Optimal. Leaf size=40 \[ -\frac {a^2}{d (1-\cos (c+d x))}-\frac {a^2 \log (1-\cos (c+d x))}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 40, 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, 43} \[ -\frac {a^2}{d (1-\cos (c+d x))}-\frac {a^2 \log (1-\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3879
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+a \sec (c+d x))^2 \, dx &=-\frac {a^4 \operatorname {Subst}\left (\int \frac {x}{(a-a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^4 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 (-1+x)^2}+\frac {1}{a^2 (-1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^2}{d (1-\cos (c+d x))}-\frac {a^2 \log (1-\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 56, normalized size = 1.40 \[ \frac {a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right ) \left (-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \cos (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 48, normalized size = 1.20 \[ \frac {a^{2} - {\left (a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{d \cos \left (d x + c\right ) - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 111, normalized size = 2.78 \[ -\frac {2 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 2 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {{\left (a^{2} + \frac {2 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 51, normalized size = 1.28 \[ \frac {a^{2} \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a^{2}}{d \left (-1+\sec \left (d x +c \right )\right )}-\frac {a^{2} \ln \left (-1+\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 34, normalized size = 0.85 \[ -\frac {a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {a^{2}}{\cos \left (d x + c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 50, normalized size = 1.25 \[ -\frac {a^2\,\left (\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{d} \]
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 \[ a^{2} \left (\int 2 \cot ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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