Optimal. Leaf size=136 \[ -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {3 a \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rubi [A] time = 0.17, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2611, 3768, 3770, 2607, 14} \[ -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {3 a \cot (c+d x) \csc (c+d x)}{128 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rule 2611
Rule 2838
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+a \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {1}{8} (3 a) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a \operatorname {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{16} a \int \csc ^5(c+d x) \, dx+\frac {a \operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{64} (3 a) \int \csc ^3(c+d x) \, dx\\ &=-\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {3 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {1}{128} (3 a) \int \csc (c+d x) \, dx\\ &=-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {3 a \cot (c+d x) \csc (c+d x)}{128 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 279, normalized size = 2.05 \[ -\frac {2 a \cot (c+d x)}{35 d}-\frac {a \csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {3 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {a \sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {3 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {a \cot (c+d x) \csc ^6(c+d x)}{7 d}+\frac {8 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{35 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 239, normalized size = 1.76 \[ \frac {210 \, a \cos \left (d x + c\right )^{7} - 770 \, a \cos \left (d x + c\right )^{5} - 770 \, a \cos \left (d x + c\right )^{3} + 210 \, a \cos \left (d x + c\right ) - 105 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 256 \, {\left (2 \, a \cos \left (d x + c\right )^{7} - 7 \, a \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 201, normalized size = 1.48 \[ \frac {35 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 80 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 112 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 560 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1680 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 1680 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {4566 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1680 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 560 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 112 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 35 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{71680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 182, normalized size = 1.34 \[ -\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {2 a \left (\cos ^{5}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{5}}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{6}}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{64 d \sin \left (d x +c \right )^{4}}+\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}+\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{128 d}+\frac {3 a \cos \left (d x +c \right )}{128 d}+\frac {3 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 138, normalized size = 1.01 \[ \frac {35 \, a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {256 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.22, size = 337, normalized size = 2.48 \[ \frac {a\,\left (35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+80\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}-80\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-1680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+1680\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{71680\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]
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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