Optimal. Leaf size=182 \[ \frac {3 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {85 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^3 x}{2} \]
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Rubi [A] time = 0.25, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2872, 3770, 3767, 8, 3768, 2638, 2635} \[ \frac {3 a^3 \cos (c+d x)}{d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {85 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^3 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 2872
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (8 a^9 \csc (c+d x)+6 a^9 \csc ^2(c+d x)-6 a^9 \csc ^3(c+d x)-8 a^9 \csc ^4(c+d x)+3 a^9 \csc ^6(c+d x)+a^9 \csc ^7(c+d x)-3 a^9 \sin (c+d x)-a^9 \sin ^2(c+d x)\right ) \, dx}{a^6}\\ &=a^3 \int \csc ^7(c+d x) \, dx-a^3 \int \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^6(c+d x) \, dx-\left (3 a^3\right ) \int \sin (c+d x) \, dx+\left (6 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx+\left (8 a^3\right ) \int \csc (c+d x) \, dx-\left (8 a^3\right ) \int \csc ^4(c+d x) \, dx\\ &=-\frac {8 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} a^3 \int 1 \, dx+\frac {1}{6} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx-\left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (6 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {\left (8 a^3\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {a^3 x}{2}-\frac {5 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{8} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {a^3 x}{2}-\frac {5 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{16} \left (5 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac {a^3 x}{2}-\frac {85 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac {3 a^3 \cos (c+d x)}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.86, size = 289, normalized size = 1.59 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (-960 (c+d x)+480 \sin (2 (c+d x))+5760 \cos (c+d x)+2176 \tan \left (\frac {1}{2} (c+d x)\right )-2176 \cot \left (\frac {1}{2} (c+d x)\right )-5 \csc ^6\left (\frac {1}{2} (c+d x)\right )-30 \csc ^4\left (\frac {1}{2} (c+d x)\right )+1290 \csc ^2\left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right )+30 \sec ^4\left (\frac {1}{2} (c+d x)\right )-1290 \sec ^2\left (\frac {1}{2} (c+d x)\right )+10200 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-10200 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-18 \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+206 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )-3296 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+36 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )\right )}{1920 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 316, normalized size = 1.74 \[ -\frac {240 \, a^{3} d x \cos \left (d x + c\right )^{6} - 1440 \, a^{3} \cos \left (d x + c\right )^{7} - 720 \, a^{3} d x \cos \left (d x + c\right )^{4} + 5610 \, a^{3} \cos \left (d x + c\right )^{5} + 720 \, a^{3} d x \cos \left (d x + c\right )^{2} - 6800 \, a^{3} \cos \left (d x + c\right )^{3} - 240 \, a^{3} d x + 2550 \, a^{3} \cos \left (d x + c\right ) + 1275 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 1275 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{7} + 23 \, a^{3} \cos \left (d x + c\right )^{5} - 35 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 307, normalized size = 1.69 \[ \frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 340 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1215 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 960 \, {\left (d x + c\right )} a^{3} + 10200 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 1800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {1920 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {24990 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1800 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1215 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 340 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 316, normalized size = 1.74 \[ \frac {4 a^{3} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {17 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d}-\frac {3 a^{3} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{d}-\frac {3 a^{3} \cot \left (d x +c \right )}{d}-\frac {17 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{4}}+\frac {17 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}+\frac {4 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )}+\frac {5 a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {5 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}-\frac {a^{3} x}{2}+\frac {85 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{48 d}+\frac {85 a^{3} \cos \left (d x +c \right )}{16 d}+\frac {85 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {a^{3} c}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 275, normalized size = 1.51 \[ \frac {80 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} - 96 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 5 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 90 \, a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.94, size = 396, normalized size = 2.18 \[ \frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {81\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {85\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,d}+\frac {a^3\,\mathrm {atan}\left (\frac {a^6}{\frac {85\,a^6}{8}+a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {85\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,\left (\frac {85\,a^6}{8}+a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}-\frac {124\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-\frac {849\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+\frac {134\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {927\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+\frac {578\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}-\frac {112\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {134\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}+\frac {6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {a^3}{6}}{d\,\left (64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}+\frac {15\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \]
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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