Optimal. Leaf size=231 \[ -\frac {a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac {\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac {\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac {b \left (2 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3}{8} b x \left (12 a^2-b^2\right )-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d} \]
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Rubi [A] time = 0.69, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2893, 3049, 3033, 3023, 2735, 3770} \[ -\frac {a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac {\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac {\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac {b \left (2 a^2-21 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3}{8} b x \left (12 a^2-b^2\right )-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2893
Rule 3023
Rule 3033
Rule 3049
Rule 3770
Rubi steps
\begin {align*} \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^3 \, dx &=-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^3 \left (3 \left (a^2-2 b^2\right )+3 a b \sin (c+d x)-2 \left (a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac {\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^2 \left (12 a \left (a^2-2 b^2\right )+18 a^2 b \sin (c+d x)-6 a \left (a^2-6 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{8 a^2}\\ &=-\frac {\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac {\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x)) \left (36 a^2 \left (a^2-2 b^2\right )+78 a^3 b \sin (c+d x)-6 a^2 \left (2 a^2-21 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=-\frac {b \left (2 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac {\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac {\int \csc (c+d x) \left (72 a^3 \left (a^2-2 b^2\right )+18 a^2 b \left (12 a^2-b^2\right ) \sin (c+d x)-24 a^3 \left (a^2-17 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{48 a^2}\\ &=-\frac {a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac {b \left (2 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac {\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac {\int \csc (c+d x) \left (72 a^3 \left (a^2-2 b^2\right )+18 a^2 b \left (12 a^2-b^2\right ) \sin (c+d x)\right ) \, dx}{48 a^2}\\ &=-\frac {3}{8} b \left (12 a^2-b^2\right ) x-\frac {a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac {b \left (2 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac {\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}-\frac {1}{2} \left (3 a \left (a^2-2 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=-\frac {3}{8} b \left (12 a^2-b^2\right ) x+\frac {3 a \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \left (a^2-17 b^2\right ) \cos (c+d x)}{2 d}-\frac {b \left (2 a^2-21 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {\left (a^2-6 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{4 a d}-\frac {\left (a^2-4 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^4}{a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^4}{2 a d}\\ \end {align*}
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Mathematica [A] time = 6.16, size = 252, normalized size = 1.09 \[ -\frac {3 \left (a^3-2 a b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 \left (a^3-2 a b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {3 b \left (b^2-12 a^2\right ) (c+d x)}{8 d}+\frac {b \left (b^2-3 a^2\right ) \sin (2 (c+d x))}{4 d}-\frac {a \left (4 a^2-15 b^2\right ) \cos (c+d x)}{4 d}+\frac {3 a^2 b \tan \left (\frac {1}{2} (c+d x)\right )}{2 d}-\frac {3 a^2 b \cot \left (\frac {1}{2} (c+d x)\right )}{2 d}+\frac {a b^2 \cos (3 (c+d x))}{4 d}+\frac {b^3 \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 260, normalized size = 1.13 \[ \frac {8 \, a b^{2} \cos \left (d x + c\right )^{5} - 3 \, {\left (12 \, a^{2} b - b^{3}\right )} d x \cos \left (d x + c\right )^{2} - 8 \, {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (12 \, a^{2} b - b^{3}\right )} d x + 12 \, {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right ) - 6 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 6 \, {\left (a^{3} - 2 \, a b^{2} - {\left (a^{3} - 2 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (2 \, b^{3} \cos \left (d x + c\right )^{5} - {\left (12 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (12 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 400, normalized size = 1.73 \[ \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, {\left (12 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )} - 12 \, {\left (a^{3} - 2 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {18 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, {\left (12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 48 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 96 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 80 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{3} + 32 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 279, normalized size = 1.21 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{2 d}-\frac {3 a^{3} \cos \left (d x +c \right )}{2 d}-\frac {3 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {3 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {3 a^{2} b \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{d}-\frac {9 a^{2} b \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}-\frac {9 a^{2} b x}{2}-\frac {9 a^{2} b c}{2 d}+\frac {a \,b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \cos \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}+\frac {b^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4 d}+\frac {3 b^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {3 b^{3} x}{8}+\frac {3 b^{3} c}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 186, normalized size = 0.81 \[ -\frac {48 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{2} b - 16 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a b^{2} - {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{3} - 8 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.59, size = 718, normalized size = 3.11 \[ \frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (32\,a\,b^2-10\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (48\,a\,b^2-\frac {17\,a^3}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (80\,a\,b^2-27\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (96\,a\,b^2-26\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (6\,a^2\,b-5\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (12\,a^2\,b-3\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (36\,a^2\,b-5\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (48\,a^2\,b+3\,b^3\right )-\frac {a^3}{2}-6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,a\,b^2-\frac {3\,a^3}{2}\right )}{d}+\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {3\,b\,\mathrm {atan}\left (\frac {\frac {3\,b\,\left (12\,a^2-b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-3\,a^3\right )-9\,a^2\,b+\frac {3\,b^3}{4}-\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^2-b^2\right )\,9{}\mathrm {i}}{4}\right )}{8}+\frac {3\,b\,\left (12\,a^2-b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-3\,a^3\right )-9\,a^2\,b+\frac {3\,b^3}{4}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^2-b^2\right )\,9{}\mathrm {i}}{4}\right )}{8}}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (81\,a^4\,b^2-\frac {27\,a^2\,b^4}{2}+\frac {9\,b^6}{16}\right )+\frac {9\,a\,b^5}{2}+27\,a^5\,b-\frac {225\,a^3\,b^3}{4}-\frac {b\,\left (12\,a^2-b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-3\,a^3\right )-9\,a^2\,b+\frac {3\,b^3}{4}-\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^2-b^2\right )\,9{}\mathrm {i}}{4}\right )\,3{}\mathrm {i}}{8}+\frac {b\,\left (12\,a^2-b^2\right )\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a\,b^2-3\,a^3\right )-9\,a^2\,b+\frac {3\,b^3}{4}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^2-b^2\right )\,9{}\mathrm {i}}{4}\right )\,3{}\mathrm {i}}{8}}\right )\,\left (12\,a^2-b^2\right )}{4\,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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