Optimal. Leaf size=157 \[ \frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot (c+d x)}{d}-\frac {25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x \]
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Rubi [A] time = 0.23, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2872, 3770, 3767, 8, 3768, 2638} \[ \frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot (c+d x)}{d}-\frac {25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2638
Rule 2872
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\int \left (-2 a^8+2 a^8 \csc (c+d x)+6 a^8 \csc ^2(c+d x)-6 a^8 \csc ^4(c+d x)-2 a^8 \csc ^5(c+d x)+2 a^8 \csc ^6(c+d x)+a^8 \csc ^7(c+d x)-a^8 \sin (c+d x)\right ) \, dx}{a^6}\\ &=-2 a^2 x+a^2 \int \csc ^7(c+d x) \, dx-a^2 \int \sin (c+d x) \, dx+\left (2 a^2\right ) \int \csc (c+d x) \, dx-\left (2 a^2\right ) \int \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \csc ^6(c+d x) \, dx+\left (6 a^2\right ) \int \csc ^2(c+d x) \, dx-\left (6 a^2\right ) \int \csc ^4(c+d x) \, dx\\ &=-2 a^2 x-\frac {2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^2 \cos (c+d x)}{d}+\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{2 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{6} \left (5 a^2\right ) \int \csc ^5(c+d x) \, dx-\frac {1}{2} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (6 a^2\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac {\left (6 a^2\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-2 a^2 x-\frac {2 a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{8} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{4} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=-2 a^2 x-\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{16} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=-2 a^2 x-\frac {25 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 1.61, size = 270, normalized size = 1.72 \[ -\frac {a^2 \sin (c+d x) (\sin (c+d x)+1)^2 \left (-1920 \cot (c+d x)+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (5 \csc (c+d x)+12)-2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (15 \csc (c+d x)+82)+\csc ^2\left (\frac {1}{2} (c+d x)\right ) (1472-210 \csc (c+d x))-2 (327 \cos (c+d x)+92 \cos (2 (c+d x))+241) \sec ^6\left (\frac {1}{2} (c+d x)\right )-320 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^7(c+d x)+480 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)+840 \sin ^2\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+120 \csc (c+d x) \left (32 (c+d x)-25 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+25 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{1920 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 303, normalized size = 1.93 \[ -\frac {960 \, a^{2} d x \cos \left (d x + c\right )^{6} - 480 \, a^{2} \cos \left (d x + c\right )^{7} - 2880 \, a^{2} d x \cos \left (d x + c\right )^{4} + 1650 \, a^{2} \cos \left (d x + c\right )^{5} + 2880 \, a^{2} d x \cos \left (d x + c\right )^{2} - 2000 \, a^{2} \cos \left (d x + c\right )^{3} - 960 \, a^{2} d x + 750 \, a^{2} \cos \left (d x + c\right ) + 375 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 375 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (23 \, a^{2} \cos \left (d x + c\right )^{5} - 35 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 259, normalized size = 1.65 \[ \frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3840 \, {\left (d x + c\right )} a^{2} + 3000 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 2640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3840 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {7350 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 205, normalized size = 1.31 \[ -\frac {5 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{4}}+\frac {5 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}+\frac {5 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d}+\frac {25 a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{48 d}+\frac {25 a^{2} \cos \left (d x +c \right )}{16 d}+\frac {25 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {2 a^{2} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {2 a^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {2 a^{2} \cot \left (d x +c \right )}{d}-2 a^{2} x -\frac {2 a^{2} c}{d}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 220, normalized size = 1.40 \[ -\frac {64 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 5 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, a^{2} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.01, size = 657, normalized size = 4.18 \[ \frac {5\,a^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-5\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+24\,a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-24\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-10\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-256\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-270\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+2360\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-255\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4095\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-2360\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+270\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+256\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+10\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3000\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+3000\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+7680\,a^2\,\mathrm {atan}\left (\frac {32\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-25\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+32\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+7680\,a^2\,\mathrm {atan}\left (\frac {32\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-25\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+32\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )} \]
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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