Optimal. Leaf size=96 \[ -\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d} \]
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Rubi [A] time = 0.14, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2838, 2607, 30, 2611, 3770} \[ -\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 2838
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^6(c+d x) \csc (c+d x) \, dx+a \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {1}{6} (5 a) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac {a \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a \cot ^7(c+d x)}{7 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {1}{8} (5 a) \int \cot ^2(c+d x) \csc (c+d x) \, dx\\ &=-\frac {a \cot ^7(c+d x)}{7 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {1}{16} (5 a) \int \csc (c+d x) \, dx\\ &=\frac {5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 175, normalized size = 1.82 \[ -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {11 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {11 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 210, normalized size = 2.19 \[ \frac {96 \, a \cos \left (d x + c\right )^{7} + 105 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 14 \, {\left (33 \, a \cos \left (d x + c\right )^{5} - 40 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 228, normalized size = 2.38 \[ \frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 840 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {2178 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{2688 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 152, normalized size = 1.58 \[ -\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}}+\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{4}}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{16 d}-\frac {5 a \left (\cos ^{3}\left (d x +c \right )\right )}{48 d}-\frac {5 a \cos \left (d x +c \right )}{16 d}-\frac {5 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 106, normalized size = 1.10 \[ \frac {7 \, a {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {96 \, a}{\tan \left (d x + c\right )^{7}}}{672 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.94, size = 385, normalized size = 4.01 \[ -\frac {a\,\left (3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}{2688\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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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