Optimal. Leaf size=199 \[ \frac{24 \pi ^4 x^3 e^{x z+\frac{x}{2}}}{x^4+20 \pi ^2 x^2+64 \pi ^4}+\frac{x^5 e^{x z+\frac{x}{2}} \sin ^4(\pi z)}{x^2+16 \pi ^2}+\frac{12 \pi ^2 x^5 e^{x z+\frac{x}{2}} \sin ^2(\pi z)}{x^4+20 \pi ^2 x^2+64 \pi ^4}-\frac{4 \pi x^4 e^{x z+\frac{x}{2}} \sin ^3(\pi z) \cos (\pi z)}{x^2+16 \pi ^2}-\frac{24 \pi ^3 x^4 e^{x z+\frac{x}{2}} \sin (\pi z) \cos (\pi z)}{x^4+20 \pi ^2 x^2+64 \pi ^4} \]
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Rubi [A] time = 0.101416, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {12, 4434, 2194} \[ \frac{24 \pi ^4 x^3 e^{x z+\frac{x}{2}}}{x^4+20 \pi ^2 x^2+64 \pi ^4}+\frac{x^5 e^{x z+\frac{x}{2}} \sin ^4(\pi z)}{x^2+16 \pi ^2}+\frac{12 \pi ^2 x^5 e^{x z+\frac{x}{2}} \sin ^2(\pi z)}{x^4+20 \pi ^2 x^2+64 \pi ^4}-\frac{4 \pi x^4 e^{x z+\frac{x}{2}} \sin ^3(\pi z) \cos (\pi z)}{x^2+16 \pi ^2}-\frac{24 \pi ^3 x^4 e^{x z+\frac{x}{2}} \sin (\pi z) \cos (\pi z)}{x^4+20 \pi ^2 x^2+64 \pi ^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 4434
Rule 2194
Rubi steps
\begin{align*} \int e^{\frac{x}{2}+x z} x^4 \sin ^4(\pi z) \, dz &=x^4 \int e^{\frac{x}{2}+x z} \sin ^4(\pi z) \, dz\\ &=-\frac{4 e^{\frac{x}{2}+x z} \pi x^4 \cos (\pi z) \sin ^3(\pi z)}{16 \pi ^2+x^2}+\frac{e^{\frac{x}{2}+x z} x^5 \sin ^4(\pi z)}{16 \pi ^2+x^2}+\frac{\left (12 \pi ^2 x^4\right ) \int e^{\frac{x}{2}+x z} \sin ^2(\pi z) \, dz}{16 \pi ^2+x^2}\\ &=-\frac{24 e^{\frac{x}{2}+x z} \pi ^3 x^4 \cos (\pi z) \sin (\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}+\frac{12 e^{\frac{x}{2}+x z} \pi ^2 x^5 \sin ^2(\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}-\frac{4 e^{\frac{x}{2}+x z} \pi x^4 \cos (\pi z) \sin ^3(\pi z)}{16 \pi ^2+x^2}+\frac{e^{\frac{x}{2}+x z} x^5 \sin ^4(\pi z)}{16 \pi ^2+x^2}+\frac{\left (24 \pi ^4 x^4\right ) \int e^{\frac{x}{2}+x z} \, dz}{64 \pi ^4+20 \pi ^2 x^2+x^4}\\ &=\frac{24 e^{\frac{x}{2}+x z} \pi ^4 x^3}{64 \pi ^4+20 \pi ^2 x^2+x^4}-\frac{24 e^{\frac{x}{2}+x z} \pi ^3 x^4 \cos (\pi z) \sin (\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}+\frac{12 e^{\frac{x}{2}+x z} \pi ^2 x^5 \sin ^2(\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}-\frac{4 e^{\frac{x}{2}+x z} \pi x^4 \cos (\pi z) \sin ^3(\pi z)}{16 \pi ^2+x^2}+\frac{e^{\frac{x}{2}+x z} x^5 \sin ^4(\pi z)}{16 \pi ^2+x^2}\\ \end{align*}
