Optimal. Leaf size=108 \[ \frac{311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac{385 \tan ^{-1}\left (\frac{\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac{385 x}{32768} \]
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Rubi [A] time = 0.0949558, 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} \[ \frac{311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac{385 \tan ^{-1}\left (\frac{\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac{385 x}{32768} \]
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 \cos (c+d x))^4} \, dx &=\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}-\frac{1}{48} \int \frac{15+6 \cos (c+d x)}{(-5+3 \cos (c+d x))^3} \, dx\\ &=\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac{\int \frac{186+75 \cos (c+d x)}{(-5+3 \cos (c+d x))^2} \, dx}{1536}\\ &=\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac{311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}-\frac{\int \frac{1155}{-5+3 \cos (c+d x)} \, dx}{24576}\\ &=\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac{311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}-\frac{385 \int \frac{1}{-5+3 \cos (c+d x)} \, dx}{8192}\\ &=\frac{385 x}{32768}+\frac{385 \tan ^{-1}\left (\frac{\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac{311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.025745, size = 66, normalized size = 0.61 \[ \frac{770 \tan ^{-1}\left (2 \tan \left (\frac{1}{2} (c+d x)\right )\right )-\frac{9 (4883 \sin (c+d x)-2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(3 \cos (c+d x)-5)^3}}{32768 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 116, normalized size = 1.1 \begin{align*}{\frac{369}{512\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{-3}}+{\frac{117}{256\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{-3}}+{\frac{639}{8192\,d}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{-3}}+{\frac{385}{16384\,d}\arctan \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.08888, size = 205, normalized size = 1.9 \begin{align*} \frac{\frac{18 \,{\left (\frac{71 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{416 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{656 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac{12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{64 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1} + 385 \, \arctan \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{16384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65929, size = 362, normalized size = 3.35 \begin{align*} -\frac{385 \,{\left (27 \, \cos \left (d x + c\right )^{3} - 135 \, \cos \left (d x + c\right )^{2} + 225 \, \cos \left (d x + c\right ) - 125\right )} \arctan \left (\frac{5 \, \cos \left (d x + c\right ) - 3}{4 \, \sin \left (d x + c\right )}\right ) + 36 \,{\left (311 \, \cos \left (d x + c\right )^{2} - 1170 \, \cos \left (d x + c\right ) + 1143\right )} \sin \left (d x + c\right )}{32768 \,{\left (27 \, d \cos \left (d x + c\right )^{3} - 135 \, d \cos \left (d x + c\right )^{2} + 225 \, d \cos \left (d x + c\right ) - 125 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.6756, size = 597, normalized size = 5.53 \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 = 1.15696, size = 122, normalized size = 1.13 \begin{align*} \frac{385 \, d x + 385 \, c + \frac{36 \,{\left (656 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 416 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 71 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3}} - 770 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) - 3}\right )}{32768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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