Optimal. Leaf size=199 \[ \frac {24 e^{\frac {x}{2}+x z} \pi ^4 x^3}{64 \pi ^4+20 \pi ^2 x^2+x^4}-\frac {24 e^{\frac {x}{2}+x z} \pi ^3 x^4 \cos (\pi z) \sin (\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}+\frac {12 e^{\frac {x}{2}+x z} \pi ^2 x^5 \sin ^2(\pi z)}{64 \pi ^4+20 \pi ^2 x^2+x^4}-\frac {4 e^{\frac {x}{2}+x z} \pi x^4 \cos (\pi z) \sin ^3(\pi z)}{16 \pi ^2+x^2}+\frac {e^{\frac {x}{2}+x z} x^5 \sin ^4(\pi z)}{16 \pi ^2+x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 199, normalized size of antiderivative = 1.00, 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, 4519, 2225}
\begin {gather*} \frac {x^5 e^{x z+\frac {x}{2}} \sin ^4(\pi z)}{x^2+16 \pi ^2}-\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}+\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 {24 \pi ^4 x^3 e^{x z+\frac {x}{2}}}{x^4+20 \pi ^2 x^2+64 \pi ^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2225
Rule 4519
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 = 0.15, size = 136, normalized size = 0.68 \begin {gather*} \frac {e^{x \left (\frac {1}{2}+z\right )} x^4 \left (192 \pi ^4+60 \pi ^2 x^2+3 x^4-4 x^2 \left (16 \pi ^2+x^2\right ) \cos (2 \pi z)+x^2 \left (4 \pi ^2+x^2\right ) \cos (4 \pi z)-128 \pi ^3 x \sin (2 \pi z)-8 \pi x^3 \sin (2 \pi z)+16 \pi ^3 x \sin (4 \pi z)+4 \pi x^3 \sin (4 \pi z)\right )}{8 \left (64 \pi ^4 x+20 \pi ^2 x^3+x^5\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 164.25, size = 422, normalized size = 2.12 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x^4 \left (\frac {I}{16}-\frac {\text {Pi} z}{16}+\frac {I \text {Pi} z \text {Sin}\left [4 \text {Pi} z\right ]}{16}+\frac {\text {Pi} z \text {Cos}\left [2 \text {Pi} z\right ]^2}{8}-\frac {I \text {Cos}\left [2 \text {Pi} z\right ]}{6}-\frac {\text {Sin}\left [4 \text {Pi} z\right ]}{64}+\frac {I \text {Cos}\left [2 \text {Pi} z\right ]^2}{16}+\frac {\text {Sin}\left [2 \text {Pi} z\right ]}{12}\right ) E^{-4 I \text {Pi} z}}{\text {Pi}},x\text {==}0\text {$\vert $$\vert $}x\text {==}-4 I \text {Pi}\right \},\left \{\frac {x^4 \left (-5 I+I 6 \text {Pi} z \text {Sin}\left [2 \text {Pi} z\right ]+6 \text {Pi} z \text {Cos}\left [2 \text {Pi} z\right ]-2 \text {Sin}\left [2 \text {Pi} z\right ]-\text {Sin}\left [4 \text {Pi} z\right ]+I 5 \text {Cos}\left [2 \text {Pi} z\right ]+I \text {Cos}\left [2 \text {Pi} z\right ]^2\right ) E^{-2 I \text {Pi} z}}{24 \text {Pi}},x\text {==}-2 I \text {Pi}\right \},\left \{\frac {x^4 \left (5 I-6 I \text {Pi} z \text {Sin}\left [2 \text {Pi} z\right ]+6 \text {Pi} z \text {Cos}\left [2 \text {Pi} z\right ]-5 I \text {Cos}\left [2 \text {Pi} z\right ]-2 \text {Sin}\left [2 \text {Pi} z\right ]-I \text {Cos}\left [2 \text {Pi} z\right ]^2-\text {Sin}\left [4 \text {Pi} z\right ]\right ) E^{I 