Optimal. Leaf size=82 \[ \frac {m e^{m x} \sin ^3(x)}{m^2+9}-\frac {3 e^{m x} \sin ^2(x) \cos (x)}{m^2+9}+\frac {6 m e^{m x} \sin (x)}{m^4+10 m^2+9}-\frac {6 e^{m x} \cos (x)}{m^4+10 m^2+9} \]
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Rubi [A] time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4434, 4432} \[ \frac {m e^{m x} \sin ^3(x)}{m^2+9}+\frac {6 m e^{m x} \sin (x)}{m^4+10 m^2+9}-\frac {6 e^{m x} \cos (x)}{m^4+10 m^2+9}-\frac {3 e^{m x} \sin ^2(x) \cos (x)}{m^2+9} \]
Antiderivative was successfully verified.
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Rule 4432
Rule 4434
Rubi steps
\begin {align*} \int e^{m x} \sin ^3(x) \, dx &=-\frac {3 e^{m x} \cos (x) \sin ^2(x)}{9+m^2}+\frac {e^{m x} m \sin ^3(x)}{9+m^2}+\frac {6 \int e^{m x} \sin (x) \, dx}{9+m^2}\\ &=-\frac {6 e^{m x} \cos (x)}{9+10 m^2+m^4}+\frac {6 e^{m x} m \sin (x)}{9+10 m^2+m^4}-\frac {3 e^{m x} \cos (x) \sin ^2(x)}{9+m^2}+\frac {e^{m x} m \sin ^3(x)}{9+m^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 64, normalized size = 0.78 \[ \frac {e^{m x} \left (-3 \left (m^2+9\right ) \cos (x)+3 \left (m^2+1\right ) \cos (3 x)-2 m \sin (x) \left (\left (m^2+1\right ) \cos (2 x)-m^2-13\right )\right )}{4 \left (m^4+10 m^2+9\right )} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{m x} \sin ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.12, size = 65, normalized size = 0.79 \[ \frac {{\left (m^{3} - {\left (m^{3} + m\right )} \cos \relax (x)^{2} + 7 \, m\right )} e^{\left (m x\right )} \sin \relax (x) + 3 \, {\left ({\left (m^{2} + 1\right )} \cos \relax (x)^{3} - {\left (m^{2} + 3\right )} \cos \relax (x)\right )} e^{\left (m x\right )}}{m^{4} + 10 \, m^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 63, normalized size = 0.77 \[ -\frac {1}{4} \, {\left (\frac {m \sin \left (3 \, x\right )}{m^{2} + 9} - \frac {3 \, \cos \left (3 \, x\right )}{m^{2} + 9}\right )} e^{\left (m x\right )} + \frac {3}{4} \, {\left (\frac {m \sin \relax (x)}{m^{2} + 1} - \frac {\cos \relax (x)}{m^{2} + 1}\right )} e^{\left (m x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.11, size = 66, normalized size = 0.80
method | result | size |
risch | \(\frac {i {\mathrm e}^{\left (3 i+m \right ) x}}{24 i+8 m}-\frac {3 i {\mathrm e}^{\left (i+m \right ) x}}{8 \left (i+m \right )}+\frac {3 i {\mathrm e}^{x \left (m -i\right )}}{8 \left (m -i\right )}-\frac {i {\mathrm e}^{x \left (m -3 i\right )}}{8 \left (m -3 i\right )}\) | \(66\) |
norman | \(\frac {-\frac {6 \,{\mathrm e}^{m x}}{m^{4}+10 m^{2}+9}+\frac {6 \,{\mathrm e}^{m x} \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{m^{4}+10 m^{2}+9}+\frac {12 m \,{\mathrm e}^{m x} \tan \left (\frac {x}{2}\right )}{m^{4}+10 m^{2}+9}+\frac {12 m \,{\mathrm e}^{m x} \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{m^{4}+10 m^{2}+9}-\frac {6 \left (2 m^{2}+3\right ) {\mathrm e}^{m x} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{m^{4}+10 m^{2}+9}+\frac {6 \left (2 m^{2}+3\right ) {\mathrm e}^{m x} \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{m^{4}+10 m^{2}+9}+\frac {8 m \left (m^{2}+4\right ) {\mathrm e}^{m x} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{m^{4}+10 