Optimal. Leaf size=81 \[ -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.12, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2834, 2606, 270, 2607, 30} \[ -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 270
Rule 2606
Rule 2607
Rule 2834
Rubi steps
\begin {align*} \int \cot ^7(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^7(c+d x) \csc ^2(c+d x) \, dx+a \int \cot ^7(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^7 \, dx,x,-\cot (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \operatorname {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot ^8(c+d x)}{8 d}+\frac {a \csc ^3(c+d x)}{3 d}-\frac {3 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)}{9 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 81, normalized size = 1.00 \[ -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {3 a \csc ^7(c+d x)}{7 d}-\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {a \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 139, normalized size = 1.72 \[ -\frac {840 \, a \cos \left (d x + c\right )^{6} - 1008 \, a \cos \left (d x + c\right )^{4} + 576 \, a \cos \left (d x + c\right )^{2} + 315 \, {\left (4 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 128 \, a}{2520 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 92, normalized size = 1.14 \[ \frac {1260 \, a \sin \left (d x + c\right )^{7} + 840 \, a \sin \left (d x + c\right )^{6} - 1890 \, a \sin \left (d x + c\right )^{5} - 1512 \, a \sin \left (d x + c\right )^{4} + 1260 \, a \sin \left (d x + c\right )^{3} + 1080 \, a \sin \left (d x + c\right )^{2} - 315 \, a \sin \left (d x + c\right ) - 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 156, normalized size = 1.93 \[ \frac {-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{8 \sin \left (d x +c \right )^{8}}+a \left (-\frac {\cos ^{8}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {\cos ^{8}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{315 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{315 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{8}\left (d x +c \right )}{63 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 92, normalized size = 1.14 \[ \frac {1260 \, a \sin \left (d x + c\right )^{7} + 840 \, a \sin \left (d x + c\right )^{6} - 1890 \, a \sin \left (d x + c\right )^{5} - 1512 \, a \sin \left (d x + c\right )^{4} + 1260 \, a \sin \left (d x + c\right )^{3} + 1080 \, a \sin \left (d x + c\right )^{2} - 315 \, a \sin \left (d x + c\right ) - 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.23, size = 92, normalized size = 1.14 \[ -\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^7}{2}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{3}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{4}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{2}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {a\,\sin \left (c+d\,x\right )}{8}+\frac {a}{9}}{d\,{\sin \left (c+d\,x\right )}^9} \]
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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