Optimal. Leaf size=169 \[ \frac {51 a^2}{32 d (1-\cos (c+d x))}+\frac {9 a^2}{64 d (\cos (c+d x)+1)}-\frac {3 a^2}{4 d (1-\cos (c+d x))^2}-\frac {a^2}{64 d (\cos (c+d x)+1)^2}+\frac {11 a^2}{48 d (1-\cos (c+d x))^3}-\frac {a^2}{32 d (1-\cos (c+d x))^4}+\frac {99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac {29 a^2 \log (\cos (c+d x)+1)}{128 d} \]
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Rubi [A] time = 0.11, antiderivative size = 169, 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} \[ \frac {51 a^2}{32 d (1-\cos (c+d x))}+\frac {9 a^2}{64 d (\cos (c+d x)+1)}-\frac {3 a^2}{4 d (1-\cos (c+d x))^2}-\frac {a^2}{64 d (\cos (c+d x)+1)^2}+\frac {11 a^2}{48 d (1-\cos (c+d x))^3}-\frac {a^2}{32 d (1-\cos (c+d x))^4}+\frac {99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac {29 a^2 \log (\cos (c+d x)+1)}{128 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \cot ^9(c+d x) (a+a \sec (c+d x))^2 \, dx &=-\frac {a^{10} \operatorname {Subst}\left (\int \frac {x^7}{(a-a x)^5 (a+a x)^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^{10} \operatorname {Subst}\left (\int \left (-\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)}-\frac {1}{32 a^8 (1+x)^3}+\frac {9}{64 a^8 (1+x)^2}-\frac {29}{128 a^8 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2}{32 d (1-\cos (c+d x))^4}+\frac {11 a^2}{48 d (1-\cos (c+d x))^3}-\frac {3 a^2}{4 d (1-\cos (c+d x))^2}+\frac {51 a^2}{32 d (1-\cos (c+d x))}-\frac {a^2}{64 d (1+\cos (c+d x))^2}+\frac {9 a^2}{64 d (1+\cos (c+d x))}+\frac {99 a^2 \log (1-\cos (c+d x))}{128 d}+\frac {29 a^2 \log (1+\cos (c+d x))}{128 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 146, normalized size = 0.86 \[ -\frac {a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (3 \csc ^8\left (\frac {1}{2} (c+d x)\right )-44 \csc ^6\left (\frac {1}{2} (c+d x)\right )+288 \csc ^4\left (\frac {1}{2} (c+d x)\right )-1224 \csc ^2\left (\frac {1}{2} (c+d x)\right )-6 \left (-\sec ^4\left (\frac {1}{2} (c+d x)\right )+18 \sec ^2\left (\frac {1}{2} (c+d x)\right )+396 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+116 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{6144 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 322, normalized size = 1.91 \[ -\frac {558 \, a^{2} \cos \left (d x + c\right )^{5} - 156 \, a^{2} \cos \left (d x + c\right )^{4} - 1268 \, a^{2} \cos \left (d x + c\right )^{3} + 676 \, a^{2} \cos \left (d x + c\right )^{2} + 686 \, a^{2} \cos \left (d x + c\right ) - 448 \, a^{2} - 87 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} + 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 297 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 2 \, a^{2} \cos \left (d x + c\right )^{5} - a^{2} \cos \left (d x + c\right )^{4} + 4 \, a^{2} \cos \left (d x + c\right )^{3} - a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{384 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} + 4 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 238, normalized size = 1.41 \[ \frac {1188 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 1536 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {96 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {6 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (3 \, a^{2} + \frac {32 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {174 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {768 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2475 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}}{1536 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 159, normalized size = 0.94 \[ -\frac {a^{2} \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a^{2}}{32 d \left (-1+\sec \left (d x +c \right )\right )^{4}}+\frac {5 a^{2}}{48 d \left (-1+\sec \left (d x +c \right )\right )^{3}}-\frac {a^{2}}{4 d \left (-1+\sec \left (d x +c \right )\right )^{2}}+\frac {21 a^{2}}{32 d \left (-1+\sec \left (d x +c \right )\right )}+\frac {99 a^{2} \ln \left (-1+\sec \left (d x +c \right )\right )}{128 d}-\frac {a^{2}}{64 d \left (1+\sec \left (d x +c \right )\right )^{2}}-\frac {7 a^{2}}{64 d \left (1+\sec \left (d x +c \right )\right )}+\frac {29 a^{2} \ln \left (1+\sec \left (d x +c \right )\right )}{128 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 165, normalized size = 0.98 \[ \frac {87 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) + 297 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (279 \, a^{2} \cos \left (d x + c\right )^{5} - 78 \, a^{2} \cos \left (d x + c\right )^{4} - 634 \, a^{2} \cos \left (d x + c\right )^{3} + 338 \, a^{2} \cos \left (d x + c\right )^{2} + 343 \, a^{2} \cos \left (d x + c\right ) - 224 \, a^{2}\right )}}{\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 149, normalized size = 0.88 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {99\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (32\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {29\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {a^2}{8}\right )}{64\,d}-\frac {a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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