Optimal. Leaf size=73 \[ -\frac {\csc ^6(c+d x)}{6 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {\csc ^2(c+d x)}{2 a^2 d} \]
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Rubi [A] time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2707, 75} \[ -\frac {\csc ^6(c+d x)}{6 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {\csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 75
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^4}{x^7}-\frac {2 a^3}{x^6}+\frac {2 a}{x^4}-\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^6(c+d x)}{6 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 73, normalized size = 1.00 \[ -\frac {\csc ^6(c+d x)}{6 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {\csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 94, normalized size = 1.29 \[ -\frac {15 \, \cos \left (d x + c\right )^{4} - 30 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 10}{30 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 46, normalized size = 0.63 \[ \frac {15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 12 \, \sin \left (d x + c\right ) - 5}{30 \, a^{2} d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 49, normalized size = 0.67 \[ \frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {2}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {2}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 46, normalized size = 0.63 \[ \frac {15 \, \sin \left (d x + c\right )^{4} - 20 \, \sin \left (d x + c\right )^{3} + 12 \, \sin \left (d x + c\right ) - 5}{30 \, a^{2} d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.37, size = 46, normalized size = 0.63 \[ \frac {15\,{\sin \left (c+d\,x\right )}^4-20\,{\sin \left (c+d\,x\right )}^3+12\,\sin \left (c+d\,x\right )-5}{30\,a^2\,d\,{\sin \left (c+d\,x\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cot ^{7}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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