Optimal. Leaf size=115 \[ -\frac {a \sin (c+d x)}{d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^4(c+d x)}{4 d}+\frac {a \csc ^3(c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {3 a \csc (c+d x)}{d}-\frac {a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2707, 88} \[ -\frac {a \sin (c+d x)}{d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^4(c+d x)}{4 d}+\frac {a \csc ^3(c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {3 a \csc (c+d x)}{d}-\frac {a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2707
Rubi steps
\begin {align*} \int \cot ^7(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+\frac {a^7}{x^7}+\frac {a^6}{x^6}-\frac {3 a^5}{x^5}-\frac {3 a^4}{x^4}+\frac {3 a^3}{x^3}+\frac {3 a^2}{x^2}-\frac {a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {3 a \csc (c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 111, normalized size = 0.97 \[ -\frac {a \sin (c+d x)}{d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^3(c+d x)}{d}-\frac {3 a \csc (c+d x)}{d}-\frac {a \left (2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 158, normalized size = 1.37 \[ \frac {90 \, a \cos \left (d x + c\right )^{4} - 135 \, a \cos \left (d x + c\right )^{2} - 60 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 12 \, {\left (5 \, a \cos \left (d x + c\right )^{6} - 30 \, a \cos \left (d x + c\right )^{4} + 40 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) + 55 \, a}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.46, size = 104, normalized size = 0.90 \[ -\frac {60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a \sin \left (d x + c\right ) - \frac {147 \, a \sin \left (d x + c\right )^{6} - 180 \, a \sin \left (d x + c\right )^{5} - 90 \, a \sin \left (d x + c\right )^{4} + 60 \, a \sin \left (d x + c\right )^{3} + 45 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right ) - 10 \, a}{\sin \left (d x + c\right )^{6}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 195, normalized size = 1.70 \[ -\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}+\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{3}}-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {16 a \sin \left (d x +c \right )}{5 d}-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{d}-\frac {6 \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{5 d}-\frac {8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{5 d}-\frac {a \left (\cot ^{6}\left (d x +c \right )\right )}{6 d}+\frac {a \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 91, normalized size = 0.79 \[ -\frac {60 \, a \log \left (\sin \left (d x + c\right )\right ) + 60 \, a \sin \left (d x + c\right ) + \frac {180 \, a \sin \left (d x + c\right )^{5} + 90 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 45 \, a \sin \left (d x + c\right )^{2} + 12 \, a \sin \left (d x + c\right ) + 10 \, a}{\sin \left (d x + c\right )^{6}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.37, size = 267, normalized size = 2.32 \[ \frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}-\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {19\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,\left (1920\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-1920\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{1920\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {51\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128}+\frac {35\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\frac {25\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{80}-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{384}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{160}+\frac {a}{384}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \sin {\left (c + d x \right )} \cot ^{7}{\left (c + d x \right )}\, dx + \int \cot ^{7}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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