Optimal. Leaf size=88 \[ -\frac {a \cot ^3(c+d x)}{3 d}+\frac {a \cot (c+d x)}{d}-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}+a x \]
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Rubi [A] time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2838, 2611, 3770, 3473, 8} \[ -\frac {a \cot ^3(c+d x)}{3 d}+\frac {a \cot (c+d x)}{d}-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}+a x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2611
Rule 2838
Rule 3473
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^4(c+d x) \, dx+a \int \cot ^4(c+d x) \csc (c+d x) \, dx\\ &=-\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^3(c+d x) \csc (c+d x)}{4 d}-\frac {1}{4} (3 a) \int \cot ^2(c+d x) \csc (c+d x) \, dx-a \int \cot ^2(c+d x) \, dx\\ &=\frac {a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {1}{8} (3 a) \int \csc (c+d x) \, dx+a \int 1 \, dx\\ &=a x-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc (c+d x)}{4 d}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 153, normalized size = 1.74 \[ -\frac {a \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 180, normalized size = 2.05 \[ \frac {48 \, a d x \cos \left (d x + c\right )^{4} - 96 \, a d x \cos \left (d x + c\right )^{2} - 30 \, a \cos \left (d x + c\right )^{3} + 48 \, a d x + 18 \, a \cos \left (d x + c\right ) - 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (4 \, a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 153, normalized size = 1.74 \[ \frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 192 \, {\left (d x + c\right )} a + 72 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {150 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 128, normalized size = 1.45 \[ -\frac {a \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a \cot \left (d x +c \right )}{d}+a x +\frac {c a}{d}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}+\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}+\frac {3 a \cos \left (d x +c \right )}{8 d}+\frac {3 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 107, normalized size = 1.22 \[ \frac {16 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a - 3 \, a {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.92, size = 217, normalized size = 2.47 \[ \frac {2\,a\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}+\frac {5\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d} \]
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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