Optimal. Leaf size=146 \[ \frac {a^2}{32 d (a \cos (c+d x)+a)^4}-\frac {1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac {1}{32 d \left (a^2 \cos (c+d x)+a^2\right )}+\frac {\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac {a}{48 d (a \cos (c+d x)+a)^3}-\frac {1}{64 d (a-a \cos (c+d x))^2}-\frac {1}{32 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.22, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3872, 2836, 12, 88, 206} \[ \frac {a^2}{32 d (a \cos (c+d x)+a)^4}-\frac {1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac {1}{32 d \left (a^2 \cos (c+d x)+a^2\right )}+\frac {\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac {a}{48 d (a \cos (c+d x)+a)^3}-\frac {1}{64 d (a-a \cos (c+d x))^2}-\frac {1}{32 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 206
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cot ^2(c+d x) \csc ^3(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {a^5 \operatorname {Subst}\left (\int \frac {x^2}{a^2 (-a-x)^3 (-a+x)^5} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {x^2}{(-a-x)^3 (-a+x)^5} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {1}{8 a (a-x)^5}-\frac {1}{16 a^2 (a-x)^4}-\frac {1}{16 a^3 (a-x)^3}-\frac {1}{32 a^4 (a-x)^2}+\frac {1}{32 a^3 (a+x)^3}+\frac {1}{64 a^4 (a+x)^2}-\frac {1}{64 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {1}{64 d (a-a \cos (c+d x))^2}+\frac {a^2}{32 d (a+a \cos (c+d x))^4}-\frac {a}{48 d (a+a \cos (c+d x))^3}-\frac {1}{32 d (a+a \cos (c+d x))^2}-\frac {1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac {1}{32 d \left (a^2+a^2 \cos (c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,-a \cos (c+d x)\right )}{64 a d}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac {1}{64 d (a-a \cos (c+d x))^2}+\frac {a^2}{32 d (a+a \cos (c+d x))^4}-\frac {a}{48 d (a+a \cos (c+d x))^3}-\frac {1}{32 d (a+a \cos (c+d x))^2}-\frac {1}{64 d \left (a^2-a^2 \cos (c+d x)\right )}-\frac {1}{32 d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 152, normalized size = 1.04 \[ -\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (6 \csc ^4\left (\frac {1}{2} (c+d x)\right )+12 \csc ^2\left (\frac {1}{2} (c+d x)\right )-3 \sec ^8\left (\frac {1}{2} (c+d x)\right )+4 \sec ^6\left (\frac {1}{2} (c+d x)\right )+12 \sec ^4\left (\frac {1}{2} (c+d x)\right )+24 \sec ^2\left (\frac {1}{2} (c+d x)\right )+24 \left (\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{384 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 283, normalized size = 1.94 \[ -\frac {6 \, \cos \left (d x + c\right )^{5} + 12 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - 20 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 70 \, \cos \left (d x + c\right ) + 32}{384 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 207, normalized size = 1.42 \[ \frac {\frac {6 \, {\left (\frac {4 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {12 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac {\frac {48 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {6 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{8}}}{1536 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 144, normalized size = 0.99 \[ -\frac {1}{64 d \,a^{2} \left (-1+\cos \left (d x +c \right )\right )^{2}}+\frac {1}{64 d \,a^{2} \left (-1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{128 d \,a^{2}}+\frac {1}{32 d \,a^{2} \left (1+\cos \left (d x +c \right )\right )^{4}}-\frac {1}{48 d \,a^{2} \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {1}{32 d \,a^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {1}{32 d \,a^{2} \left (1+\cos \left (d x +c \right )\right )}+\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{128 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 167, normalized size = 1.14 \[ -\frac {\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 6 \, \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 10 \, \cos \left (d x + c\right )^{2} + 35 \, \cos \left (d x + c\right ) + 16\right )}}{a^{2} \cos \left (d x + c\right )^{6} + 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}} - \frac {3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 152, normalized size = 1.04 \[ \frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{64\,a^2\,d}-\frac {\frac {{\cos \left (c+d\,x\right )}^5}{64}+\frac {{\cos \left (c+d\,x\right )}^4}{32}-\frac {{\cos \left (c+d\,x\right )}^3}{96}-\frac {5\,{\cos \left (c+d\,x\right )}^2}{96}+\frac {35\,\cos \left (c+d\,x\right )}{192}+\frac {1}{12}}{d\,\left (a^2\,{\cos \left (c+d\,x\right )}^6+2\,a^2\,{\cos \left (c+d\,x\right )}^5-a^2\,{\cos \left (c+d\,x\right )}^4-4\,a^2\,{\cos \left (c+d\,x\right )}^3-a^2\,{\cos \left (c+d\,x\right )}^2+2\,a^2\,\cos \left (c+d\,x\right )+a^2\right )} \]
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 \[ \frac {\int \frac {\csc ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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