Optimal. Leaf size=63 \[ \frac{\log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]
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Rubi [A] time = 0.0232539, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2660, 616, 31} \[ \frac{\log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 2660
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{3+5 \sin (c+d x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{3+10 x+3 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+3 x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{9+3 x} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}\\ &=-\frac{\log \left (3+\tan \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}+\frac{\log \left (1+3 \tan \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0228457, size = 63, normalized size = 1. \[ \frac{\log \left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 38, normalized size = 0.6 \begin{align*}{\frac{1}{4\,d}\ln \left ( 3\,\tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }-{\frac{1}{4\,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 = 0.962064, size = 66, normalized size = 1.05 \begin{align*} \frac{\log \left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96185, size = 128, normalized size = 2.03 \begin{align*} -\frac{\log \left (4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - \log \left (-4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.728546, size = 42, normalized size = 0.67 \begin{align*} \begin{cases} \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + \frac{1}{3} \right )}}{4 d} - \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 3 \right )}}{4 d} & \text{for}\: d \neq 0 \\\frac{x}{5 \sin{\left (c \right )} + 3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18728, size = 49, normalized size = 0.78 \begin{align*} \frac{\log \left ({\left | 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \right |}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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