Optimal. Leaf size=82 \[ -\frac {3 a \cos (c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac {3 a \tanh ^{-1}(\cos (c+d x))}{2 d}+a x \]
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Rubi [A] time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2710, 2592, 288, 321, 206, 3473, 8} \[ -\frac {3 a \cos (c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac {3 a \tanh ^{-1}(\cos (c+d x))}{2 d}+a x \]
Antiderivative was successfully verified.
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Rule 8
Rule 206
Rule 288
Rule 321
Rule 2592
Rule 2710
Rule 3473
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx &=\int \left (a \cos (c+d x) \cot ^3(c+d x)+a \cot ^4(c+d x)\right ) \, dx\\ &=a \int \cos (c+d x) \cot ^3(c+d x) \, dx+a \int \cot ^4(c+d x) \, dx\\ &=-\frac {a \cot ^3(c+d x)}{3 d}-a \int \cot ^2(c+d x) \, dx-\frac {a \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+a \int 1 \, dx+\frac {(3 a) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d}\\ &=a x-\frac {3 a \cos (c+d x)}{2 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d}\\ &=a x+\frac {3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a \cos (c+d x)}{2 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a \cot ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 125, normalized size = 1.52 \[ -\frac {a \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d}-\frac {a \cos (c+d x)}{d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 160, normalized size = 1.95 \[ \frac {16 \, a \cos \left (d x + c\right )^{3} + 9 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 9 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 12 \, a \cos \left (d x + c\right ) + 6 \, {\left (2 \, a d x \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a d x + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 141, normalized size = 1.72 \[ \frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, {\left (d x + c\right )} a - 36 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {48 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {66 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 106, normalized size = 1.29 \[ -\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{2 d}-\frac {3 a \cos \left (d x +c \right )}{2 d}-\frac {3 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {a \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a \cot \left (d x +c \right )}{d}+a x +\frac {c a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 92, normalized size = 1.12 \[ \frac {4 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a + 3 \, a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.60, size = 228, normalized size = 2.78 \[ \frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {2\,a\,\mathrm {atan}\left (\frac {4\,a^2}{6\,a^2+4\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {6\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6\,a^2+4\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \sin {\left (c + d x \right )} \cot ^{4}{\left (c + d x \right )}\, dx + \int \cot ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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