Optimal. Leaf size=115 \[ -\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin (c+d x)}{d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {3 a \csc (c+d x)}{d}+\frac {3 a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2836, 12, 88} \[ -\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin (c+d x)}{d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {3 a \csc (c+d x)}{d}+\frac {3 a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos (c+d x) \cot ^6(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^6 (a-x)^3 (a+x)^4}{x^6} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x^6} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a+\frac {a^7}{x^6}+\frac {a^6}{x^5}-\frac {3 a^5}{x^4}-\frac {3 a^4}{x^3}+\frac {3 a^3}{x^2}+\frac {3 a^2}{x}-x\right ) \, dx,x,a \sin (c+d x)\right )}{a 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 {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {3 a \log (\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 102, normalized size = 0.89 \[ -\frac {a \sin (c+d x)}{d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^3(c+d x)}{d}-\frac {3 a \csc (c+d x)}{d}+\frac {a \left (-2 \sin ^2(c+d x)-\csc ^4(c+d x)+6 \csc ^2(c+d x)+12 \log (\sin (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 157, normalized size = 1.37 \[ \frac {20 \, a \cos \left (d x + c\right )^{6} - 120 \, a \cos \left (d x + c\right )^{4} + 160 \, a \cos \left (d x + c\right )^{2} + 60 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 5 \, {\left (2 \, a \cos \left (d x + c\right )^{6} - 5 \, a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + 4 \, a\right )} \sin \left (d x + c\right ) - 64 \, a}{20 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 103, normalized size = 0.90 \[ -\frac {10 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 20 \, a \sin \left (d x + c\right ) + \frac {137 \, a \sin \left (d x + c\right )^{5} + 60 \, a \sin \left (d x + c\right )^{4} - 30 \, a \sin \left (d x + c\right )^{3} - 20 \, a \sin \left (d x + c\right )^{2} + 5 \, a \sin \left (d x + c\right ) + 4 \, a}{\sin \left (d x + c\right )^{5}}}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 239, normalized size = 2.08 \[ -\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}+\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{2 d}+\frac {3 a \left (\cos ^{4}\left (d x +c \right )\right )}{4 d}+\frac {3 a \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}+\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{3}}-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {16 a \sin \left (d x +c \right )}{5 d}-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{d}-\frac {6 a \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}-\frac {8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 91, normalized size = 0.79 \[ -\frac {10 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left (\sin \left (d x + c\right )\right ) + 20 \, a \sin \left (d x + c\right ) + \frac {60 \, a \sin \left (d x + c\right )^{4} - 30 \, a \sin \left (d x + c\right )^{3} - 20 \, a \sin \left (d x + c\right )^{2} + 5 \, a \sin \left (d x + c\right ) + 4 \, a}{\sin \left (d x + c\right )^{5}}}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.10, size = 281, normalized size = 2.44 \[ \frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {3\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {19\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {102\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+54\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+137\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {39\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}+\frac {161\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a}{5}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )} \]
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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