Optimal. Leaf size=165 \[ \frac {9}{64 a^2 d (1-\cos (c+d x))}+\frac {51}{32 a^2 d (\cos (c+d x)+1)}-\frac {1}{64 a^2 d (1-\cos (c+d x))^2}-\frac {3}{4 a^2 d (\cos (c+d x)+1)^2}+\frac {11}{48 a^2 d (\cos (c+d x)+1)^3}-\frac {1}{32 a^2 d (\cos (c+d x)+1)^4}+\frac {29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac {99 \log (\cos (c+d x)+1)}{128 a^2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 165, 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} \[ \frac {9}{64 a^2 d (1-\cos (c+d x))}+\frac {51}{32 a^2 d (\cos (c+d x)+1)}-\frac {1}{64 a^2 d (1-\cos (c+d x))^2}-\frac {3}{4 a^2 d (\cos (c+d x)+1)^2}+\frac {11}{48 a^2 d (\cos (c+d x)+1)^3}-\frac {1}{32 a^2 d (\cos (c+d x)+1)^4}+\frac {29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac {99 \log (\cos (c+d x)+1)}{128 a^2 d} \]
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))^2} \, dx &=-\frac {a^6 \operatorname {Subst}\left (\int \frac {x^7}{(a-a x)^3 (a+a x)^5} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^6 \operatorname {Subst}\left (\int \left (-\frac {1}{32 a^8 (-1+x)^3}-\frac {9}{64 a^8 (-1+x)^2}-\frac {29}{128 a^8 (-1+x)}-\frac {1}{8 a^8 (1+x)^5}+\frac {11}{16 a^8 (1+x)^4}-\frac {3}{2 a^8 (1+x)^3}+\frac {51}{32 a^8 (1+x)^2}-\frac {99}{128 a^8 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {1}{64 a^2 d (1-\cos (c+d x))^2}+\frac {9}{64 a^2 d (1-\cos (c+d x))}-\frac {1}{32 a^2 d (1+\cos (c+d x))^4}+\frac {11}{48 a^2 d (1+\cos (c+d x))^3}-\frac {3}{4 a^2 d (1+\cos (c+d x))^2}+\frac {51}{32 a^2 d (1+\cos (c+d x))}+\frac {29 \log (1-\cos (c+d x))}{128 a^2 d}+\frac {99 \log (1+\cos (c+d x))}{128 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 154, normalized size = 0.93 \[ -\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (6 \csc ^4\left (\frac {1}{2} (c+d x)\right )-108 \csc ^2\left (\frac {1}{2} (c+d x)\right )+3 \sec ^8\left (\frac {1}{2} (c+d x)\right )-44 \sec ^6\left (\frac {1}{2} (c+d x)\right )+288 \sec ^4\left (\frac {1}{2} (c+d x)\right )-1224 \sec ^2\left (\frac {1}{2} (c+d x)\right )-24 \left (29 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+99 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{384 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 283, normalized size = 1.72 \[ \frac {558 \, \cos \left (d x + c\right )^{5} + 156 \, \cos \left (d x + c\right )^{4} - 1268 \, \cos \left (d x + c\right )^{3} - 676 \, \cos \left (d x + c\right )^{2} + 297 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 87 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 686 \, \cos \left (d x + c\right ) + 448}{384 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 236, normalized size = 1.43 \[ -\frac {\frac {6 \, {\left (\frac {16 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {87 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {348 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac {1536 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} + \frac {\frac {768 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {174 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {32 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{8}}}{1536 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.02, size = 144, normalized size = 0.87 \[ -\frac {1}{64 a^{2} d \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {9}{64 a^{2} d \left (-1+\cos \left (d x +c \right )\right )}+\frac {29 \ln \left (-1+\cos \left (d x +c \right )\right )}{128 a^{2} d}-\frac {1}{32 a^{2} d \left (1+\cos \left (d x +c \right )\right )^{4}}+\frac {11}{48 a^{2} d \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {3}{4 a^{2} d \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {51}{32 a^{2} d \left (1+\cos \left (d x +c \right )\right )}+\frac {99 \ln \left (1+\cos \left (d x +c \right )\right )}{128 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 167, normalized size = 1.01 \[ \frac {\frac {2 \, {\left (279 \, \cos \left (d x + c\right )^{5} + 78 \, \cos \left (d x + c\right )^{4} - 634 \, \cos \left (d x + c\right )^{3} - 338 \, \cos \left (d x + c\right )^{2} + 343 \, \cos \left (d x + c\right ) + 224\right )}}{a^{2} \cos \left (d x + c\right )^{6} + 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}} + \frac {297 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {87 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 151, normalized size = 0.92 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,a^2\,d}-\frac {29\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,a^2\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{48\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{512\,a^2\,d}+\frac {29\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a^2\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {1}{4}\right )}{64\,a^2\,d} \]
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 \[ \frac {\int \frac {\cot ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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