2 \text {Pi} z}}{24 \text {Pi}},x\text {==}I 2 \text {Pi}\right \},\left \{\frac {x^4 \left (-\frac {I}{16}-\frac {I \text {Pi} z \text {Sin}\left [4 \text {Pi} z\right ]}{16}-\frac {\text {Pi} z}{16}+\frac {\text {Pi} z \text {Cos}\left [2 \text {Pi} z\right ]^2}{8}-\frac {I \text {Cos}\left [2 \text {Pi} z\right ]^2}{16}-\frac {\text {Sin}\left [4 \text {Pi} z\right ]}{64}+\frac {I \text {Cos}\left [2 \text {Pi} z\right ]}{6}+\frac {\text {Sin}\left [2 \text {Pi} z\right ]}{12}\right ) E^{I 4 \text {Pi} z}}{\text {Pi}},x\text {==}I 4 \text {Pi}\right \},\left \{\frac {x^3 \left (48 \text {Pi}^4+x \left (2 x^3 \text {Sin}\left [\text {Pi} z\right ]^4-2 \text {Pi} x^2 \text {Sin}\left [2 \text {Pi} z\right ]+\text {Pi} x^2 \text {Sin}\left [4 \text {Pi} z\right ]+32 \text {Pi}^2 x \text {Sin}\left [\text {Pi} z\right ]^4-32 \text {Pi}^3 \text {Sin}\left [2 \text {Pi} z\right ]+4 \text {Pi}^3 \text {Sin}\left [4 \text {Pi} z\right ]\right )+3 \text {Pi}^2 x^2 \left (1-\text {Cos}\left [4 \text {Pi} z\right ]\right )\right ) E^{\frac {x \left (1+2 z\right )}{2}}}{2 \left (64 \text {Pi}^4+x^4+20 \text {Pi}^2 x^2\right )},\text {True}\right \}\right \}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 128, normalized size = 0.64
method | result | size |
default | \(-\frac {x^{4} \left (-\frac {3 \,{\mathrm e}^{\frac {1}{2} x +x z}}{x}-\frac {x \,{\mathrm e}^{\frac {1}{2} x +x z} \cos \left (4 \pi z \right )}{16 \pi ^{2}+x^{2}}-\frac {4 \pi \,{\mathrm e}^{\frac {1}{2} x +x z} \sin \left (4 \pi z \right )}{16 \pi ^{2}+x^{2}}+\frac {4 x \,{\mathrm e}^{\frac {1}{2} x +x z} \cos \left (2 \pi z \right )}{4 \pi ^{2}+x^{2}}+\frac {8 \pi \,{\mathrm e}^{\frac {1}{2} x +x z} \sin \left (2 \pi z \right )}{4 \pi ^{2}+x^{2}}\right )}{8}\) | \(128\) |
risch | \(\frac {3 x^{3} {\mathrm e}^{\frac {x \left (1+2 z \right )}{2}}}{8}+\frac {x^{5} {\mathrm e}^{\frac {1}{2} x +x z} \cos \left (4 \pi z \right )}{128 \pi ^{2}+8 x^{2}}+\frac {x^{4} {\mathrm e}^{\frac {1}{2} x +x z} \pi \sin \left (4 \pi z \right )}{32 \pi ^{2}+2 x^{2}}-\frac {x^{5} {\mathrm e}^{\frac {1}{2} x +x z} \cos \left (2 \pi z \right )}{2 \left (4 \pi ^{2}+x^{2}\right )}-\frac {x^{4} {\mathrm e}^{\frac {1}{2} x +x z} \pi \sin \left (2 \pi z \right )}{4 \pi ^{2}+x^{2}}\) | \(134\) |
norman | \(\frac {\frac {24 \,{\mathrm e}^{\frac {1}{2} x +x z} \pi ^{4} x^{3}}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}-\frac {48 \pi ^{3} x^{4} {\mathrm e}^{\frac {1}{2} x +x z} \tan \left (\frac {\pi z}{2}\right )}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}+\frac {48 \pi ^{3} x^{4} {\mathrm e}^{\frac {1}{2} x +x z} \left (\tan ^{7}\left (\frac {\pi z}{2}\right )\right )}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}+\frac {16 \left (9 \pi ^{4}+10 \pi ^{2} x^{2}+x^{4}\right ) x^{3} {\mathrm e}^{\frac {1}{2} x +x z} \left (\tan ^{4}\left (\frac {\pi z}{2}\right )\right )}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}+\frac {24 \,{\mathrm e}^{\frac {1}{2} x +x z} \pi ^{4} x^{3} \left (\tan ^{8}\left (\frac {\pi z}{2}\right )\right )}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}-\frac {16 \pi \,x^{4} \left (11 \pi ^{2}+2 x^{2}\right ) {\mathrm e}^{\frac {1}{2} x +x z} \left (\tan ^{3}\left (\frac {\pi z}{2}\right )\right )}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}+\frac {16 \pi \,x^{4} \left (11 \pi ^{2}+2 x^{2}\right ) {\mathrm e}^{\frac {1}{2} x +x z} \left (\tan ^{5}\left (\frac {\pi z}{2}\right )\right )}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}+\frac {48 x^{3} \left (2 \pi ^{2}+x^{2}\right ) \pi ^{2} {\mathrm e}^{\frac {1}{2} x +x z} \left (\tan ^{2}\left (\frac {\pi z}{2}\right )\right )}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}+\frac {48 x^{3} \left (2 \pi ^{2}+x^{2}\right ) \pi ^{2} {\mathrm e}^{\frac {1}{2} x +x z} \left (\tan ^{6}\left (\frac {\pi z}{2}\right )\right )}{64 \pi ^{4}+20 \pi ^{2} x^{2}+x^{4}}}{\left (1+\tan ^{2}\left (\frac {\pi z}{2}\right )\right )^{4}}\) | \(433\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 160, normalized size = 0.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.32, size = 145, normalized size = 0.73 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 135.78, size = 1277, normalized size = 6.42
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 127, normalized size = 0.64 \begin {gather*} x^{4} \left (\frac {3 \mathrm {e}^{\frac {2 x z+x}{2}}}{8 x}+\mathrm {e}^{\frac {2 x z+x}{2}} \left (-\frac {2 x \cos \left (2 \pi z\right )}{\left (2 x\right )^{2}+\left (4 \pi \right )^{2}}-\frac {4 \pi \sin \left (2 \pi z\right )}{\left (2 x\right )^{2}+\left (4 \pi \right )^{2}}\right )+\mathrm {e}^{\frac {2 x z+x}{2}} \left (\frac {8 x \cos \left (4 \pi z\right )}{\left (8 x\right )^{2}+\left (32 \pi \right )^{2}}+\frac {32 \pi \sin \left (4 \pi z\right )}{\left (8 x\right )^{2}+\left (32 \pi \right )^{2}}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.11, size = 140, normalized size = 0.70 \begin {gather*} \frac {x^3\,{\mathrm {e}}^{\frac {x}{2}+x\,z}\,\left (24\,\Pi ^4-\frac {x^4\,\cos \left (2\,\Pi \,z\right )}{2}+\frac {x^4\,\cos \left (4\,\Pi \,z\right )}{8}+\frac {3\,x^4}{8}+\frac {15\,\Pi ^2\,x^2}{2}-\Pi \,x^3\,\sin \left (2\,\Pi \,z\right )-16\,\Pi ^3\,x\,\sin \left (2\,\Pi \,z\right )+\frac {\Pi \,x^3\,\sin \left (4\,\Pi \,z\right )}{2}+2\,\Pi ^3\,x\,\sin \left (4\,\Pi \,z\right )-8\,\Pi ^2\,x^2\,\cos \left (2\,\Pi \,z\right )+\frac {\Pi ^2\,x^2\,\cos \left (4\,\Pi \,z\right )}{2}\right )}{64\,\Pi ^4+20\,\Pi ^2\,x^2+x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________