Optimal. Leaf size=84 \[ -\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}+\frac {\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{15 d}-\frac {\cot (c+d x) (8 a \sec (c+d x)+15 a)}{15 d}-a x \]
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Rubi [A] time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)}{5 d}+\frac {\cot ^3(c+d x) (4 a \sec (c+d x)+5 a)}{15 d}-\frac {\cot (c+d x) (8 a \sec (c+d x)+15 a)}{15 d}-a x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3882
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac {1}{5} \int \cot ^4(c+d x) (-5 a-4 a \sec (c+d x)) \, dx\\ &=-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 a \sec (c+d x))}{15 d}+\frac {1}{15} \int \cot ^2(c+d x) (15 a+8 a \sec (c+d x)) \, dx\\ &=-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 a \sec (c+d x))}{15 d}-\frac {\cot (c+d x) (15 a+8 a \sec (c+d x))}{15 d}+\frac {1}{15} \int -15 a \, dx\\ &=-a x-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))}{5 d}+\frac {\cot ^3(c+d x) (5 a+4 a \sec (c+d x))}{15 d}-\frac {\cot (c+d x) (15 a+8 a \sec (c+d x))}{15 d}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 79, normalized size = 0.94 \[ -\frac {a \cot ^5(c+d x) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\tan ^2(c+d x)\right )}{5 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 139, normalized size = 1.65 \[ -\frac {23 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{3} - 27 \, a \cos \left (d x + c\right )^{2} + 7 \, a \cos \left (d x + c\right ) + 15 \, {\left (a d x \cos \left (d x + c\right )^{3} - a d x \cos \left (d x + c\right )^{2} - a d x \cos \left (d x + c\right ) + a d x\right )} \sin \left (d x + c\right ) + 8 \, a}{15 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 83, normalized size = 0.99 \[ -\frac {5 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, {\left (d x + c\right )} a - 90 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3 \, {\left (80 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 129, normalized size = 1.54 \[ \frac {a \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )+a \left (-\frac {\cos ^{6}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{15 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{5 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 79, normalized size = 0.94 \[ -\frac {{\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a + \frac {{\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} + 3\right )} a}{\sin \left (d x + c\right )^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 156, normalized size = 1.86 \[ -\frac {a\,\left (3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-90\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-30\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )\right )}{240\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
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 \[ a \left (\int \cot ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{6}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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