Optimal. Leaf size=81 \[ \frac {5}{4 a^2 d (\cos (c+d x)+1)}-\frac {1}{4 a^2 d (\cos (c+d x)+1)^2}+\frac {\log (1-\cos (c+d x))}{8 a^2 d}+\frac {7 \log (\cos (c+d x)+1)}{8 a^2 d} \]
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Rubi [A] time = 0.06, antiderivative size = 81, 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} \[ \frac {5}{4 a^2 d (\cos (c+d x)+1)}-\frac {1}{4 a^2 d (\cos (c+d x)+1)^2}+\frac {\log (1-\cos (c+d x))}{8 a^2 d}+\frac {7 \log (\cos (c+d x)+1)}{8 a^2 d} \]
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))^2} \, dx &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {x^3}{(a-a x) (a+a x)^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (-\frac {1}{8 a^4 (-1+x)}-\frac {1}{2 a^4 (1+x)^3}+\frac {5}{4 a^4 (1+x)^2}-\frac {7}{8 a^4 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {1}{4 a^2 d (1+\cos (c+d x))^2}+\frac {5}{4 a^2 d (1+\cos (c+d x))}+\frac {\log (1-\cos (c+d x))}{8 a^2 d}+\frac {7 \log (1+\cos (c+d x))}{8 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 83, normalized size = 1.02 \[ \frac {\sec ^2(c+d x) \left (10 \cos ^2\left (\frac {1}{2} (c+d x)\right )+4 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+7 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-1\right )}{4 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 106, normalized size = 1.31 \[ \frac {7 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 10 \, \cos \left (d x + c\right ) + 8}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 117, normalized size = 1.44 \[ \frac {\frac {2 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} - \frac {16 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac {\frac {8 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{4}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 72, normalized size = 0.89 \[ \frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{8 a^{2} d}-\frac {1}{4 a^{2} d \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {5}{4 a^{2} d \left (1+\cos \left (d x +c \right )\right )}+\frac {7 \ln \left (1+\cos \left (d x +c \right )\right )}{8 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 74, normalized size = 0.91 \[ \frac {\frac {2 \, {\left (5 \, \cos \left (d x + c\right ) + 4\right )}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}} + \frac {7 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 62, normalized size = 0.77 \[ \frac {\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4}-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16}}{a^2\,d} \]
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 \[ \frac {\int \frac {\cot {\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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