Optimal. Leaf size=42 \[ -\frac {2 a \cos (c+d x)+a}{6 d (1-\cos (c+d x)) (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.13, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2836, 12, 81} \[ -\frac {2 a \cos (c+d x)+a}{6 d (1-\cos (c+d x)) (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 81
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cot ^2(c+d x) \csc (c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {x^2}{a^2 (-a-x)^2 (-a+x)^4} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {x^2}{(-a-x)^2 (-a+x)^4} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a+2 a \cos (c+d x)}{6 d (1-\cos (c+d x)) (a+a \cos (c+d x))^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 38, normalized size = 0.90 \[ -\frac {(2 \cos (c+d x)+1) \csc ^2(c+d x)}{6 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 60, normalized size = 1.43 \[ \frac {2 \, \cos \left (d x + c\right ) + 1}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 82, normalized size = 1.95 \[ \frac {\frac {3 \, {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}} + \frac {\frac {6 \, a^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{6}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 57, normalized size = 1.36 \[ \frac {\frac {1}{-16+16 \cos \left (d x +c \right )}+\frac {1}{12 \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {1}{8 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {1}{16 \left (1+\cos \left (d x +c \right )\right )}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 59, normalized size = 1.40 \[ \frac {2 \, \cos \left (d x + c\right ) + 1}{6 \, {\left (a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right ) - a^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 58, normalized size = 1.38 \[ -\frac {\frac {\cos \left (c+d\,x\right )}{3}+\frac {1}{6}}{d\,\left (-a^2\,{\cos \left (c+d\,x\right )}^4-2\,a^2\,{\cos \left (c+d\,x\right )}^3+2\,a^2\,\cos \left (c+d\,x\right )+a^2\right )} \]
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 \[ \frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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