Optimal. Leaf size=162 \[ -\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-a^2 x \]
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Rubi [A] time = 0.23, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 3473, 8, 2611, 3770, 2607, 30} \[ -\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-a^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3473
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x)+2 a^2 \cot ^6(c+d x) \csc (c+d x)+a^2 \cot ^6(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc (c+d x) \, dx\\ &=-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}-a^2 \int \cot ^4(c+d x) \, dx-\frac {1}{3} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac {a^2 \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}+a^2 \int \cot ^2(c+d x) \, dx+\frac {1}{4} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx\\ &=-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}-\frac {1}{8} \left (5 a^2\right ) \int \csc (c+d x) \, dx-a^2 \int 1 \, dx\\ &=-a^2 x+\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 1.09, size = 262, normalized size = 1.62 \[ \frac {a^2 \left (9344 \tan \left (\frac {1}{2} (c+d x)\right )-9344 \cot \left (\frac {1}{2} (c+d x)\right )-4620 \csc ^2\left (\frac {1}{2} (c+d x)\right )+70 \sec ^6\left (\frac {1}{2} (c+d x)\right )-840 \sec ^4\left (\frac {1}{2} (c+d x)\right )+4620 \sec ^2\left (\frac {1}{2} (c+d x)\right )-8400 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+8400 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\frac {15}{2} \sin (c+d x) \csc ^8\left (\frac {1}{2} (c+d x)\right )+(33 \sin (c+d x)-70) \csc ^6\left (\frac {1}{2} (c+d x)\right )+(289 \sin (c+d x)+840) \csc ^4\left (\frac {1}{2} (c+d x)\right )-4624 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+15 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )-66 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )-13440 c-13440 d x\right )}{13440 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 323, normalized size = 1.99 \[ -\frac {2336 \, a^{2} \cos \left (d x + c\right )^{7} - 6496 \, a^{2} \cos \left (d x + c\right )^{5} + 5600 \, a^{2} \cos \left (d x + c\right )^{3} - 1680 \, a^{2} \cos \left (d x + c\right ) - 525 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 525 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (24 \, a^{2} d x \cos \left (d x + c\right )^{6} - 72 \, a^{2} d x \cos \left (d x + c\right )^{4} - 33 \, a^{2} \cos \left (d x + c\right )^{5} + 72 \, a^{2} d x \cos \left (d x + c\right )^{2} + 40 \, a^{2} \cos \left (d x + c\right )^{3} - 24 \, a^{2} d x - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 270, normalized size = 1.67 \[ \frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 665 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3150 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13440 \, {\left (d x + c\right )} a^{2} - 8400 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 8715 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {21780 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8715 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3150 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 665 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 229, normalized size = 1.41 \[ -\frac {a^{2} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a^{2} \cot \left (d x +c \right )}{d}-a^{2} x -\frac {a^{2} c}{d}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{6}}+\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{12 d \sin \left (d x +c \right )^{4}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d}-\frac {5 a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{24 d}-\frac {5 a^{2} \cos \left (d x +c \right )}{8 d}-\frac {5 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 154, normalized size = 0.95 \[ -\frac {112 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 35 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {240 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.84, size = 351, normalized size = 2.17 \[ \frac {19\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {15\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}-\frac {19\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {2\,a^2\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {5\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}-\frac {83\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {83\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \]
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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