Optimal. Leaf size=138 \[ -\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}-\frac{279 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}+\frac{279 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d} \]
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Rubi [A] time = 0.113442, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2664, 2754, 12, 2659, 207} \[ -\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}-\frac{279 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}+\frac{279 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 2754
Rule 12
Rule 2659
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{(3-5 \cos (c+d x))^4} \, dx &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{1}{48} \int \frac{-9-10 \cos (c+d x)}{(3-5 \cos (c+d x))^3} \, dx\\ &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}+\frac{\int \frac{154+75 \cos (c+d x)}{(3-5 \cos (c+d x))^2} \, dx}{1536}\\ &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac{\int -\frac{837}{3-5 \cos (c+d x)} \, dx}{24576}\\ &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}-\frac{279 \int \frac{1}{3-5 \cos (c+d x)} \, dx}{8192}\\ &=-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}-\frac{279 \operatorname{Subst}\left (\int \frac{1}{-2+8 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{4096 d}\\ &=-\frac{279 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}+\frac{279 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{32768 d}-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}\\ \end{align*}
Mathematica [B] time = 0.231309, size = 288, normalized size = 2.09 \[ \frac{226140 \sin (c+d x)-190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))-104625 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )+467046 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )-765855 \cos (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+104625 \cos (3 (c+d x)) \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-467046 \log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{393216 d (5 \cos (c+d x)-3)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 160, normalized size = 1.2 \begin{align*} -{\frac{125}{49152\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-3}}+{\frac{25}{8192\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-2}}-{\frac{295}{32768\,d} \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-1}}-{\frac{279}{32768\,d}\ln \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }-{\frac{125}{49152\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-3}}-{\frac{25}{8192\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-2}}-{\frac{295}{32768\,d} \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}}+{\frac{279}{32768\,d}\ln \left ( 1+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 = 1.84675, size = 239, normalized size = 1.73 \begin{align*} -\frac{\frac{20 \,{\left (\frac{447 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{1696 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{2832 \, \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} - 837 \, \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 837 \, \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{98304 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68231, size = 522, normalized size = 3.78 \begin{align*} \frac{837 \,{\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\right )} \log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) - 837 \,{\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\right )} \log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) + 40 \,{\left (4975 \, \cos \left (d x + c\right )^{2} - 4770 \, \cos \left (d x + c\right ) + 1583\right )} \sin \left (d x + c\right )}{196608 \,{\left (125 \, d \cos \left (d x + c\right )^{3} - 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) - 27 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.0992, size = 831, normalized size = 6.02 \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 = 1.15496, size = 131, normalized size = 0.95 \begin{align*} -\frac{\frac{20 \,{\left (2832 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1696 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 447 \, \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}} - 837 \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 837 \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{98304 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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