m^{2}+9}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 73, normalized size = 0.89 \[ \frac {3 \, {\left (m^{2} + 1\right )} \cos \left (3 \, x\right ) e^{\left (m x\right )} - 3 \, {\left (m^{2} + 9\right )} \cos \relax (x) e^{\left (m x\right )} - {\left (m^{3} + m\right )} e^{\left (m x\right )} \sin \left (3 \, x\right ) + 3 \, {\left (m^{3} + 9 \, m\right )} e^{\left (m x\right )} \sin \relax (x)}{4 \, {\left (m^{4} + 10 \, m^{2} + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 47, normalized size = 0.57 \[ -\frac {{\mathrm {e}}^{m\,x}\,\left (\frac {3\,\left (\cos \relax (x)-m\,\sin \relax (x)\right )}{m^2+1}-\frac {3\,\cos \left (3\,x\right )-m\,\sin \left (3\,x\right )}{m^2+9}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.65, size = 638, normalized size = 7.78 \[ \begin {cases} \frac {x e^{- 3 i x} \sin ^{3}{\relax (x )}}{8} - \frac {3 i x e^{- 3 i x} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{8} - \frac {3 x e^{- 3 i x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{8} + \frac {i x e^{- 3 i x} \cos ^{3}{\relax (x )}}{8} + \frac {7 i e^{- 3 i x} \sin ^{3}{\relax (x )}}{24} + \frac {i e^{- 3 i x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{4} + \frac {e^{- 3 i x} \cos ^{3}{\relax (x )}}{8} & \text {for}\: m = - 3 i \\\frac {3 x e^{- i x} \sin ^{3}{\relax (x )}}{8} - \frac {3 i x e^{- i x} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{8} + \frac {3 x e^{- i x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{8} - \frac {3 i x e^{- i x} \cos ^{3}{\relax (x )}}{8} + \frac {5 i e^{- i x} \sin ^{3}{\relax (x )}}{8} + \frac {3 i e^{- i x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{4} + \frac {3 e^{- i x} \cos ^{3}{\relax (x )}}{8} & \text {for}\: m = - i \\\frac {3 x e^{i x} \sin ^{3}{\relax (x )}}{8} + \frac {3 i x e^{i x} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{8} + \frac {3 x e^{i x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{8} + \frac {3 i x e^{i x} \cos ^{3}{\relax (x )}}{8} - \frac {5 i e^{i x} \sin ^{3}{\relax (x )}}{8} - \frac {3 i e^{i x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{4} + \frac {3 e^{i x} \cos ^{3}{\relax (x )}}{8} & \text {for}\: m = i \\\frac {x e^{3 i x} \sin ^{3}{\relax (x )}}{8} + \frac {3 i x e^{3 i x} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{8} - \frac {3 x e^{3 i x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{8} - \frac {i x e^{3 i x} \cos ^{3}{\relax (x )}}{8} - \frac {7 i e^{3 i x} \sin ^{3}{\relax (x )}}{24} - \frac {i e^{3 i x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{4} + \frac {e^{3 i x} \cos ^{3}{\relax (x )}}{8} & \text {for}\: m = 3 i \\\frac {m^{3} e^{m x} \sin ^{3}{\relax (x )}}{m^{4} + 10 m^{2} + 9} - \frac {3 m^{2} e^{m x} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{m^{4} + 10 m^{2} + 9} + \frac {7 m e^{m x} \sin ^{3}{\relax (x )}}{m^{4} + 10 m^{2} + 9} + \frac {6 m e^{m x} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{m^{4} + 10 m^{2} + 9} - \frac {9 e^{m x} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{m^{4} + 10 m^{2} + 9} - \frac {6 e^{m x} \cos ^{3}{\relax (x )}}{m^{4} + 10 m^{2} + 9} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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