Optimal. Leaf size=126 \[ -\frac {1}{32 d \left (a^3-a^3 \cos (c+d x)\right )}-\frac {1}{16 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {\tanh ^{-1}(\cos (c+d x))}{32 a^3 d}-\frac {a}{16 d (a \cos (c+d x)+a)^4}+\frac {1}{6 d (a \cos (c+d x)+a)^3}-\frac {3}{32 a d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.13, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2707, 88, 206} \[ -\frac {1}{32 d \left (a^3-a^3 \cos (c+d x)\right )}-\frac {1}{16 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {\tanh ^{-1}(\cos (c+d x))}{32 a^3 d}-\frac {a}{16 d (a \cos (c+d x)+a)^4}+\frac {1}{6 d (a \cos (c+d x)+a)^3}-\frac {3}{32 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 206
Rule 2707
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cot ^3(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(-a-x)^2 (-a+x)^5} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{4 (a-x)^5}+\frac {1}{2 (a-x)^4}-\frac {3}{16 a (a-x)^3}-\frac {1}{16 a^2 (a-x)^2}+\frac {1}{32 a^2 (a+x)^2}-\frac {1}{32 a^2 \left (a^2-x^2\right )}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a}{16 d (a+a \cos (c+d x))^4}+\frac {1}{6 d (a+a \cos (c+d x))^3}-\frac {3}{32 a d (a+a \cos (c+d x))^2}-\frac {1}{32 d \left (a^3-a^3 \cos (c+d x)\right )}-\frac {1}{16 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 )}{32 a^2 d}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{32 a^3 d}-\frac {a}{16 d (a+a \cos (c+d x))^4}+\frac {1}{6 d (a+a \cos (c+d x))^3}-\frac {3}{32 a d (a+a \cos (c+d x))^2}-\frac {1}{32 d \left (a^3-a^3 \cos (c+d x)\right )}-\frac {1}{16 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 138, normalized size = 1.10 \[ -\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (12 \csc ^2\left (\frac {1}{2} (c+d x)\right )+3 \sec ^8\left (\frac {1}{2} (c+d x)\right )-16 \sec ^6\left (\frac {1}{2} (c+d x)\right )+18 \sec ^4\left (\frac {1}{2} (c+d x)\right )+24 \sec ^2\left (\frac {1}{2} (c+d x)\right )+24 \left (\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{96 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 240, normalized size = 1.90 \[ -\frac {6 \, \cos \left (d x + c\right )^{4} + 18 \, \cos \left (d x + c\right )^{3} - 50 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 54 \, \cos \left (d x + c\right ) - 16}{192 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 182, normalized size = 1.44 \[ \frac {\frac {12 \, {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {12 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {\frac {24 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3 \, a^{9} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{12}}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.86, size = 126, normalized size = 1.00 \[ \frac {1}{32 d \,a^{3} \left (-1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{64 d \,a^{3}}-\frac {1}{16 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{4}}+\frac {1}{6 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {3}{32 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {1}{16 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{64 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 146, normalized size = 1.16 \[ -\frac {\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 9 \, \cos \left (d x + c\right )^{3} - 25 \, \cos \left (d x + c\right )^{2} - 27 \, \cos \left (d x + c\right ) - 8\right )}}{a^{3} \cos \left (d x + c\right )^{5} + 3 \, a^{3} \cos \left (d x + c\right )^{4} + 2 \, a^{3} \cos \left (d x + c\right )^{3} - 2 \, a^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}} - \frac {3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 130, normalized size = 1.03 \[ \frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{32\,a^3\,d}-\frac {-\frac {{\cos \left (c+d\,x\right )}^4}{32}-\frac {3\,{\cos \left (c+d\,x\right )}^3}{32}+\frac {25\,{\cos \left (c+d\,x\right )}^2}{96}+\frac {9\,\cos \left (c+d\,x\right )}{32}+\frac {1}{12}}{d\,\left (-a^3\,{\cos \left (c+d\,x\right )}^5-3\,a^3\,{\cos \left (c+d\,x\right )}^4-2\,a^3\,{\cos \left (c+d\,x\right )}^3+2\,a^3\,{\cos \left (c+d\,x\right )}^2+3\,a^3\,\cos \left (c+d\,x\right )+a^3\right )} \]
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 \[ \frac {\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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