Optimal. Leaf size=95 \[ \frac {3 a}{4 d (1-\cos (c+d x))}+\frac {a}{8 d (\cos (c+d x)+1)}-\frac {a}{8 d (1-\cos (c+d x))^2}+\frac {11 a \log (1-\cos (c+d x))}{16 d}+\frac {5 a \log (\cos (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.06, antiderivative size = 95, 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} \[ \frac {3 a}{4 d (1-\cos (c+d x))}+\frac {a}{8 d (\cos (c+d x)+1)}-\frac {a}{8 d (1-\cos (c+d x))^2}+\frac {11 a \log (1-\cos (c+d x))}{16 d}+\frac {5 a \log (\cos (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac {a^6 \operatorname {Subst}\left (\int \frac {x^4}{(a-a x)^3 (a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^6 \operatorname {Subst}\left (\int \left (-\frac {1}{4 a^5 (-1+x)^3}-\frac {3}{4 a^5 (-1+x)^2}-\frac {11}{16 a^5 (-1+x)}+\frac {1}{8 a^5 (1+x)^2}-\frac {5}{16 a^5 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a}{8 d (1-\cos (c+d x))^2}+\frac {3 a}{4 d (1-\cos (c+d x))}+\frac {a}{8 d (1+\cos (c+d x))}+\frac {11 a \log (1-\cos (c+d x))}{16 d}+\frac {5 a \log (1+\cos (c+d x))}{16 d}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 127, normalized size = 1.34 \[ \frac {a \left (-16 \cot ^4(c+d x)+32 \cot ^2(c+d x)-\csc ^4\left (\frac {1}{2} (c+d x)\right )+10 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )-10 \sec ^2\left (\frac {1}{2} (c+d x)\right )+24 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+64 \log (\tan (c+d x))-24 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+64 \log (\cos (c+d x))\right )}{64 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 150, normalized size = 1.58 \[ -\frac {10 \, a \cos \left (d x + c\right )^{2} + 6 \, a \cos \left (d x + c\right ) - 5 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 11 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 149, normalized size = 1.57 \[ \frac {22 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {{\left (a + \frac {10 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {33 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 93, normalized size = 0.98 \[ -\frac {a \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a}{8 d \left (-1+\sec \left (d x +c \right )\right )^{2}}+\frac {a}{2 d \left (-1+\sec \left (d x +c \right )\right )}+\frac {11 a \ln \left (-1+\sec \left (d x +c \right )\right )}{16 d}-\frac {a}{8 d \left (1+\sec \left (d x +c \right )\right )}+\frac {5 a \ln \left (1+\sec \left (d x +c \right )\right )}{16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 86, normalized size = 0.91 \[ \frac {5 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (5 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) - 6 \, a\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 88, normalized size = 0.93 \[ \frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a}{4}-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}\right )}{8\,d}+\frac {11\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \]
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 \[ a \left (\int \cot ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{5}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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