Optimal. Leaf size=101 \[ \frac {17}{8 a^3 d (\cos (c+d x)+1)}-\frac {7}{8 a^3 d (\cos (c+d x)+1)^2}+\frac {1}{6 a^3 d (\cos (c+d x)+1)^3}+\frac {\log (1-\cos (c+d x))}{16 a^3 d}+\frac {15 \log (\cos (c+d x)+1)}{16 a^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3879, 88} \[ \frac {17}{8 a^3 d (\cos (c+d x)+1)}-\frac {7}{8 a^3 d (\cos (c+d x)+1)^2}+\frac {1}{6 a^3 d (\cos (c+d x)+1)^3}+\frac {\log (1-\cos (c+d x))}{16 a^3 d}+\frac {15 \log (\cos (c+d x)+1)}{16 a^3 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {x^4}{(a-a x) (a+a x)^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (-\frac {1}{16 a^5 (-1+x)}+\frac {1}{2 a^5 (1+x)^4}-\frac {7}{4 a^5 (1+x)^3}+\frac {17}{8 a^5 (1+x)^2}-\frac {15}{16 a^5 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {1}{6 a^3 d (1+\cos (c+d x))^3}-\frac {7}{8 a^3 d (1+\cos (c+d x))^2}+\frac {17}{8 a^3 d (1+\cos (c+d x))}+\frac {\log (1-\cos (c+d x))}{16 a^3 d}+\frac {15 \log (1+\cos (c+d x))}{16 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 97, normalized size = 0.96 \[ \frac {\sec ^3(c+d x) \left (102 \cos ^4\left (\frac {1}{2} (c+d x)\right )-21 \cos ^2\left (\frac {1}{2} (c+d x)\right )+12 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+15 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+2\right )}{12 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 151, normalized size = 1.50 \[ \frac {102 \, \cos \left (d x + c\right )^{2} + 45 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 162 \, \cos \left (d x + c\right ) + 68}{48 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 143, normalized size = 1.42 \[ \frac {\frac {6 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} - \frac {96 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} - \frac {\frac {66 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {15 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{9}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.82, size = 90, normalized size = 0.89 \[ \frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{16 d \,a^{3}}+\frac {1}{6 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {7}{8 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {17}{8 d \,a^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {15 \ln \left (1+\cos \left (d x +c \right )\right )}{16 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 98, normalized size = 0.97 \[ \frac {\frac {2 \, {\left (51 \, \cos \left (d x + c\right )^{2} + 81 \, \cos \left (d x + c\right ) + 34\right )}}{a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}} + \frac {45 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 75, normalized size = 0.74 \[ \frac {\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8}-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{48}}{a^3\,d} \]
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 \[ \frac {\int \frac {\cot {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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