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 = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 3794
Rule 3919
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*}
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Mathematica [A] time = 0.33, 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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fricas [B] time = 0.47, size = 124, normalized size = 2.18 \[ \frac {{\left (3 \, d x - 5\right )} \cos \left (d x + c\right )^{2} - 6 \, d x - {\left (3 \, d x + 4\right )} \cos \left (d x + c\right ) - {\left (6 \, d x + {\left (3 \, d x + 5\right )} \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) + 1}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d - {\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 60, normalized size = 1.05 \[ \frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 83, normalized size = 1.46 \[ -\frac {4}{3 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {2}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 142, normalized size = 2.49 \[ \frac {2 \, {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac {3 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 52, normalized size = 0.91 \[ \frac {x}{a^2}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {8}{3}}{a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{\csc ^{2}{\left (c + d x \right )} + 2 \csc {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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