Optimal. Leaf size=131 \[ -\frac {a \cot ^7(c+d x)}{7 d}-\frac {3 a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^3(c+d x)}{d}-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3872, 2838, 2621, 302, 207, 3767} \[ -\frac {a \cot ^7(c+d x)}{7 d}-\frac {3 a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^3(c+d x)}{d}-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 302
Rule 2621
Rule 2838
Rule 3767
Rule 3872
Rubi steps
\begin {align*} \int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx &=-\int (-a-a \cos (c+d x)) \csc ^8(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^8(c+d x) \, dx+a \int \csc ^8(c+d x) \sec (c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {x^8}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{d}-\frac {3 a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \operatorname {Subst}\left (\int \left (1+x^2+x^4+x^6+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{d}-\frac {3 a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{d}-\frac {3 a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 113, normalized size = 0.86 \[ -\frac {a \csc ^7(c+d x) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};\sin ^2(c+d x)\right )}{7 d}-\frac {16 a \cot (c+d x)}{35 d}-\frac {a \cot (c+d x) \csc ^6(c+d x)}{7 d}-\frac {6 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac {8 a \cot (c+d x) \csc ^2(c+d x)}{35 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 281, normalized size = 2.15 \[ -\frac {96 \, a \cos \left (d x + c\right )^{6} + 114 \, a \cos \left (d x + c\right )^{5} - 450 \, a \cos \left (d x + c\right )^{4} - 250 \, a \cos \left (d x + c\right )^{3} + 670 \, a \cos \left (d x + c\right )^{2} - 105 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 142 \, a \cos \left (d x + c\right ) - 352 \, a}{210 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 136, normalized size = 1.04 \[ -\frac {21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6720 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 6720 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 3045 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1015 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 168 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 149, normalized size = 1.14 \[ -\frac {16 a \cot \left (d x +c \right )}{35 d}-\frac {a \cot \left (d x +c \right ) \left (\csc ^{6}\left (d x +c \right )\right )}{7 d}-\frac {6 a \cot \left (d x +c \right ) \left (\csc ^{4}\left (d x +c \right )\right )}{35 d}-\frac {8 a \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{35 d}-\frac {a}{7 d \sin \left (d x +c \right )^{7}}-\frac {a}{5 d \sin \left (d x +c \right )^{5}}-\frac {a}{3 d \sin \left (d x +c \right )^{3}}-\frac {a}{d \sin \left (d x +c \right )}+\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 116, normalized size = 0.89 \[ -\frac {a {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} + 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} + 15\right )}}{\sin \left (d x + c\right )^{7}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {6 \, {\left (35 \, \tan \left (d x + c\right )^{6} + 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} + 5\right )} a}{\tan \left (d x + c\right )^{7}}}{210 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 128, normalized size = 0.98 \[ \frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {29\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (64\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {a}{7}\right )}{64\,d} \]
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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