Optimal. Leaf size=108 \[ -\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{385 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}+\frac{385 x}{32768} \]
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Rubi [A] time = 0.0945206, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2664, 2754, 12, 2658} \[ -\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{385 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}+\frac{385 x}{32768} \]
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))^4} \, dx &=-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{1}{48} \int \frac{15+6 \sin (c+d x)}{(-5+3 \sin (c+d x))^3} \, dx\\ &=-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}+\frac{\int \frac{186+75 \sin (c+d x)}{(-5+3 \sin (c+d x))^2} \, dx}{1536}\\ &=-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac{\int \frac{1155}{-5+3 \sin (c+d x)} \, dx}{24576}\\ &=-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac{385 \int \frac{1}{-5+3 \sin (c+d x)} \, dx}{8192}\\ &=\frac{385 x}{32768}-\frac{385 \tan ^{-1}\left (\frac{\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}-\frac{\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac{25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac{311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.108153, size = 133, normalized size = 1.23 \[ \frac{\frac{305091 \sin (c+d x)-105300 \sin (2 (c+d x))-8397 \sin (3 (c+d x))+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))-239470}{2 (3 \sin (c+d x)-5)^3}-1925 \tan ^{-1}\left (\frac{2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}\right )}{81920 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 272, normalized size = 2.5 \begin{align*}{\frac{39933}{20480\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-3}}-{\frac{672723}{102400\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4} \left ( 5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-6\,\tan \left ( 1/2\,dx+c/2 \right ) +5 \right ) ^{-3}}+{\frac{2870073}{256000\,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 ) ^{-3}}-{\frac{604899}{51200\,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 ) ^{-3}}+{\frac{145233}{20480\,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 ) ^{-3}}-{\frac{10287}{4096\,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 ) ^{-3}}+{\frac{385}{16384\,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 = 1.47087, size = 342, normalized size = 3.17 \begin{align*} -\frac{\frac{36 \,{\left (\frac{403425 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{672110 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{637794 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{373735 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{110925 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 142875\right )}}{\frac{450 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{915 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1116 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{915 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{450 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{125 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 125} - 48125 \, \arctan \left (\frac{5 \, \sin \left (d x + c\right )}{4 \,{\left (\cos \left (d x + c\right ) + 1\right )}} - \frac{3}{4}\right )}{2048000 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02043, size = 383, normalized size = 3.55 \begin{align*} -\frac{11196 \, \cos \left (d x + c\right )^{3} - 385 \,{\left (135 \, \cos \left (d x + c\right )^{2} - 9 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 28\right )} \sin \left (d x + c\right ) - 260\right )} \arctan \left (\frac{5 \, \sin \left (d x + c\right ) - 3}{4 \, \cos \left (d x + c\right )}\right ) + 42120 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 52344 \, \cos \left (d x + c\right )}{32768 \,{\left (135 \, d \cos \left (d x + c\right )^{2} - 9 \,{\left (3 \, d \cos \left (d x + c\right )^{2} - 28 \, d\right )} \sin \left (d x + c\right ) - 260 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 72.4124, size = 1690, normalized size = 15.65 \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 = 1.13794, size = 200, normalized size = 1.85 \begin{align*} \frac{48125 \, d x + 48125 \, c + \frac{72 \,{\left (110925 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 373735 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 637794 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 672110 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 403425 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 142875\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 )}^{3}} + 96250 \, \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 )}{4096000 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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