Optimal. Leaf size=198 \[ -\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {4 a^4 \cos (c+d x)}{d}-\frac {a^4 \cot ^5(c+d x)}{5 d}-\frac {5 a^4 \cot ^3(c+d x)}{3 d}+\frac {10 a^4 \cot (c+d x)}{d}+\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {15 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {5 a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac {5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {97 a^4 x}{8} \]
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Rubi [A] time = 0.43, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2709, 3767, 8, 3768, 3770, 2638, 2635, 2633} \[ -\frac {4 a^4 \cos ^3(c+d x)}{3 d}-\frac {4 a^4 \cos (c+d x)}{d}-\frac {a^4 \cot ^5(c+d x)}{5 d}-\frac {5 a^4 \cot ^3(c+d x)}{3 d}+\frac {10 a^4 \cot (c+d x)}{d}+\frac {a^4 \sin ^3(c+d x) \cos (c+d x)}{4 d}+\frac {15 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {5 a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac {5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {97 a^4 x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2709
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\int \left (14 a^{10}-14 a^{10} \csc ^2(c+d x)-8 a^{10} \csc ^3(c+d x)+3 a^{10} \csc ^4(c+d x)+4 a^{10} \csc ^5(c+d x)+a^{10} \csc ^6(c+d x)+8 a^{10} \sin (c+d x)-3 a^{10} \sin ^2(c+d x)-4 a^{10} \sin ^3(c+d x)-a^{10} \sin ^4(c+d x)\right ) \, dx}{a^6}\\ &=14 a^4 x+a^4 \int \csc ^6(c+d x) \, dx-a^4 \int \sin ^4(c+d x) \, dx+\left (3 a^4\right ) \int \csc ^4(c+d x) \, dx-\left (3 a^4\right ) \int \sin ^2(c+d x) \, dx+\left (4 a^4\right ) \int \csc ^5(c+d x) \, dx-\left (4 a^4\right ) \int \sin ^3(c+d x) \, dx-\left (8 a^4\right ) \int \csc ^3(c+d x) \, dx+\left (8 a^4\right ) \int \sin (c+d x) \, dx-\left (14 a^4\right ) \int \csc ^2(c+d x) \, dx\\ &=14 a^4 x-\frac {8 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac {3 a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac {1}{4} \left (3 a^4\right ) \int \sin ^2(c+d x) \, dx-\frac {1}{2} \left (3 a^4\right ) \int 1 \, dx+\left (3 a^4\right ) \int \csc ^3(c+d x) \, dx-\left (4 a^4\right ) \int \csc (c+d x) \, dx-\frac {a^4 \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (3 a^4\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}+\frac {\left (4 a^4\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (14 a^4\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=\frac {25 a^4 x}{2}+\frac {4 a^4 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}-\frac {4 a^4 \cos ^3(c+d x)}{3 d}+\frac {10 a^4 \cot (c+d x)}{d}-\frac {5 a^4 \cot ^3(c+d x)}{3 d}-\frac {a^4 \cot ^5(c+d x)}{5 d}+\frac {5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac {15 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}-\frac {1}{8} \left (3 a^4\right ) \int 1 \, dx+\frac {1}{2} \left (3 a^4\right ) \int \csc (c+d x) \, dx\\ &=\frac {97 a^4 x}{8}+\frac {5 a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {4 a^4 \cos (c+d x)}{d}-\frac {4 a^4 \cos ^3(c+d x)}{3 d}+\frac {10 a^4 \cot (c+d x)}{d}-\frac {5 a^4 \cot ^3(c+d x)}{3 d}-\frac {a^4 \cot ^5(c+d x)}{5 d}+\frac {5 a^4 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{d}+\frac {15 a^4 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 1.58, size = 283, normalized size = 1.43 \[ \frac {a^4 (\sin (c+d x)+1)^4 \left (5820 (c+d x)+480 \sin (2 (c+d x))-15 \sin (4 (c+d x))-2400 \cos (c+d x)-160 \cos (3 (c+d x))-2752 \tan \left (\frac {1}{2} (c+d x)\right )+2752 \cot \left (\frac {1}{2} (c+d x)\right )-30 \csc ^4\left (\frac {1}{2} (c+d x)\right )+300 \csc ^2\left (\frac {1}{2} (c+d x)\right )+30 \sec ^4\left (\frac {1}{2} (c+d x)\right )-300 \sec ^2\left (\frac {1}{2} (c+d x)\right )-1200 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1200 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\frac {3}{2} \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+96 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)-\frac {79}{2} \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+632 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)\right )}{480 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 291, normalized size = 1.47 \[ \frac {30 \, a^{4} \cos \left (d x + c\right )^{9} - 345 \, a^{4} \cos \left (d x + c\right )^{7} + 2231 \, a^{4} \cos \left (d x + c\right )^{5} - 3395 \, a^{4} \cos \left (d x + c\right )^{3} + 1455 \, a^{4} \cos \left (d x + c\right ) + 150 \, {\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 150 \, {\left (a^{4} \cos \left (d x + c\right )^{4} - 2 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 5 \, {\left (32 \, a^{4} \cos \left (d x + c\right )^{7} - 291 \, a^{4} d x \cos \left (d x + c\right )^{4} + 32 \, a^{4} \cos \left (d x + c\right )^{5} + 582 \, a^{4} d x \cos \left (d x + c\right )^{2} - 100 \, a^{4} \cos \left (d x + c\right )^{3} - 291 \, a^{4} d x + 60 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.39, size = 339, normalized size = 1.71 \[ \frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 30 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 85 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5820 \, {\left (d x + c\right )} a^{4} - 1200 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 2670 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {40 \, {\left (45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 192 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 69 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 384 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 69 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 320 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 128 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}} + \frac {2740 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2670 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 85 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 293, normalized size = 1.48 \[ \frac {97 a^{4} x}{8}+\frac {7 a^{4} \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{d}+\frac {97 a^{4} c}{8 d}-\frac {a^{4} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d}-\frac {5 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{6 d}-\frac {a^{4} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{4}}-\frac {2 a^{4} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{3}}+\frac {7 a^{4} \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}+\frac {35 a^{4} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4 d}+\frac {105 a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}-\frac {a^{4} \left (\cos ^{7}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {a^{4} \cot \left (d x +c \right )}{d}-\frac {5 a^{4} \cos \left (d x +c \right )}{2 d}-\frac {a^{4} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{4} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {5 a^{4} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 313, normalized size = 1.58 \[ -\frac {40 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} + 15 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} - 120 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{4} + 8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{4} + 30 \, a^{4} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.85, size = 454, normalized size = 2.29 \[ \frac {17\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{16\,d}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {5\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {97\,a^4\,\mathrm {atan}\left (\frac {9409\,a^8}{16\,\left (\frac {485\,a^8}{4}+\frac {9409\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}-\frac {485\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {485\,a^8}{4}+\frac {9409\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}-\frac {-58\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+496\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-\frac {1567\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+962\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-\frac {18437\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{15}+\frac {2296\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {3986\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{5}+\frac {868\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {2312\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}-8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {97\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{5}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+192\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+128\,{\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 {89\,a^4\,\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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