Optimal. Leaf size=19 \[ \sin (x)+\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5} \]
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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 = {3269, 200}
\begin {gather*} \frac {\sin ^5(x)}{5}+\frac {2 \sin ^3(x)}{3}+\sin (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 3269
Rubi steps
\begin {align*} \int \cos (x) \left (1+\sin ^2(x)\right )^2 \, dx &=\text {Subst}\left (\int \left (1+x^2\right )^2 \, dx,x,\sin (x)\right )\\ &=\text {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 \begin {gather*} \sin (x)+\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 16, normalized size = 0.84
method | result | size |
derivativedivides | \(\sin \left (x \right )+\frac {2 \left (\sin ^{3}\left (x \right )\right )}{3}+\frac {\left (\sin ^{5}\left (x \right )\right )}{5}\) | \(16\) |
default | \(\sin \left (x \right )+\frac {2 \left (\sin ^{3}\left (x \right )\right )}{3}+\frac {\left (\sin ^{5}\left (x \right )\right )}{5}\) | \(16\) |
risch | \(\frac {13 \sin \left (x \right )}{8}+\frac {\sin \left (5 x \right )}{80}-\frac {11 \sin \left (3 x \right )}{48}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.47, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{5} \, \sin \left (x\right )^{5} + \frac {2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.84, size = 18, normalized size = 0.95 \begin {gather*} \frac {1}{15} \, {\left (3 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2} + 28\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 17, normalized size = 0.89 \begin {gather*} \frac {\sin ^{5}{\left (x \right )}}{5} + \frac {2 \sin ^{3}{\left (x \right )}}{3} + \sin {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{5} \, \sin \left (x\right )^{5} + \frac {2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end {gather*}
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 \begin {gather*} \frac {{\sin \left (x\right )}^5}{5}+\frac {2\,{\sin \left (x\right )}^3}{3}+\sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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