Optimal. Leaf size=113 \[ \frac{45 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)}-\frac{5 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}-\frac{43 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}+\frac{43 \log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]
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Rubi [A] time = 0.0778509, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2664, 2754, 12, 2660, 616, 31} \[ \frac{45 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)}-\frac{5 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}-\frac{43 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}+\frac{43 \log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 2754
Rule 12
Rule 2660
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(3+5 \sin (c+d x))^3} \, dx &=-\frac{5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac{1}{32} \int \frac{-6+5 \sin (c+d x)}{(3+5 \sin (c+d x))^2} \, dx\\ &=-\frac{5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac{45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}+\frac{1}{512} \int \frac{43}{3+5 \sin (c+d x)} \, dx\\ &=-\frac{5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac{45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}+\frac{43}{512} \int \frac{1}{3+5 \sin (c+d x)} \, dx\\ &=-\frac{5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac{45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}+\frac{43 \operatorname{Subst}\left (\int \frac{1}{3+10 x+3 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{256 d}\\ &=-\frac{5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac{45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}+\frac{129 \operatorname{Subst}\left (\int \frac{1}{1+3 x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}-\frac{129 \operatorname{Subst}\left (\int \frac{1}{9+3 x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}\\ &=-\frac{43 \log \left (3+\tan \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}+\frac{43 \log \left (1+3 \tan \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d}-\frac{5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac{45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.465063, size = 180, normalized size = 1.59 \[ \frac{\sin \left (\frac{1}{2} (c+d x)\right ) \left (-\frac{180}{3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}-\frac{60}{\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )}\right )+\frac{40}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{40}{\left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-43 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )+43 \log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2048 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 114, normalized size = 1. \begin{align*} -{\frac{25}{1152\,d} \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}}+{\frac{155}{4608\,d} \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-1}}+{\frac{43}{2048\,d}\ln \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }+{\frac{25}{128\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +3 \right ) ^{-2}}-{\frac{15}{512\,d} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +3 \right ) ^{-1}}-{\frac{43}{2048\,d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +3 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980263, size = 263, normalized size = 2.33 \begin{align*} \frac{\frac{40 \,{\left (\frac{735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{649 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{75 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 99\right )}}{\frac{60 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{118 \, \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{9 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 9} + 387 \, \log \left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - 387 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}{18432 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76575, size = 390, normalized size = 3.45 \begin{align*} -\frac{43 \,{\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right ) - 34\right )} \log \left (4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 43 \,{\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right ) - 34\right )} \log \left (-4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) + 1800 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 440 \, \cos \left (d x + c\right )}{4096 \,{\left (25 \, 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 = 5.98672, size = 1227, normalized size = 10.86 \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 = 1.1981, size = 144, normalized size = 1.27 \begin{align*} -\frac{\frac{40 \,{\left (75 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 649 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 735 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 99\right )}}{{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3\right )}^{2}} - 387 \, \log \left ({\left | 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 387 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \right |}\right )}{18432 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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