Optimal. Leaf size=21 \[ \frac {1}{4} \sin ^4(x+1)-\frac {1}{6} \sin ^6(x+1) \]
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Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2564, 14} \[ \frac {1}{4} \sin ^4(x+1)-\frac {1}{6} \sin ^6(x+1) \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rubi steps
\begin {align*} \int \cos ^3(1+x) \sin ^3(1+x) \, dx &=\operatorname {Subst}\left (\int x^3 \left (1-x^2\right ) \, dx,x,\sin (1+x)\right )\\ &=\operatorname {Subst}\left (\int \left (x^3-x^5\right ) \, dx,x,\sin (1+x)\right )\\ &=\frac {1}{4} \sin ^4(1+x)-\frac {1}{6} \sin ^6(1+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.19 \[ \frac {1}{8} \left (\frac {1}{24} \cos (6 (x+1))-\frac {3}{8} \cos (2 (x+1))\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 17, normalized size = 0.81 \[ \frac {1}{6} \, \cos \left (x + 1\right )^{6} - \frac {1}{4} \, \cos \left (x + 1\right )^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 17, normalized size = 0.81 \[ -\frac {1}{6} \, \sin \left (x + 1\right )^{6} + \frac {1}{4} \, \sin \left (x + 1\right )^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 24, normalized size = 1.14 \[ -\frac {\left (\cos ^{4}\left (1+x \right )\right ) \left (\sin ^{2}\left (1+x \right )\right )}{6}-\frac {\left (\cos ^{4}\left (1+x \right )\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 17, normalized size = 0.81 \[ -\frac {1}{6} \, \sin \left (x + 1\right )^{6} + \frac {1}{4} \, \sin \left (x + 1\right )^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 18, normalized size = 0.86 \[ -\frac {{\sin \left (x+1\right )}^4\,\left (2\,{\sin \left (x+1\right )}^2-3\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.78, size = 24, normalized size = 1.14 \[ - \frac {\sin ^{2}{\left (x + 1 \right )} \cos ^{4}{\left (x + 1 \right )}}{4} - \frac {\cos ^{6}{\left (x + 1 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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