Optimal. Leaf size=97 \[ -\frac {a \cot ^{10}(c+d x)}{10 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^{11}(c+d x)}{11 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2834, 2606, 270, 2607, 14} \[ -\frac {a \cot ^{10}(c+d x)}{10 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^{11}(c+d x)}{11 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2834
Rubi steps
\begin {align*} \int \cot ^7(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^7(c+d x) \csc ^4(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^4 \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 ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int \left (x^7+x^9\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int \left (-x^4+3 x^6-3 x^8+x^{10}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^{10}(c+d x)}{10 d}+\frac {a \csc ^5(c+d x)}{5 d}-\frac {3 a \csc ^7(c+d x)}{7 d}+\frac {a \csc ^9(c+d x)}{3 d}-\frac {a \csc ^{11}(c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 86, normalized size = 0.89 \[ -\frac {a \csc ^4(c+d x) \left (840 \csc ^7(c+d x)+924 \csc ^6(c+d x)-3080 \csc ^5(c+d x)-3465 \csc ^4(c+d x)+3960 \csc ^3(c+d x)+4620 \csc ^2(c+d x)-1848 \csc (c+d x)-2310\right )}{9240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 152, normalized size = 1.57 \[ \frac {1848 \, a \cos \left (d x + c\right )^{6} - 1584 \, a \cos \left (d x + c\right )^{4} + 704 \, a \cos \left (d x + c\right )^{2} + 231 \, {\left (10 \, a \cos \left (d x + c\right )^{6} - 10 \, a \cos \left (d x + c\right )^{4} + 5 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 128 \, a}{9240 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 92, normalized size = 0.95 \[ \frac {2310 \, a \sin \left (d x + c\right )^{7} + 1848 \, a \sin \left (d x + c\right )^{6} - 4620 \, a \sin \left (d x + c\right )^{5} - 3960 \, a \sin \left (d x + c\right )^{4} + 3465 \, a \sin \left (d x + c\right )^{3} + 3080 \, a \sin \left (d x + c\right )^{2} - 924 \, a \sin \left (d x + c\right ) - 840 \, a}{9240 \, d \sin \left (d x + c\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.39, size = 194, normalized size = 2.00 \[ \frac {a \left (-\frac {\cos ^{8}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos ^{8}\left (d x +c \right )}{40 \sin \left (d x +c \right )^{8}}\right )+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {\cos ^{8}\left (d x +c \right )}{33 \sin \left (d x +c \right )^{9}}-\frac {\cos ^{8}\left (d x +c \right )}{231 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{1155 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{1155 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{8}\left (d x +c \right )}{231 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{231}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 92, normalized size = 0.95 \[ \frac {2310 \, a \sin \left (d x + c\right )^{7} + 1848 \, a \sin \left (d x + c\right )^{6} - 4620 \, a \sin \left (d x + c\right )^{5} - 3960 \, a \sin \left (d x + c\right )^{4} + 3465 \, a \sin \left (d x + c\right )^{3} + 3080 \, a \sin \left (d x + c\right )^{2} - 924 \, a \sin \left (d x + c\right ) - 840 \, a}{9240 \, d \sin \left (d x + c\right )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.28, size = 92, normalized size = 0.95 \[ -\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{4}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{2}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{7}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{8}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{3}+\frac {a\,\sin \left (c+d\,x\right )}{10}+\frac {a}{11}}{d\,{\sin \left (c+d\,x\right )}^{11}} \]
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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