Optimal. Leaf size=83 \[ \frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}+\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}+\frac{59 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{1024 d}-\frac{59 x}{2048} \]
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Rubi [A] time = 0.0631747, antiderivative size = 83, normalized size of antiderivative = 1., 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} \[ \frac{45 \cos (c+d x)}{512 d (5-3 \sin (c+d x))}+\frac{3 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}+\frac{59 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{1024 d}-\frac{59 x}{2048} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 2754
Rule 12
Rule 2658
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*}
Mathematica [A] time = 0.278905, size = 113, normalized size = 1.36 \[ \frac{\frac{546 \cos (c+d x)+9 (60 \sin (c+d x)-15 \sin (2 (c+d x))+9 \cos (2 (c+d x))-59)}{(5-3 \sin (c+d x))^2}+59 \tan ^{-1}\left (\frac{2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}\right )}{1024 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 184, normalized size = 2.2 \begin{align*} -{\frac{963}{1280\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-2}}+{\frac{11739}{6400\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-2}}-{\frac{2313}{1280\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-2}}+{\frac{273}{256\,d} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-2}}-{\frac{59}{1024\,d}\arctan \left ({\frac{5}{4}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{3}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46139, size = 234, normalized size = 2.82 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00305, size = 273, normalized size = 3.29 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.9146, size = 915, normalized size = 11.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22392, size = 165, normalized size = 1.99 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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