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 = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, 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 12
Rule 88
Rule 2836
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*}
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Mathematica [A] time = 0.24, 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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fricas [A] time = 0.57, size = 166, normalized size = 1.29 \[ \frac {15015 \, a \cos \left (d x + c\right )^{6} - 9009 \, a \cos \left (d x + c\right )^{4} + 3003 \, a \cos \left (d x + c\right )^{2} + 40 \, {\left (429 \, a \cos \left (d x + c\right )^{6} - 286 \, a \cos \left (d x + c\right )^{4} + 104 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right ) - 429 \, a}{120120 \, {\left (d \cos \left (d x + c\right )^{14} - 7 \, d \cos \left (d x + c\right )^{12} + 21 \, d \cos \left (d x + c\right )^{10} - 35 \, d \cos \left (d x + c\right )^{8} + 35 \, d \cos \left (d x + c\right )^{6} - 21 \, d \cos \left (d x + c\right )^{4} + 7 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 92, normalized size = 0.71 \[ \frac {17160 \, a \sin \left (d x + c\right )^{7} + 15015 \, a \sin \left (d x + c\right )^{6} - 40040 \, a \sin \left (d x + c\right )^{5} - 36036 \, a \sin \left (d x + c\right )^{4} + 32760 \, a \sin \left (d x + c\right )^{3} + 30030 \, a \sin \left (d x + c\right )^{2} - 9240 \, a \sin \left (d x + c\right ) - 8580 \, a}{120120 \, d \sin \left (d x + c\right )^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 248, normalized size = 1.92 \[ \frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{13 \sin \left (d x +c \right )^{13}}-\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{143 \sin \left (d x +c \right )^{11}}-\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{429 \sin \left (d x +c \right )^{9}}-\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{3003 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{3003 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{3003 \sin \left (d x +c \right )^{3}}+\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{3003 \sin \left (d x +c \right )}+\frac {5 \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{3003}\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{14 \sin \left (d x +c \right )^{14}}-\frac {\cos ^{8}\left (d x +c \right )}{28 \sin \left (d x +c \right )^{12}}-\frac {\cos ^{8}\left (d x +c \right )}{70 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{8}\left (d x +c \right )}{280 \sin \left (d x +c \right )^{8}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 92, normalized size = 0.71 \[ \frac {17160 \, a \sin \left (d x + c\right )^{7} + 15015 \, a \sin \left (d x + c\right )^{6} - 40040 \, a \sin \left (d x + c\right )^{5} - 36036 \, a \sin \left (d x + c\right )^{4} + 32760 \, a \sin \left (d x + c\right )^{3} + 30030 \, a \sin \left (d x + c\right )^{2} - 9240 \, a \sin \left (d x + c\right ) - 8580 \, a}{120120 \, d \sin \left (d x + c\right )^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.24, size = 92, normalized size = 0.71 \[ -\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{3}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{10}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{11}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{4}+\frac {a\,\sin \left (c+d\,x\right )}{13}+\frac {a}{14}}{d\,{\sin \left (c+d\,x\right )}^{14}} \]
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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