Optimal. Leaf size=145 \[ \frac {1}{4 a d (1-\cos (c+d x))}+\frac {15}{16 a d (\cos (c+d x)+1)}-\frac {1}{32 a d (1-\cos (c+d x))^2}-\frac {9}{32 a d (\cos (c+d x)+1)^2}+\frac {1}{24 a d (\cos (c+d x)+1)^3}+\frac {11 \log (1-\cos (c+d x))}{32 a d}+\frac {21 \log (\cos (c+d x)+1)}{32 a d} \]
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Rubi [A] time = 0.10, antiderivative size = 145, 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} \[ \frac {1}{4 a d (1-\cos (c+d x))}+\frac {15}{16 a d (\cos (c+d x)+1)}-\frac {1}{32 a d (1-\cos (c+d x))^2}-\frac {9}{32 a d (\cos (c+d x)+1)^2}+\frac {1}{24 a d (\cos (c+d x)+1)^3}+\frac {11 \log (1-\cos (c+d x))}{32 a d}+\frac {21 \log (\cos (c+d x)+1)}{32 a d} \]
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)} \, dx &=-\frac {a^6 \operatorname {Subst}\left (\int \frac {x^6}{(a-a x)^3 (a+a x)^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^6 \operatorname {Subst}\left (\int \left (-\frac {1}{16 a^7 (-1+x)^3}-\frac {1}{4 a^7 (-1+x)^2}-\frac {11}{32 a^7 (-1+x)}+\frac {1}{8 a^7 (1+x)^4}-\frac {9}{16 a^7 (1+x)^3}+\frac {15}{16 a^7 (1+x)^2}-\frac {21}{32 a^7 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {1}{32 a d (1-\cos (c+d x))^2}+\frac {1}{4 a d (1-\cos (c+d x))}+\frac {1}{24 a d (1+\cos (c+d x))^3}-\frac {9}{32 a d (1+\cos (c+d x))^2}+\frac {15}{16 a d (1+\cos (c+d x))}+\frac {11 \log (1-\cos (c+d x))}{32 a d}+\frac {21 \log (1+\cos (c+d x))}{32 a d}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 135, normalized size = 0.93 \[ -\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (3 \csc ^4\left (\frac {1}{2} (c+d x)\right )-48 \csc ^2\left (\frac {1}{2} (c+d x)\right )-2 \sec ^6\left (\frac {1}{2} (c+d x)\right )+27 \sec ^4\left (\frac {1}{2} (c+d x)\right )-180 \sec ^2\left (\frac {1}{2} (c+d x)\right )-264 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-504 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{192 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 217, normalized size = 1.50 \[ \frac {66 \, \cos \left (d x + c\right )^{4} - 78 \, \cos \left (d x + c\right )^{3} - 158 \, \cos \left (d x + c\right )^{2} + 63 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 33 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 58 \, \cos \left (d x + c\right ) + 88}{96 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 211, normalized size = 1.46 \[ -\frac {\frac {3 \, {\left (\frac {14 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {66 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {132 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} + \frac {384 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} + \frac {\frac {132 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {21 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 126, normalized size = 0.87 \[ -\frac {1}{32 a d \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {1}{4 a d \left (-1+\cos \left (d x +c \right )\right )}+\frac {11 \ln \left (-1+\cos \left (d x +c \right )\right )}{32 d a}+\frac {1}{24 a d \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {9}{32 a d \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {15}{16 a d \left (1+\cos \left (d x +c \right )\right )}+\frac {21 \ln \left (1+\cos \left (d x +c \right )\right )}{32 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 130, normalized size = 0.90 \[ \frac {\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{4} - 39 \, \cos \left (d x + c\right )^{3} - 79 \, \cos \left (d x + c\right )^{2} + 29 \, \cos \left (d x + c\right ) + 44\right )}}{a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + a} + \frac {63 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac {33 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 132, normalized size = 0.91 \[ \frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32\,a\,d}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,a\,d}+\frac {11\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,a\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {1}{4}\right )}{32\,a\,d} \]
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 \[ \frac {\int \frac {\cot ^{5}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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