Optimal. Leaf size=54 \[ \frac{24 e^x}{85}+\frac{1}{17} e^x \sin ^4(x)+\frac{12}{85} e^x \sin ^2(x)-\frac{4}{17} e^x \sin ^3(x) \cos (x)-\frac{24}{85} e^x \sin (x) \cos (x) \]
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Rubi [A] time = 0.0256499, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4434, 2194} \[ \frac{24 e^x}{85}+\frac{1}{17} e^x \sin ^4(x)+\frac{12}{85} e^x \sin ^2(x)-\frac{4}{17} e^x \sin ^3(x) \cos (x)-\frac{24}{85} e^x \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 4434
Rule 2194
Rubi steps
\begin{align*} \int e^x \sin ^4(x) \, dx &=-\frac{4}{17} e^x \cos (x) \sin ^3(x)+\frac{1}{17} e^x \sin ^4(x)+\frac{12}{17} \int e^x \sin ^2(x) \, dx\\ &=-\frac{24}{85} e^x \cos (x) \sin (x)+\frac{12}{85} e^x \sin ^2(x)-\frac{4}{17} e^x \cos (x) \sin ^3(x)+\frac{1}{17} e^x \sin ^4(x)+\frac{24 \int e^x \, dx}{85}\\ &=\frac{24 e^x}{85}-\frac{24}{85} e^x \cos (x) \sin (x)+\frac{12}{85} e^x \sin ^2(x)-\frac{4}{17} e^x \cos (x) \sin ^3(x)+\frac{1}{17} e^x \sin ^4(x)\\ \end{align*}
Mathematica [A] time = 0.0376441, size = 33, normalized size = 0.61 \[ \frac{1}{680} e^x (-136 \sin (2 x)+20 \sin (4 x)-68 \cos (2 x)+5 \cos (4 x)+255) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 34, normalized size = 0.6 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) -4\,\cos \left ( x \right ) \right ){{\rm e}^{x}} \left ( \sin \left ( x \right ) \right ) ^{3}}{17}}+{\frac{ \left ( 12\,\sin \left ( x \right ) -24\,\cos \left ( x \right ) \right ){{\rm e}^{x}}\sin \left ( x \right ) }{85}}+{\frac{24\,{{\rm e}^{x}}}{85}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02925, size = 50, normalized size = 0.93 \begin{align*} \frac{1}{136} \, \cos \left (4 \, x\right ) e^{x} - \frac{1}{10} \, \cos \left (2 \, x\right ) e^{x} + \frac{1}{34} \, e^{x} \sin \left (4 \, x\right ) - \frac{1}{5} \, e^{x} \sin \left (2 \, x\right ) + \frac{3}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.462996, size = 115, normalized size = 2.13 \begin{align*} \frac{4}{85} \,{\left (5 \, \cos \left (x\right )^{3} - 11 \, \cos \left (x\right )\right )} e^{x} \sin \left (x\right ) + \frac{1}{85} \,{\left (5 \, \cos \left (x\right )^{4} - 22 \, \cos \left (x\right )^{2} + 41\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.4161, size = 70, normalized size = 1.3 \begin{align*} \frac{41 e^{x} \sin ^{4}{\left (x \right )}}{85} - \frac{44 e^{x} \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{85} + \frac{12 e^{x} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{17} - \frac{24 e^{x} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{85} + \frac{24 e^{x} \cos ^{4}{\left (x \right )}}{85} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14763, size = 47, normalized size = 0.87 \begin{align*} \frac{1}{136} \,{\left (\cos \left (4 \, x\right ) + 4 \, \sin \left (4 \, x\right )\right )} e^{x} - \frac{1}{10} \,{\left (\cos \left (2 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} e^{x} + \frac{3}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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