Optimal. Leaf size=103 \[ -\frac {1}{8 a d (1-\cos (c+d x))}+\frac {1}{8 a d (1+\cos (c+d x))^2}-\frac {3}{4 a d (1+\cos (c+d x))}-\frac {5 \log (1-\cos (c+d x))}{16 a d}-\frac {11 \log (1+\cos (c+d x))}{16 a d} \]
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Rubi [A]
time = 0.06, antiderivative size = 103, 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 = {3964, 90}
\begin {gather*} -\frac {1}{8 a d (1-\cos (c+d x))}-\frac {3}{4 a d (\cos (c+d x)+1)}+\frac {1}{8 a d (\cos (c+d x)+1)^2}-\frac {5 \log (1-\cos (c+d x))}{16 a d}-\frac {11 \log (\cos (c+d x)+1)}{16 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 3964
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac {a^4 \text {Subst}\left (\int \frac {x^4}{(a-a x)^2 (a+a x)^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^4 \text {Subst}\left (\int \left (\frac {1}{8 a^5 (-1+x)^2}+\frac {5}{16 a^5 (-1+x)}+\frac {1}{4 a^5 (1+x)^3}-\frac {3}{4 a^5 (1+x)^2}+\frac {11}{16 a^5 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {1}{8 a d (1-\cos (c+d x))}+\frac {1}{8 a d (1+\cos (c+d x))^2}-\frac {3}{4 a d (1+\cos (c+d x))}-\frac {5 \log (1-\cos (c+d x))}{16 a d}-\frac {11 \log (1+\cos (c+d x))}{16 a d}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 107, normalized size = 1.04 \begin {gather*} -\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+44 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+20 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 \sec ^2\left (\frac {1}{2} (c+d x)\right )-\sec ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sec (c+d x)}{16 a d (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 67, normalized size = 0.65
method | result | size |
derivativedivides | \(\frac {\frac {1}{-8+8 \cos \left (d x +c \right )}-\frac {5 \ln \left (-1+\cos \left (d x +c \right )\right )}{16}+\frac {1}{8 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {3}{4 \left (1+\cos \left (d x +c \right )\right )}-\frac {11 \ln \left (1+\cos \left (d x +c \right )\right )}{16}}{d a}\) | \(67\) |
default | \(\frac {\frac {1}{-8+8 \cos \left (d x +c \right )}-\frac {5 \ln \left (-1+\cos \left (d x +c \right )\right )}{16}+\frac {1}{8 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {3}{4 \left (1+\cos \left (d x +c \right )\right )}-\frac {11 \ln \left (1+\cos \left (d x +c \right )\right )}{16}}{d a}\) | \(67\) |
risch | \(\frac {i x}{a}+\frac {2 i c}{a d}-\frac {5 \,{\mathrm e}^{5 i \left (d x +c \right )}-6 \,{\mathrm e}^{4 i \left (d x +c \right )}-14 \,{\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}+5 \,{\mathrm e}^{i \left (d x +c \right )}}{4 d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{2}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a d}-\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a d}\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 91, normalized size = 0.88 \begin {gather*} -\frac {\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) - 6\right )}}{a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a} + \frac {11 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac {5 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.54, size = 139, normalized size = 1.35 \begin {gather*} -\frac {10 \, \cos \left (d x + c\right )^{2} + 11 \, {\left (\cos \left (d x + c\right )^{3} + \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 ) + 5 \, {\left (\cos \left (d x + c\right )^{3} + \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 ) - 6 \, \cos \left (d x + c\right ) - 12}{16 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 157, normalized size = 1.52 \begin {gather*} \frac {\frac {2 \, {\left (\frac {5 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{a {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {10 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} + \frac {32 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} + \frac {\frac {10 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{2}}}{32 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.34, size = 76, normalized size = 0.74 \begin {gather*} -\frac {\frac {5\,\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 {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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