Mathematica [A] time = 0.208029, size = 136, normalized size = 0.68 \[ \frac{x^4 e^{x \left (z+\frac{1}{2}\right )} \left (-8 \pi x^3 \sin (2 \pi z)+4 \pi x^3 \sin (4 \pi z)-4 \left (x^2+16 \pi ^2\right ) x^2 \cos (2 \pi z)+\left (x^2+4 \pi ^2\right ) x^2 \cos (4 \pi z)+3 x^4+60 \pi ^2 x^2-128 \pi ^3 x \sin (2 \pi z)+16 \pi ^3 x \sin (4 \pi z)+192 \pi ^4\right )}{8 \left (x^5+20 \pi ^2 x^3+64 \pi ^4 x\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 127, normalized size = 0.6 \begin{align*}{\frac{{x}^{4}}{8} \left ( 3\,{\frac{{{\rm e}^{x/2+xz}}}{x}}+{\frac{x\cos \left ( 4\,\pi \,z \right ) }{16\,{\pi }^{2}+{x}^{2}}{{\rm e}^{{\frac{x}{2}}+xz}}}+4\,{\frac{\pi \,{{\rm e}^{x/2+xz}}\sin \left ( 4\,\pi \,z \right ) }{16\,{\pi }^{2}+{x}^{2}}}-4\,{\frac{x{{\rm e}^{x/2+xz}}\cos \left ( 2\,\pi \,z \right ) }{4\,{\pi }^{2}+{x}^{2}}}-8\,{\frac{\pi \,{{\rm e}^{x/2+xz}}\sin \left ( 2\,\pi \,z \right ) }{4\,{\pi }^{2}+{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02902, size = 216, normalized size = 1.09 \begin{align*} \frac{{\left ({\left (4 \, \pi ^{2} x^{2} + x^{4}\right )} \cos \left (4 \, \pi z\right ) e^{\left (x z + \frac{1}{2} \, x\right )} - 4 \,{\left (16 \, \pi ^{2} x^{2} + x^{4}\right )} \cos \left (2 \, \pi z\right ) e^{\left (x z + \frac{1}{2} \, x\right )} + 4 \,{\left (4 \, \pi ^{3} x + \pi x^{3}\right )} e^{\left (x z + \frac{1}{2} \, x\right )} \sin \left (4 \, \pi z\right ) - 8 \,{\left (16 \, \pi ^{3} x + \pi x^{3}\right )} e^{\left (x z + \frac{1}{2} \, x\right )} \sin \left (2 \, \pi z\right ) + 3 \,{\left (64 \, \pi ^{4} + 20 \, \pi ^{2} x^{2} + x^{4}\right )} e^{\left (x z + \frac{1}{2} \, x\right )}\right )} x^{4}}{8 \,{\left (64 \, \pi ^{4} x + 20 \, \pi ^{2} x^{3} + x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1752, size = 342, normalized size = 1.72 \begin{align*} \frac{4 \,{\left ({\left (4 \, \pi ^{3} x^{4} + \pi x^{6}\right )} \cos \left (\pi z\right )^{3} -{\left (10 \, \pi ^{3} x^{4} + \pi x^{6}\right )} \cos \left (\pi z\right )\right )} e^{\left (x z + \frac{1}{2} \, x\right )} \sin \left (\pi z\right ) +{\left (24 \, \pi ^{4} x^{3} + 16 \, \pi ^{2} x^{5} + x^{7} +{\left (4 \, \pi ^{2} x^{5} + x^{7}\right )} \cos \left (\pi z\right )^{4} - 2 \,{\left (10 \, \pi ^{2} x^{5} + x^{7}\right )} \cos \left (\pi z\right )^{2}\right )} e^{\left (x z + \frac{1}{2} \, x\right )}}{64 \, \pi ^{4} + 20 \, \pi ^{2} x^{2} + x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09004, size = 154, normalized size = 0.77 \begin{align*} \frac{1}{8} \,{\left ({\left (\frac{x \cos \left (4 \, \pi z\right )}{16 \, \pi ^{2} + x^{2}} + \frac{4 \, \pi \sin \left (4 \, \pi z\right )}{16 \, \pi ^{2} + x^{2}}\right )} e^{\left (x z + \frac{1}{2} \, x\right )} - 4 \,{\left (\frac{x \cos \left (2 \, \pi z\right )}{4 \, \pi ^{2} + x^{2}} + \frac{2 \, \pi \sin \left (2 \, \pi z\right )}{4 \, \pi ^{2} + x^{2}}\right )} e^{\left (x z + \frac{1}{2} \, x\right )} + \frac{3 \, e^{\left (x z + \frac{1}{2} \, x\right )}}{x}\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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