Optimal. Leaf size=97 \[ -\frac {a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{10}(c+d x)}{10 d}+\frac {2 a \csc ^9(c+d x)}{9 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.08, antiderivative size = 97, 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} \[ -\frac {a \csc ^{11}(c+d x)}{11 d}-\frac {a \csc ^{10}(c+d x)}{10 d}+\frac {2 a \csc ^9(c+d x)}{9 d}+\frac {a \csc ^8(c+d x)}{4 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cot ^5(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^{12} (a-x)^2 (a+x)^3}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^3}{x^{12}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {a^5}{x^{12}}+\frac {a^4}{x^{11}}-\frac {2 a^3}{x^{10}}-\frac {2 a^2}{x^9}+\frac {a}{x^8}+\frac {1}{x^7}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^8(c+d x)}{4 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^{10}(c+d x)}{10 d}-\frac {a \csc ^{11}(c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 88, normalized size = 0.91 \[ -\frac {a \csc ^{11}(c+d x)}{11 d}+\frac {2 a \csc ^9(c+d x)}{9 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \left (6 \csc ^{10}(c+d x)-15 \csc ^8(c+d x)+10 \csc ^6(c+d x)\right )}{60 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 128, normalized size = 1.32 \[ \frac {1980 \, a \cos \left (d x + c\right )^{4} - 880 \, a \cos \left (d x + c\right )^{2} + 231 \, {\left (10 \, a \cos \left (d x + c\right )^{4} - 5 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 160 \, a}{13860 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 70, normalized size = 0.72 \[ -\frac {2310 \, a \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, a \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, a \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 202, normalized size = 2.08 \[ \frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{6}\left (d x +c \right )}{20 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{6}\left (d x +c \right )}{60 \sin \left (d x +c \right )^{6}}\right )+a \left (-\frac {\cos ^{6}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {5 \left (\cos ^{6}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {5 \left (\cos ^{6}\left (d x +c \right )\right )}{231 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{6}\left (d x +c \right )}{231 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{693 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{231 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{231}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 70, normalized size = 0.72 \[ -\frac {2310 \, a \sin \left (d x + c\right )^{5} + 1980 \, a \sin \left (d x + c\right )^{4} - 3465 \, a \sin \left (d x + c\right )^{3} - 3080 \, a \sin \left (d x + c\right )^{2} + 1386 \, a \sin \left (d x + c\right ) + 1260 \, a}{13860 \, d \sin \left (d x + c\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.98, size = 70, normalized size = 0.72 \[ -\frac {\frac {a\,{\sin \left (c+d\,x\right )}^5}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{4}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^2}{9}+\frac {a\,\sin \left (c+d\,x\right )}{10}+\frac {a}{11}}{d\,{\sin \left (c+d\,x\right )}^{11}} \]
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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