Optimal. Leaf size=57 \[ -\frac {a}{2 d (1-\cos (c+d x))}-\frac {3 a \log (1-\cos (c+d x))}{4 d}-\frac {a \log (\cos (c+d x)+1)}{4 d} \]
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Rubi [A] time = 0.04, antiderivative size = 57, 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} \[ -\frac {a}{2 d (1-\cos (c+d x))}-\frac {3 a \log (1-\cos (c+d x))}{4 d}-\frac {a \log (\cos (c+d x)+1)}{4 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac {a^4 \operatorname {Subst}\left (\int \frac {x^2}{(a-a x)^2 (a+a x)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^4 \operatorname {Subst}\left (\int \left (\frac {1}{2 a^3 (-1+x)^2}+\frac {3}{4 a^3 (-1+x)}+\frac {1}{4 a^3 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a}{2 d (1-\cos (c+d x))}-\frac {3 a \log (1-\cos (c+d x))}{4 d}-\frac {a \log (1+\cos (c+d x))}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.78, size = 114, normalized size = 2.00 \[ -\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 69, normalized size = 1.21 \[ -\frac {{\left (a \cos \left (d x + c\right ) - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (a \cos \left (d x + c\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, a}{4 \, {\left (d \cos \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 103, normalized size = 1.81 \[ -\frac {3 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 4 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {{\left (a + \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 60, normalized size = 1.05 \[ \frac {a \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a}{2 d \left (-1+\sec \left (d x +c \right )\right )}-\frac {3 a \ln \left (-1+\sec \left (d x +c \right )\right )}{4 d}-\frac {a \ln \left (1+\sec \left (d x +c \right )\right )}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 42, normalized size = 0.74 \[ -\frac {a \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, a}{\cos \left (d x + c\right ) - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 46, normalized size = 0.81 \[ -\frac {a\,\left ({\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{4\,d} \]
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 \[ a \left (\int \cot ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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