Optimal. Leaf size=81 \[ -\frac {45 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)}-\frac {3 \cos (c+d x)}{32 d (3 \sin (c+d x)+5)^2}-\frac {59 \tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{1024 d}-\frac {59 x}{2048} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2664, 2754, 12, 2658} \[ -\frac {45 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)}-\frac {3 \cos (c+d x)}{32 d (3 \sin (c+d x)+5)^2}-\frac {59 \tan ^{-1}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{1024 d}-\frac {59 x}{2048} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2658
Rule 2664
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(-5-3 \sin (c+d x))^3} \, dx &=-\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}-\frac {1}{32} \int \frac {10-3 \sin (c+d x)}{(-5-3 \sin (c+d x))^2} \, dx\\ &=-\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}-\frac {45 \cos (c+d x)}{512 d (5+3 \sin (c+d x))}+\frac {1}{512} \int \frac {59}{-5-3 \sin (c+d x)} \, dx\\ &=-\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}-\frac {45 \cos (c+d x)}{512 d (5+3 \sin (c+d x))}+\frac {59}{512} \int \frac {1}{-5-3 \sin (c+d x)} \, dx\\ &=-\frac {59 x}{2048}-\frac {59 \tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{1024 d}-\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}-\frac {45 \cos (c+d x)}{512 d (5+3 \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 114, normalized size = 1.41 \[ \frac {\frac {3 (3 (60 \sin (c+d x)-15 \sin (2 (c+d x))-9 \cos (2 (c+d x))+59)-182 \cos (c+d x))}{(3 \sin (c+d x)+5)^2}-59 \tan ^{-1}\left (\frac {2 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )}{1024 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 94, normalized size = 1.16 \[ -\frac {59 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right ) - 34\right )} \arctan \left (\frac {5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right ) - 540 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 1092 \, \cos \left (d x + c\right )}{2048 \, {\left (9 \, d \cos \left (d x + c\right )^{2} - 30 \, d \sin \left (d x + c\right ) - 34 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.43, size = 121, normalized size = 1.49 \[ -\frac {1475 \, d x + 1475 \, c + \frac {24 \, {\left (1605 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3913 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3855 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2275\right )}}{{\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}^{2}} + 2950 \, \arctan \left (-\frac {3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{51200 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 184, normalized size = 2.27 \[ -\frac {963 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1280 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{2}}-\frac {11739 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6400 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{2}}-\frac {2313 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1280 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{2}}-\frac {273}{256 d \left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{2}}-\frac {59 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{1024 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 173, normalized size = 2.14 \[ -\frac {\frac {12 \, {\left (\frac {3855 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3913 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1605 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 2275\right )}}{\frac {60 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {86 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {60 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {25 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 25} + 1475 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {3}{4}\right )}{25600 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.65, size = 112, normalized size = 1.38 \[ \frac {59\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{1024\,d}-\frac {59\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {3}{4}\right )}{1024\,d}-\frac {\frac {963\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1280}+\frac {11739\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6400}+\frac {2313\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1280}+\frac {273}{256}}{d\,{\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+5\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.75, size = 921, normalized size = 11.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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