Optimal. Leaf size=57 \[ \frac{4 \cot (c+d x)}{3 a^2 d (\csc (c+d x)+1)}+\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a \csc (c+d x)+a)^2} \]
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Rubi [A] time = 0.066061, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3777, 3919, 3794} \[ \frac{4 \cot (c+d x)}{3 a^2 d (\csc (c+d x)+1)}+\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a \csc (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3777
Rule 3919
Rule 3794
Rubi steps
\begin{align*} \int \frac{1}{(a+a \csc (c+d x))^2} \, dx &=\frac{\cot (c+d x)}{3 d (a+a \csc (c+d x))^2}-\frac{\int \frac{-3 a+a \csc (c+d x)}{a+a \csc (c+d x)} \, dx}{3 a^2}\\ &=\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a+a \csc (c+d x))^2}-\frac{4 \int \frac{\csc (c+d x)}{a+a \csc (c+d x)} \, dx}{3 a}\\ &=\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a+a \csc (c+d x))^2}+\frac{4 \cot (c+d x)}{3 d \left (a^2+a^2 \csc (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.324959, size = 108, normalized size = 1.89 \[ \frac{3 (3 c+3 d x-4) \cos \left (\frac{1}{2} (c+d x)\right )+(-3 c-3 d x+10) \cos \left (\frac{3}{2} (c+d x)\right )+6 \sin \left (\frac{1}{2} (c+d x)\right ) ((c+d x) \cos (c+d x)+2 c+2 d x-3)}{6 a^2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 83, normalized size = 1.5 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}}-{\frac{4}{3\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}+2\,{\frac{1}{d{a}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+2\,{\frac{1}{d{a}^{2} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.47041, size = 192, normalized size = 3.37 \begin{align*} \frac{2 \,{\left (\frac{\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac{3 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac{3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.480864, size = 306, normalized size = 5.37 \begin{align*} \frac{{\left (3 \, d x - 5\right )} \cos \left (d x + c\right )^{2} - 6 \, d x -{\left (3 \, d x + 4\right )} \cos \left (d x + c\right ) -{\left (6 \, d x +{\left (3 \, d x + 5\right )} \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) + 1}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d -{\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\csc ^{2}{\left (c + d x \right )} + 2 \csc{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33023, size = 81, normalized size = 1.42 \begin{align*} \frac{\frac{3 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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