Optimal. Leaf size=175 \[ -\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {25 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {13 a^3 x}{2} \]
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Rubi [A] time = 0.30, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2709, 3770, 3767, 8, 3768, 2635, 2633} \[ -\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {25 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {13 a^3 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2709
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\int \left (8 a^9+6 a^9 \csc (c+d x)-6 a^9 \csc ^2(c+d x)-8 a^9 \csc ^3(c+d x)+3 a^9 \csc ^5(c+d x)+a^9 \csc ^6(c+d x)-3 a^9 \sin ^2(c+d x)-a^9 \sin ^3(c+d x)\right ) \, dx}{a^6}\\ &=8 a^3 x+a^3 \int \csc ^6(c+d x) \, dx-a^3 \int \sin ^3(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^5(c+d x) \, dx-\left (3 a^3\right ) \int \sin ^2(c+d x) \, dx+\left (6 a^3\right ) \int \csc (c+d x) \, dx-\left (6 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (8 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=8 a^3 x-\frac {6 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {4 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} \left (3 a^3\right ) \int 1 \, dx+\frac {1}{4} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^3\right ) \int \csc (c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^3 \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (6 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=\frac {13 a^3 x}{2}-\frac {2 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{8} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=\frac {13 a^3 x}{2}-\frac {25 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {5 a^3 \cot (c+d x)}{d}-\frac {2 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {23 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.71, size = 271, normalized size = 1.55 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (6240 (c+d x)+720 \sin (2 (c+d x))+720 \cos (c+d x)-80 \cos (3 (c+d x))-2624 \tan \left (\frac {1}{2} (c+d x)\right )+2624 \cot \left (\frac {1}{2} (c+d x)\right )-45 \csc ^4\left (\frac {1}{2} (c+d x)\right )+690 \csc ^2\left (\frac {1}{2} (c+d x)\right )+45 \sec ^4\left (\frac {1}{2} (c+d x)\right )-690 \sec ^2\left (\frac {1}{2} (c+d x)\right )+3000 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-3000 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3 \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )-19 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+304 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+6 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )\right )}{960 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 278, normalized size = 1.59 \[ -\frac {360 \, a^{3} \cos \left (d x + c\right )^{7} - 2392 \, a^{3} \cos \left (d x + c\right )^{5} + 3640 \, a^{3} \cos \left (d x + c\right )^{3} - 1560 \, a^{3} \cos \left (d x + c\right ) + 375 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 375 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{7} - 156 \, a^{3} d x \cos \left (d x + c\right )^{4} - 40 \, a^{3} \cos \left (d x + c\right )^{5} + 312 \, a^{3} d x \cos \left (d x + c\right )^{2} + 125 \, a^{3} \cos \left (d x + c\right )^{3} - 156 \, a^{3} d x - 75 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 276, normalized size = 1.58 \[ \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6240 \, {\left (d x + c\right )} a^{3} + 3000 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 2580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {320 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}} - \frac {6850 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 50 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 293, normalized size = 1.67 \[ \frac {5 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}+\frac {5 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d}+\frac {25 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{24 d}+\frac {25 a^{3} \cos \left (d x +c \right )}{8 d}+\frac {25 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{3}}+\frac {4 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}+\frac {4 a^{3} \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}+\frac {5 a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}+\frac {15 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {13 a^{3} x}{2}+\frac {13 a^{3} c}{2 d}-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}-\frac {a^{3} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a^{3} \cot \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 250, normalized size = 1.43 \[ -\frac {20 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 120 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} + 16 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 45 \, a^{3} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.92, size = 408, normalized size = 2.33 \[ \frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {25\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}+\frac {13\,a^3\,\mathrm {atan}\left (\frac {169\,a^6}{\frac {325\,a^6}{4}-169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {325\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {325\,a^6}{4}-169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}+\frac {-10\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+20\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {769\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {373\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+\frac {1744\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {589\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{6}+\frac {402\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {34\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {a^3}{5}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+96\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}-\frac {43\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \]
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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