Optimal. Leaf size=124 \[ \frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {\tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac {\cot (c+d x) \csc (c+d x)}{16 a d} \]
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Rubi [A] time = 0.18, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2839, 2611, 3768, 3770, 2607, 14} \[ \frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {\tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac {\cot (c+d x) \csc (c+d x)}{16 a d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rule 2611
Rule 2839
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a}+\frac {\int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a}\\ &=-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\int \csc ^5(c+d x) \, dx}{6 a}-\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\int \csc ^3(c+d x) \, dx}{8 a}-\frac {\operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot (c+d x) \csc (c+d x)}{16 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\int \csc (c+d x) \, dx}{16 a}\\ &=\frac {\tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot (c+d x) \csc (c+d x)}{16 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 229, normalized size = 1.85 \[ -\frac {\csc ^6(c+d x) \left (-480 \sin (2 (c+d x))-192 \sin (4 (c+d x))+32 \sin (6 (c+d x))+1140 \cos (c+d x)+170 \cos (3 (c+d x))-30 \cos (5 (c+d x))+150 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+225 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-90 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+15 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-150 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-225 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+90 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-15 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{7680 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 188, normalized size = 1.52 \[ -\frac {30 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 15 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (2 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 30 \, \cos \left (d x + c\right )}{480 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 216, normalized size = 1.74 \[ -\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {5 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 20 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {294 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 20 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 246, normalized size = 1.98 \[ \frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 a d}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 a d}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{96 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}-\frac {1}{384 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {1}{16 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a d}+\frac {1}{160 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{96 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 274, normalized size = 2.21 \[ \frac {\frac {\frac {120 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {12 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {120 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a \sin \left (d x + c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.68, size = 339, normalized size = 2.73 \[ -\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+12\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+20\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-20\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]
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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