Optimal. Leaf size=200 \[ -\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {55 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {25 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]
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Rubi [A] time = 0.36, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2611, 3770, 2607, 30, 3768, 14} \[ -\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {55 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {25 a^3 \cot (c+d x) \csc (c+d x)}{128 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^3(c+d x)+a^3 \cot ^6(c+d x) \csc ^4(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc (c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{6} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx-\frac {1}{8} \left (15 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {3 a^3 \cot ^7(c+d x)}{7 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{8} \left (5 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\frac {1}{16} \left (15 a^3\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {5 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{16} \left (5 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac {5 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {25 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} \left (15 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac {55 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {25 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^3 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^3 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {15 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{16 d}-\frac {3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\\ \end {align*}
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Mathematica [B] time = 0.14, size = 459, normalized size = 2.30 \[ a^3 \left (-\frac {29 \tan \left (\frac {1}{2} (c+d x)\right )}{126 d}+\frac {29 \cot \left (\frac {1}{2} (c+d x)\right )}{126 d}-\frac {3 \csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {17 \csc ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}-\frac {13 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {73 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {3 \sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {17 \sec ^6\left (\frac {1}{2} (c+d x)\right )}{1536 d}+\frac {13 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {73 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {55 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}+\frac {55 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^8\left (\frac {1}{2} (c+d x)\right )}{4608 d}-\frac {53 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^6\left (\frac {1}{2} (c+d x)\right )}{32256 d}+\frac {319 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{10752 d}-\frac {4163 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32256 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^8\left (\frac {1}{2} (c+d x)\right )}{4608 d}+\frac {53 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )}{32256 d}-\frac {319 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )}{10752 d}+\frac {4163 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32256 d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 291, normalized size = 1.46 \[ \frac {7424 \, a^{3} \cos \left (d x + c\right )^{9} - 9216 \, a^{3} \cos \left (d x + c\right )^{7} + 3465 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3465 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 42 \, {\left (219 \, a^{3} \cos \left (d x + c\right )^{7} - 803 \, a^{3} \cos \left (d x + c\right )^{5} + 605 \, a^{3} \cos \left (d x + c\right )^{3} - 165 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16128 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 324, normalized size = 1.62 \[ \frac {28 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 189 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 324 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 672 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3024 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1512 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 9744 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18144 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 55440 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 16632 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {156838 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 16632 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 18144 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9744 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1512 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3024 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 324 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 189 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 28 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{129024 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 216, normalized size = 1.08 \[ -\frac {11 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{6}}+\frac {11 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{192 d \sin \left (d x +c \right )^{4}}-\frac {11 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}-\frac {11 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{128 d}-\frac {55 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{384 d}-\frac {55 a^{3} \cos \left (d x +c \right )}{128 d}-\frac {55 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {29 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{63 d \sin \left (d x +c \right )^{7}}-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{9 d \sin \left (d x +c \right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 246, normalized size = 1.23 \[ -\frac {63 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 168 \, a^{3} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {6912 \, a^{3}}{\tan \left (d x + c\right )^{7}} + \frac {256 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{16128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.40, size = 357, normalized size = 1.78 \[ \frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {29\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {9\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {9\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {9\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {29\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {9\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {55\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}+\frac {33\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {33\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]
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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