Optimal. Leaf size=88 \[ \frac {\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}-\frac {\cot ^3(c+d x) (5-4 \sec (c+d x))}{15 a d}+\frac {\cot (c+d x) (15-8 \sec (c+d x))}{15 a d}+\frac {x}{a} \]
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Rubi [A] time = 0.13, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3888, 3882, 8} \[ \frac {\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}-\frac {\cot ^3(c+d x) (5-4 \sec (c+d x))}{15 a d}+\frac {\cot (c+d x) (15-8 \sec (c+d x))}{15 a d}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3882
Rule 3888
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{a+a \sec (c+d x)} \, dx &=\frac {\int \cot ^6(c+d x) (-a+a \sec (c+d x)) \, dx}{a^2}\\ &=\frac {\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}+\frac {\int \cot ^4(c+d x) (5 a-4 a \sec (c+d x)) \, dx}{5 a^2}\\ &=-\frac {\cot ^3(c+d x) (5-4 \sec (c+d x))}{15 a d}+\frac {\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}+\frac {\int \cot ^2(c+d x) (-15 a+8 a \sec (c+d x)) \, dx}{15 a^2}\\ &=\frac {\cot (c+d x) (15-8 \sec (c+d x))}{15 a d}-\frac {\cot ^3(c+d x) (5-4 \sec (c+d x))}{15 a d}+\frac {\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}+\frac {\int 15 a \, dx}{15 a^2}\\ &=\frac {x}{a}+\frac {\cot (c+d x) (15-8 \sec (c+d x))}{15 a d}-\frac {\cot ^3(c+d x) (5-4 \sec (c+d x))}{15 a d}+\frac {\cot ^5(c+d x) (1-\sec (c+d x))}{5 a d}\\ \end {align*}
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Mathematica [B] time = 0.90, size = 254, normalized size = 2.89 \[ \frac {\csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \csc ^3(c+d x) \sec (c+d x) (534 \sin (c+d x)+178 \sin (2 (c+d x))-178 \sin (3 (c+d x))-89 \sin (4 (c+d x))-520 \sin (2 c+d x)-248 \sin (c+2 d x)-120 \sin (3 c+2 d x)+248 \sin (2 c+3 d x)+120 \sin (4 c+3 d x)+184 \sin (3 c+4 d x)-360 d x \cos (2 c+d x)+120 d x \cos (c+2 d x)-120 d x \cos (3 c+2 d x)-120 d x \cos (2 c+3 d x)+120 d x \cos (4 c+3 d x)-60 d x \cos (3 c+4 d x)+60 d x \cos (5 c+4 d x)-200 \sin (c)-584 \sin (d x)+360 d x \cos (d x))}{1920 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 134, normalized size = 1.52 \[ \frac {23 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{3} - 27 \, \cos \left (d x + c\right )^{2} + 15 \, {\left (d x \cos \left (d x + c\right )^{3} + d x \cos \left (d x + c\right )^{2} - d x \cos \left (d x + c\right ) - d x\right )} \sin \left (d x + c\right ) - 7 \, \cos \left (d x + c\right ) + 8}{15 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 98, normalized size = 1.11 \[ \frac {\frac {240 \, {\left (d x + c\right )}}{a} + \frac {5 \, {\left (18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {3 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{5}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 113, normalized size = 1.28 \[ -\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{80 a d}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {1}{48 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 137, normalized size = 1.56 \[ -\frac {\frac {3 \, {\left (\frac {80 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a} - \frac {480 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {5 \, {\left (\frac {18 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a \sin \left (d x + c\right )^{3}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 158, normalized size = 1.80 \[ -\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-30\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-90\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-240\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (c+d\,x\right )}{240\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]
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 \[ \frac {\int \frac {\cot ^{4}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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