Optimal. Leaf size=114 \[ -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d} \]
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Rubi [A] time = 0.15, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2607, 14, 2611, 3768, 3770} \[ -\frac {a \cot ^7(c+d x)}{7 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d} \]
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 ^4(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+a \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {1}{2} a \int \cot ^2(c+d x) \csc ^3(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 ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{8} a \int \csc ^3(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 (c+d x)}{16 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{16} a \int \csc (c+d x) \, dx\\ &=-\frac {a \tanh ^{-1}(\cos (c+d x))}{16 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 (c+d x)}{16 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 239, normalized size = 2.10 \[ -\frac {2 a \cot (c+d x)}{35 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {a \cot (c+d x) \csc ^6(c+d x)}{7 d}+\frac {8 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{35 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 221, normalized size = 1.94 \[ -\frac {192 \, a \cos \left (d x + c\right )^{7} - 672 \, a \cos \left (d x + c\right )^{5} + 105 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 70 \, {\left (3 \, a \cos \left (d x + c\right )^{5} + 8 \, a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3360 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 229, normalized size = 2.01 \[ \frac {15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 840 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2178 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 160, normalized size = 1.40 \[ -\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{4}}+\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{2}}+\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{48 d}+\frac {a \cos \left (d x +c \right )}{16 d}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {2 a \left (\cos ^{5}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 118, normalized size = 1.04 \[ \frac {35 \, a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {96 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a}{\tan \left (d x + c\right )^{7}}}{3360 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.95, size = 385, normalized size = 3.38 \[ \frac {a\,\left (15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+35\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}{13440\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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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