Optimal. Leaf size=129 \[ -\frac{a \csc ^{14}(c+d x)}{14 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{a \csc ^{12}(c+d x)}{4 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{3 a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^8(c+d x)}{8 d}+\frac{a \csc ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.098505, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac{a \csc ^{14}(c+d x)}{14 d}-\frac{a \csc ^{13}(c+d x)}{13 d}+\frac{a \csc ^{12}(c+d x)}{4 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}-\frac{3 a \csc ^{10}(c+d x)}{10 d}-\frac{a \csc ^9(c+d x)}{3 d}+\frac{a \csc ^8(c+d x)}{8 d}+\frac{a \csc ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \cot ^7(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^{15} (a-x)^3 (a+x)^4}{x^{15}} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \frac{(a-x)^3 (a+x)^4}{x^{15}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \left (\frac{a^7}{x^{15}}+\frac{a^6}{x^{14}}-\frac{3 a^5}{x^{13}}-\frac{3 a^4}{x^{12}}+\frac{3 a^3}{x^{11}}+\frac{3 a^2}{x^{10}}-\frac{a}{x^9}-\frac{1}{x^8}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a \csc ^7(c+d x)}{7 d}+\frac{a \csc ^8(c+d x)}{8 d}-\frac{a \csc ^9(c+d x)}{3 d}-\frac{3 a \csc ^{10}(c+d x)}{10 d}+\frac{3 a \csc ^{11}(c+d x)}{11 d}+\frac{a \csc ^{12}(c+d x)}{4 d}-\frac{a \csc ^{13}(c+d x)}{13 d}-\frac{a \csc ^{14}(c+d x)}{14 d}\\ \end{align*}
Mathematica [A] time = 0.233332, size = 86, normalized size = 0.67 \[ -\frac{a \csc ^{14}(c+d x) (9940 \sin (c+d x)+41860 \sin (3 (c+d x))+20020 \sin (5 (c+d x))+8580 \sin (7 (c+d x))+129129 \cos (2 (c+d x))+54054 \cos (4 (c+d x))+15015 \cos (6 (c+d x))+76362)}{3843840 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.063, size = 248, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{13\, \left ( \sin \left ( dx+c \right ) \right ) ^{13}}}-{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{143\, \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{429\, \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3003\, \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3003\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3003\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{3003\,\sin \left ( dx+c \right ) }}+{\frac{5\,\sin \left ( dx+c \right ) }{3003} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) +a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{14\, \left ( \sin \left ( dx+c \right ) \right ) ^{14}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{28\, \left ( \sin \left ( dx+c \right ) \right ) ^{12}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{70\, \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{280\, \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01359, size = 124, normalized size = 0.96 \begin{align*} \frac{17160 \, a \sin \left (d x + c\right )^{7} + 15015 \, a \sin \left (d x + c\right )^{6} - 40040 \, a \sin \left (d x + c\right )^{5} - 36036 \, a \sin \left (d x + c\right )^{4} + 32760 \, a \sin \left (d x + c\right )^{3} + 30030 \, a \sin \left (d x + c\right )^{2} - 9240 \, a \sin \left (d x + c\right ) - 8580 \, a}{120120 \, d \sin \left (d x + c\right )^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27976, size = 460, normalized size = 3.57 \begin{align*} \frac{15015 \, a \cos \left (d x + c\right )^{6} - 9009 \, a \cos \left (d x + c\right )^{4} + 3003 \, a \cos \left (d x + c\right )^{2} + 40 \,{\left (429 \, a \cos \left (d x + c\right )^{6} - 286 \, a \cos \left (d x + c\right )^{4} + 104 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 429 \, a}{120120 \,{\left (d \cos \left (d x + c\right )^{14} - 7 \, d \cos \left (d x + c\right )^{12} + 21 \, d \cos \left (d x + c\right )^{10} - 35 \, d \cos \left (d x + c\right )^{8} + 35 \, d \cos \left (d x + c\right )^{6} - 21 \, d \cos \left (d x + c\right )^{4} + 7 \, d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37425, size = 124, normalized size = 0.96 \begin{align*} \frac{17160 \, a \sin \left (d x + c\right )^{7} + 15015 \, a \sin \left (d x + c\right )^{6} - 40040 \, a \sin \left (d x + c\right )^{5} - 36036 \, a \sin \left (d x + c\right )^{4} + 32760 \, a \sin \left (d x + c\right )^{3} + 30030 \, a \sin \left (d x + c\right )^{2} - 9240 \, a \sin \left (d x + c\right ) - 8580 \, a}{120120 \, d \sin \left (d x + c\right )^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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