Optimal. Leaf size=22 \[ \frac {(a+b \sin (c+d x))^3}{3 b d} \]
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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2747, 32}
\begin {gather*} \frac {(a+b \sin (c+d x))^3}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2747
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\text {Subst}\left (\int (a+x)^2 \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {(a+b \sin (c+d x))^3}{3 b d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(22)=44\).
time = 0.02, size = 46, normalized size = 2.09 \begin {gather*} \frac {a^2 \sin (c+d x)}{d}+\frac {a b \sin ^2(c+d x)}{d}+\frac {b^2 \sin ^3(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 21, normalized size = 0.95
method | result | size |
derivativedivides | \(\frac {\left (a +b \sin \left (d x +c \right )\right )^{3}}{3 b d}\) | \(21\) |
default | \(\frac {\left (a +b \sin \left (d x +c \right )\right )^{3}}{3 b d}\) | \(21\) |
risch | \(\frac {a^{2} \sin \left (d x +c \right )}{d}+\frac {b^{2} \sin \left (d x +c \right )}{4 d}-\frac {\sin \left (3 d x +3 c \right ) b^{2}}{12 d}-\frac {a b \cos \left (2 d x +2 c \right )}{2 d}\) | \(62\) |
norman | \(\frac {\frac {4 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (3 a^{2}+2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 20, normalized size = 0.91 \begin {gather*} \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{3 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (20) = 40\).
time = 0.35, size = 48, normalized size = 2.18 \begin {gather*} -\frac {3 \, a b \cos \left (d x + c\right )^{2} + {\left (b^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (15) = 30\).
time = 0.11, size = 53, normalized size = 2.41 \begin {gather*} \begin {cases} \frac {a^{2} \sin {\left (c + d x \right )}}{d} + \frac {a b \sin ^{2}{\left (c + d x \right )}}{d} + \frac {b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \cos {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.31, size = 20, normalized size = 0.91 \begin {gather*} \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{3 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 39, normalized size = 1.77 \begin {gather*} \frac {a^2\,\sin \left (c+d\,x\right )+a\,b\,{\sin \left (c+d\,x\right )}^2+\frac {b^2\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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