Optimal. Leaf size=19 \[ \frac {\sin ^5(x)}{5}+\frac {2 \sin ^3(x)}{3}+\sin (x) \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3190, 194} \[ \frac {\sin ^5(x)}{5}+\frac {2 \sin ^3(x)}{3}+\sin (x) \]
Antiderivative was successfully verified.
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Rule 194
Rule 3190
Rubi steps
\begin {align*} \int \cos (x) \left (1+\sin ^2(x)\right )^2 \, dx &=\operatorname {Subst}\left (\int \left (1+x^2\right )^2 \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)+\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ \frac {\sin ^5(x)}{5}+\frac {2 \sin ^3(x)}{3}+\sin (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 18, normalized size = 0.95 \[ \frac {1}{15} \, {\left (3 \, \cos \relax (x)^{4} - 16 \, \cos \relax (x)^{2} + 28\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 15, normalized size = 0.79 \[ \frac {1}{5} \, \sin \relax (x)^{5} + \frac {2}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 16, normalized size = 0.84 \[ \frac {\left (\sin ^{5}\relax (x )\right )}{5}+\frac {2 \left (\sin ^{3}\relax (x )\right )}{3}+\sin \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 15, normalized size = 0.79 \[ \frac {1}{5} \, \sin \relax (x)^{5} + \frac {2}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 15, normalized size = 0.79 \[ \frac {{\sin \relax (x)}^5}{5}+\frac {2\,{\sin \relax (x)}^3}{3}+\sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 17, normalized size = 0.89 \[ \frac {\sin ^{5}{\relax (x )}}{5} + \frac {2 \sin ^{3}{\relax (x )}}{3} + \sin {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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