) x^{3}+\left (\frac {a^{2}}{3}+\frac {b^{2}}{6}\right ) x^{3} \left (\tan ^{4}\left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right )\right )+\left (\frac {2 a^{2}}{3}+\frac {b^{2}}{3}\right ) x^{3} \left (\tan ^{2}\left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right )\right )-\frac {4 a b}{3 d}-\frac {b^{2} \tan \left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right )}{3 d}+\frac {b^{2} \left (\tan ^{3}\left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 a b \left (\tan ^{2}\left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d \,x^{3}}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 52, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, a^{2} x^{3} + \frac {{\left (2 \, d x^{3} - \sin \left (2 \, d x^{3} + 2 \, c\right )\right )} b^{2}}{12 \, d} - \frac {2 \, a b \cos \left (d x^{3} + c\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 53, normalized size = 0.88 \begin {gather*} \frac {{\left (2 \, a^{2} + b^{2}\right )} d x^{3} - b^{2} \cos \left (d x^{3} + c\right ) \sin \left (d x^{3} + c\right ) - 4 \, a b \cos \left (d x^{3} + c\right )}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 99, normalized size = 1.65 \begin {gather*} \begin {cases} \frac {a^{2} x^{3}}{3} - \frac {2 a b \cos {\left (c + d x^{3} \right )}}{3 d} + \frac {b^{2} x^{3} \sin ^{2}{\left (c + d x^{3} \right )}}{6} + \frac {b^{2} x^{3} \cos ^{2}{\left (c + d x^{3} \right )}}{6} - \frac {b^{2} \sin {\left (c + d x^{3} \right )} \cos {\left (c + d x^{3} \right )}}{6 d} & \text {for}\: d \neq 0 \\\frac {x^{3} \left (a + b \sin {\left (c \right )}\right )^{2}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.20, size = 57, normalized size = 0.95 \begin {gather*} \frac {4 \, {\left (d x^{3} + c\right )} a^{2} + {\left (2 \, d x^{3} + 2 \, c - \sin \left (2 \, d x^{3} + 2 \, c\right )\right )} b^{2} - 8 \, a b \cos \left (d x^{3} + c\right )}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.72, size = 51, normalized size = 0.85 \begin {gather*} \frac {a^2\,x^3}{3}+\frac {b^2\,x^3}{6}-\frac {b^2\,\sin \left (2\,d\,x^3+2\,c\right )}{12\,d}-\frac {2\,a\,b\,\cos \left (d\,x^3+c\right )}{3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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