Optimal. Leaf size=111 \[ -\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac {\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}+a x \]
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Rubi [A] time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac {\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}+a x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3882
Rubi steps
\begin {align*} \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx &=-\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {1}{7} \int \cot ^6(c+d x) (-7 a-6 b \sec (c+d x)) \, dx\\ &=-\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}+\frac {1}{35} \int \cot ^4(c+d x) (35 a+24 b \sec (c+d x)) \, dx\\ &=-\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac {1}{105} \int \cot ^2(c+d x) (-105 a-48 b \sec (c+d x)) \, dx\\ &=-\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}+\frac {\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac {1}{105} \int 105 a \, dx\\ &=a x-\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}+\frac {\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 92, normalized size = 0.83 \[ -\frac {a \cot ^7(c+d x) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-\tan ^2(c+d x)\right )}{7 d}-\frac {b \csc ^7(c+d x)}{7 d}+\frac {3 b \csc ^5(c+d x)}{5 d}-\frac {b \csc ^3(c+d x)}{d}+\frac {b \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 179, normalized size = 1.61 \[ \frac {176 \, a \cos \left (d x + c\right )^{7} + 105 \, b \cos \left (d x + c\right )^{6} - 406 \, a \cos \left (d x + c\right )^{5} - 210 \, b \cos \left (d x + c\right )^{4} + 350 \, a \cos \left (d x + c\right )^{3} + 168 \, b \cos \left (d x + c\right )^{2} - 105 \, a \cos \left (d x + c\right ) + 105 \, {\left (a d x \cos \left (d x + c\right )^{6} - 3 \, a d x \cos \left (d x + c\right )^{4} + 3 \, a d x \cos \left (d x + c\right )^{2} - a d x\right )} \sin \left (d x + c\right ) - 48 \, b}{105 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 225, normalized size = 2.03 \[ \frac {15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1295 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 735 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13440 \, {\left (d x + c\right )} a - 9765 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3675 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {9765 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3675 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1295 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 735 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 147 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a - 15 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.91, size = 162, normalized size = 1.46 \[ \frac {a \left (-\frac {\left (\cot ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+b \left (-\frac {\cos ^{8}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{8}\left (d x +c \right )}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 100, normalized size = 0.90 \[ \frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a + \frac {3 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} b}{\sin \left (d x + c\right )^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.62, size = 174, normalized size = 1.57 \[ a\,x+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {37\,a}{384}-\frac {7\,b}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {9\,a}{640}-\frac {7\,b}{640}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {a}{896}-\frac {b}{896}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\left (-93\,a-35\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {37\,a}{3}+7\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {9\,a}{5}-\frac {7\,b}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{7}+\frac {b}{7}\right )}{128\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {93\,a}{128}-\frac {35\,b}{128}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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