Optimal. Leaf size=185 \[ \frac {5}{64 a^3 d (1-\cos (c+d x))}+\frac {303}{128 a^3 d (\cos (c+d x)+1)}-\frac {1}{128 a^3 d (1-\cos (c+d x))^2}-\frac {99}{64 a^3 d (\cos (c+d x)+1)^2}+\frac {35}{48 a^3 d (\cos (c+d x)+1)^3}-\frac {13}{64 a^3 d (\cos (c+d x)+1)^4}+\frac {1}{40 a^3 d (\cos (c+d x)+1)^5}+\frac {37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac {219 \log (\cos (c+d x)+1)}{256 a^3 d} \]
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Rubi [A] time = 0.12, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3879, 88} \[ \frac {5}{64 a^3 d (1-\cos (c+d x))}+\frac {303}{128 a^3 d (\cos (c+d x)+1)}-\frac {1}{128 a^3 d (1-\cos (c+d x))^2}-\frac {99}{64 a^3 d (\cos (c+d x)+1)^2}+\frac {35}{48 a^3 d (\cos (c+d x)+1)^3}-\frac {13}{64 a^3 d (\cos (c+d x)+1)^4}+\frac {1}{40 a^3 d (\cos (c+d x)+1)^5}+\frac {37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac {219 \log (\cos (c+d x)+1)}{256 a^3 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac {a^6 \operatorname {Subst}\left (\int \frac {x^8}{(a-a x)^3 (a+a x)^6} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^6 \operatorname {Subst}\left (\int \left (-\frac {1}{64 a^9 (-1+x)^3}-\frac {5}{64 a^9 (-1+x)^2}-\frac {37}{256 a^9 (-1+x)}+\frac {1}{8 a^9 (1+x)^6}-\frac {13}{16 a^9 (1+x)^5}+\frac {35}{16 a^9 (1+x)^4}-\frac {99}{32 a^9 (1+x)^3}+\frac {303}{128 a^9 (1+x)^2}-\frac {219}{256 a^9 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {1}{128 a^3 d (1-\cos (c+d x))^2}+\frac {5}{64 a^3 d (1-\cos (c+d x))}+\frac {1}{40 a^3 d (1+\cos (c+d x))^5}-\frac {13}{64 a^3 d (1+\cos (c+d x))^4}+\frac {35}{48 a^3 d (1+\cos (c+d x))^3}-\frac {99}{64 a^3 d (1+\cos (c+d x))^2}+\frac {303}{128 a^3 d (1+\cos (c+d x))}+\frac {37 \log (1-\cos (c+d x))}{256 a^3 d}+\frac {219 \log (1+\cos (c+d x))}{256 a^3 d}\\ \end {align*}
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Mathematica [A] time = 1.24, size = 169, normalized size = 0.91 \[ \frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (1400 \cos ^4\left (\frac {1}{2} (c+d x)\right )-195 \cos ^2\left (\frac {1}{2} (c+d x)\right )+60 \cos ^8\left (\frac {1}{2} (c+d x)\right ) \left (10 \cot ^2\left (\frac {1}{2} (c+d x)\right )+303\right )-30 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (\cot ^4\left (\frac {1}{2} (c+d x)\right )+198\right )+120 \cos ^{10}\left (\frac {1}{2} (c+d x)\right ) \left (37 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+219 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+12\right )}{1920 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 317, normalized size = 1.71 \[ \frac {8790 \, \cos \left (d x + c\right )^{6} + 11010 \, \cos \left (d x + c\right )^{5} - 13880 \, \cos \left (d x + c\right )^{4} - 25560 \, \cos \left (d x + c\right )^{3} - 734 \, \cos \left (d x + c\right )^{2} + 3285 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 555 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 13878 \, \cos \left (d x + c\right ) + 5536}{3840 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.91, size = 261, normalized size = 1.41 \[ -\frac {\frac {30 \, {\left (\frac {18 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {111 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {2220 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {15360 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\frac {9780 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2790 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {740 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {135 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {12 \, a^{12} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{15}}}{15360 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.86, size = 162, normalized size = 0.88 \[ -\frac {1}{128 a^{3} d \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {5}{64 a^{3} d \left (-1+\cos \left (d x +c \right )\right )}+\frac {37 \ln \left (-1+\cos \left (d x +c \right )\right )}{256 d \,a^{3}}+\frac {1}{40 a^{3} d \left (1+\cos \left (d x +c \right )\right )^{5}}-\frac {13}{64 a^{3} d \left (1+\cos \left (d x +c \right )\right )^{4}}+\frac {35}{48 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {99}{64 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {303}{128 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {219 \ln \left (1+\cos \left (d x +c \right )\right )}{256 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 188, normalized size = 1.02 \[ \frac {\frac {2 \, {\left (4395 \, \cos \left (d x + c\right )^{6} + 5505 \, \cos \left (d x + c\right )^{5} - 6940 \, \cos \left (d x + c\right )^{4} - 12780 \, \cos \left (d x + c\right )^{3} - 367 \, \cos \left (d x + c\right )^{2} + 6939 \, \cos \left (d x + c\right ) + 2768\right )}}{a^{3} \cos \left (d x + c\right )^{7} + 3 \, a^{3} \cos \left (d x + c\right )^{6} + a^{3} \cos \left (d x + c\right )^{5} - 5 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}} + \frac {3285 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {555 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{3840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 170, normalized size = 0.92 \[ \frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{256\,a^3\,d}-\frac {93\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,a^3\,d}+\frac {37\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{768\,a^3\,d}-\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,a^3\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{1280\,a^3\,d}+\frac {37\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a^3\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^3\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {1}{4}\right )}{128\,a^3\,d} \]
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 \[ \frac {\int \frac {\cot ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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