Optimal. Leaf size=113 \[ -\frac {a \cot ^{12}(c+d x)}{12 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^{13}(c+d x)}{13 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {a \csc ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.13, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2834, 2606, 270, 2607, 266, 43} \[ -\frac {a \cot ^{12}(c+d x)}{12 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^{13}(c+d x)}{13 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {a \csc ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 270
Rule 2606
Rule 2607
Rule 2834
Rubi steps
\begin {align*} \int \cot ^7(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^7(c+d x) \csc ^6(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^7(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^6 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int x^7 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int x^3 (1+x)^2 \, dx,x,\cot ^2(c+d x)\right )}{2 d}-\frac {a \operatorname {Subst}\left (\int \left (-x^6+3 x^8-3 x^{10}+x^{12}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{13}(c+d x)}{13 d}-\frac {a \operatorname {Subst}\left (\int \left (x^3+2 x^4+x^5\right ) \, dx,x,\cot ^2(c+d x)\right )}{2 d}\\ &=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{5 d}-\frac {a \cot ^{12}(c+d x)}{12 d}+\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{3 d}+\frac {3 a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{13}(c+d x)}{13 d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 86, normalized size = 0.76 \[ -\frac {a \csc ^{13}(c+d x) (3003 \sin (c+d x)+24024 \sin (3 (c+d x))+10010 \sin (5 (c+d x))+5005 \sin (7 (c+d x))+70460 \cos (2 (c+d x))+28600 \cos (4 (c+d x))+8580 \cos (6 (c+d x))+40200)}{1921920 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 161, normalized size = 1.42 \[ -\frac {17160 \, a \cos \left (d x + c\right )^{6} - 11440 \, a \cos \left (d x + c\right )^{4} + 4160 \, a \cos \left (d x + c\right )^{2} + 1001 \, {\left (20 \, a \cos \left (d x + c\right )^{6} - 15 \, a \cos \left (d x + c\right )^{4} + 6 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 640 \, a}{120120 \, {\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 92, normalized size = 0.81 \[ \frac {20020 \, a \sin \left (d x + c\right )^{7} + 17160 \, a \sin \left (d x + c\right )^{6} - 45045 \, a \sin \left (d x + c\right )^{5} - 40040 \, a \sin \left (d x + c\right )^{4} + 36036 \, a \sin \left (d x + c\right )^{3} + 32760 \, a \sin \left (d x + c\right )^{2} - 10010 \, a \sin \left (d x + c\right ) - 9240 \, a}{120120 \, d \sin \left (d x + c\right )^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.39, size = 230, normalized size = 2.04 \[ \frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{12 \sin \left (d x +c \right )^{12}}-\frac {\cos ^{8}\left (d x +c \right )}{30 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{8}\left (d x +c \right )}{120 \sin \left (d x +c \right )^{8}}\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{13 \sin \left (d x +c \right )^{13}}-\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{143 \sin \left (d x +c \right )^{11}}-\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{429 \sin \left (d x +c \right )^{9}}-\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{3003 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{3003 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{3003 \sin \left (d x +c \right )^{3}}+\frac {5 \left (\cos ^{8}\left (d x +c \right )\right )}{3003 \sin \left (d x +c \right )}+\frac {5 \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{3003}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 92, normalized size = 0.81 \[ \frac {20020 \, a \sin \left (d x + c\right )^{7} + 17160 \, a \sin \left (d x + c\right )^{6} - 45045 \, a \sin \left (d x + c\right )^{5} - 40040 \, a \sin \left (d x + c\right )^{4} + 36036 \, a \sin \left (d x + c\right )^{3} + 32760 \, a \sin \left (d x + c\right )^{2} - 10010 \, a \sin \left (d x + c\right ) - 9240 \, a}{120120 \, d \sin \left (d x + c\right )^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.35, size = 92, normalized size = 0.81 \[ -\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{6}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{7}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{3}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{10}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{11}+\frac {a\,\sin \left (c+d\,x\right )}{12}+\frac {a}{13}}{d\,{\sin \left (c+d\,x\right )}^{13}} \]
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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