Optimal. Leaf size=82 \[ -\frac{7}{8 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{5}{8 a d (a \cos (c+d x)+a)^2}-\frac{1}{6 d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.151134, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3872, 2836, 12, 88, 206} \[ -\frac{7}{8 d \left (a^3 \cos (c+d x)+a^3\right )}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{5}{8 a d (a \cos (c+d x)+a)^2}-\frac{1}{6 d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2836
Rule 12
Rule 88
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc (c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac{\cos ^2(c+d x) \cot (c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{x^3}{a^3 (-a-x) (-a+x)^4} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(-a-x) (-a+x)^4} \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^2}{2 (a-x)^4}+\frac{5 a}{4 (a-x)^3}-\frac{7}{8 (a-x)^2}+\frac{1}{8 \left (a^2-x^2\right )}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac{1}{6 d (a+a \cos (c+d x))^3}+\frac{5}{8 a d (a+a \cos (c+d x))^2}-\frac{7}{8 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,-a \cos (c+d x)\right )}{8 a^2 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac{1}{6 d (a+a \cos (c+d x))^3}+\frac{5}{8 a d (a+a \cos (c+d x))^2}-\frac{7}{8 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.325845, size = 97, normalized size = 1.18 \[ -\frac{\sec ^3(c+d x) \left (42 \cos ^4\left (\frac{1}{2} (c+d x)\right )-15 \cos ^2\left (\frac{1}{2} (c+d x)\right )+12 \cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )+2\right )}{12 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 90, normalized size = 1.1 \begin{align*} -{\frac{1}{6\,d{a}^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{3}}}+{\frac{5}{8\,d{a}^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}-{\frac{7}{8\,d{a}^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{16\,d{a}^{3}}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{16\,d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00691, size = 132, normalized size = 1.61 \begin{align*} -\frac{\frac{2 \,{\left (21 \, \cos \left (d x + c\right )^{2} + 27 \, \cos \left (d x + c\right ) + 10\right )}}{a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}} + \frac{3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} - \frac{3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75499, size = 416, normalized size = 5.07 \begin{align*} -\frac{42 \, \cos \left (d x + c\right )^{2} + 3 \,{\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 3 \,{\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 54 \, \cos \left (d x + c\right ) + 20}{48 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\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{\csc{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34449, size = 153, normalized size = 1.87 \begin{align*} \frac{\frac{6 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac{\frac{18 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{9 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{2 \, a^{6}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{9}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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