Optimal. Leaf size=189 \[ -\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \left (a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d}+\frac {\left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-3 a b x \]
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Rubi [A] time = 0.48, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \left (a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d}+\frac {\left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-3 a b x \]
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))^2 \, dx &=-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x))^2 \left (3 a^2-2 b^2+2 a b \sin (c+d x)-\left (2 a^2-3 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {\int \csc (c+d x) (a+b \sin (c+d x)) \left (3 a \left (3 a^2-2 b^2\right )+11 a^2 b \sin (c+d x)-4 a \left (a^2-3 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2}\\ &=-\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {\int \csc (c+d x) \left (6 a^2 \left (3 a^2-2 b^2\right )+36 a^3 b \sin (c+d x)-2 a^2 \left (4 a^2-23 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{12 a^2}\\ &=-\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {\int \csc (c+d x) \left (6 a^2 \left (3 a^2-2 b^2\right )+36 a^3 b \sin (c+d x)\right ) \, dx}{12 a^2}\\ &=-3 a b x-\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}-\frac {1}{2} \left (3 a^2-2 b^2\right ) \int \csc (c+d x) \, dx\\ &=-3 a b x+\frac {\left (3 a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {\left (4 a^2-23 b^2\right ) \cos (c+d x)}{6 d}-\frac {b \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{3 a d}-\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{6 a^2 d}-\frac {b \cot (c+d x) (a+b \sin (c+d x))^3}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^3}{2 a d}\\ \end {align*}
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Mathematica [A] time = 3.36, size = 191, normalized size = 1.01 \[ \frac {-6 \left (4 a^2-5 b^2\right ) \cos (c+d x)+3 \left (a^2 \left (-\csc ^2\left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-12 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+12 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 a b \sin (2 (c+d x))+8 a b \tan \left (\frac {1}{2} (c+d x)\right )-8 a b \cot \left (\frac {1}{2} (c+d x)\right )-24 a b c-24 a b d x+8 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+2 b^2 \cos (3 (c+d x))}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 210, normalized size = 1.11 \[ \frac {4 \, b^{2} \cos \left (d x + c\right )^{5} - 36 \, a b d x \cos \left (d x + c\right )^{2} + 36 \, a b d x - 4 \, {\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left ({\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} + 2 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (3 \, a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} + 2 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, {\left (a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 252, normalized size = 1.33 \[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 72 \, {\left (d x + c\right )} a b + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, {\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {3 \, {\left (18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {16 \, {\left (3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2} + 4 \, b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 208, normalized size = 1.10 \[ -\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{2 d}-\frac {3 a^{2} \cos \left (d x +c \right )}{2 d}-\frac {3 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {2 a b \left (\cos ^{5}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {2 a b \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {3 a b \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}-3 a b x -\frac {3 a b c}{d}+\frac {b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b^{2} \cos \left (d x +c \right )}{d}+\frac {b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 150, normalized size = 0.79 \[ -\frac {12 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a b - 2 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} b^{2} - 3 \, a^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.44, size = 397, normalized size = 2.10 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{2}-b^2\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {17\,a^2}{2}-16\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {35\,a^2}{2}-16\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {19\,a^2}{2}-\frac {32\,b^2}{3}\right )+\frac {a^2}{2}+20\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {6\,a\,b\,\mathrm {atan}\left (\frac {36\,a^2\,b^2}{-18\,a^3\,b+36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+12\,a\,b^3}-\frac {12\,a\,b^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{-18\,a^3\,b+36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+12\,a\,b^3}+\frac {18\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{-18\,a^3\,b+36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^2+12\,a\,b^3}\right )}{d}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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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