Optimal. Leaf size=183 \[ \frac {a \left (2 a^2+15 b^2\right ) \cot (c+d x)}{15 d}+\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}+\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d} \]
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Rubi [A] time = 0.57, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2889, 3048, 3047, 3031, 3021, 2748, 3767, 8, 3770} \[ \frac {a \left (2 a^2+15 b^2\right ) \cot (c+d x)}{15 d}+\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}+\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2889
Rule 3021
Rule 3031
Rule 3047
Rule 3048
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \cot ^2(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \csc ^6(c+d x) (a+b \sin (c+d x))^3 \left (1-\sin ^2(c+d x)\right ) \, dx\\ &=-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {1}{5} \int \csc ^5(c+d x) (a+b \sin (c+d x))^2 \left (3 b-a \sin (c+d x)-4 b \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {1}{20} \int \csc ^4(c+d x) (a+b \sin (c+d x)) \left (-2 \left (2 a^2-3 b^2\right )-11 a b \sin (c+d x)-13 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {1}{60} \int \csc ^3(c+d x) \left (9 b \left (5 a^2-2 b^2\right )+4 a \left (2 a^2+15 b^2\right ) \sin (c+d x)+39 b^3 \sin ^2(c+d x)\right ) \, dx\\ &=\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {1}{120} \int \csc ^2(c+d x) \left (8 a \left (2 a^2+15 b^2\right )+15 b \left (3 a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}-\frac {1}{8} \left (b \left (3 a^2+4 b^2\right )\right ) \int \csc (c+d x) \, dx-\frac {1}{15} \left (a \left (2 a^2+15 b^2\right )\right ) \int \csc ^2(c+d x) \, dx\\ &=\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}+\frac {\left (a \left (2 a^2+15 b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{15 d}\\ &=\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {a \left (2 a^2+15 b^2\right ) \cot (c+d x)}{15 d}+\frac {3 b \left (5 a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{40 d}+\frac {a \left (2 a^2-3 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{30 d}-\frac {3 b \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{20 d}-\frac {\cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{5 d}\\ \end {align*}
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Mathematica [A] time = 1.29, size = 344, normalized size = 1.88 \[ \frac {32 \left (2 a^3+15 a b^2\right ) \cot \left (\frac {1}{2} (c+d x)\right )-64 a^3 \tan \left (\frac {1}{2} (c+d x)\right )-3 a^3 \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )-16 a^3 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+6 a^3 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )+30 \left (3 a^2 b-4 b^3\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )+a \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (\left (a^2-60 b^2\right ) \sin (c+d x)-45 a b\right )+45 a^2 b \sec ^4\left (\frac {1}{2} (c+d x)\right )-90 a^2 b \sec ^2\left (\frac {1}{2} (c+d x)\right )-360 a^2 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+360 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-480 a b^2 \tan \left (\frac {1}{2} (c+d x)\right )+960 a b^2 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+120 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )-480 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 275, normalized size = 1.50 \[ \frac {16 \, {\left (2 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left ({\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b + 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left ({\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 3 \, a^{2} b + 4 \, b^{3} - 2 \, {\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 30 \, {\left ({\left (3 \, a^{2} b - 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 290, normalized size = 1.58 \[ \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 360 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, {\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {822 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1096 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 360 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 227, normalized size = 1.24 \[ -\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}-\frac {2 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{15 d \sin \left (d x +c \right )^{3}}-\frac {3 a^{2} b \left (\cos ^{3}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}-\frac {3 a^{2} b \left (\cos ^{3}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}-\frac {3 a^{2} b \cos \left (d x +c \right )}{8 d}-\frac {3 a^{2} b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {a \,b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{3}}-\frac {b^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {b^{3} \cos \left (d x +c \right )}{2 d}-\frac {b^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 157, normalized size = 0.86 \[ -\frac {45 \, a^{2} b {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 60 \, b^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {240 \, a b^{2}}{\tan \left (d x + c\right )^{3}} + \frac {16 \, {\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.44, size = 241, normalized size = 1.32 \[ \frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^3}{96}+\frac {a\,b^2}{8}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2\,b}{8}+\frac {b^3}{2}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3}{3}+4\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^3+12\,a\,b^2\right )+\frac {a^3}{5}+\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{32\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3}{16}+\frac {3\,a\,b^2}{8}\right )}{d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d} \]
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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