Optimal. Leaf size=55 \[ -\frac {\csc ^6(c+d x)}{6 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^4(c+d x)}{4 a^2 d} \]
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Rubi [A] time = 0.10, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 43} \[ -\frac {\csc ^6(c+d x)}{6 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {\csc ^4(c+d x)}{4 a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^7 (a-x)^2}{x^7} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {(a-x)^2}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (\frac {a^2}{x^7}-\frac {2 a}{x^6}+\frac {1}{x^5}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^4(c+d x)}{4 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 = 0.08, size = 38, normalized size = 0.69 \[ -\frac {\csc ^4(c+d x) \left (10 \csc ^2(c+d x)-24 \csc (c+d x)+15\right )}{60 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 72, normalized size = 1.31 \[ -\frac {15 \, \cos \left (d x + c\right )^{2} + 24 \, \sin \left (d x + c\right ) - 25}{60 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 36, normalized size = 0.65 \[ -\frac {15 \, \sin \left (d x + c\right )^{2} - 24 \, \sin \left (d x + c\right ) + 10}{60 \, a^{2} d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 39, normalized size = 0.71 \[ \frac {-\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {2}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 36, normalized size = 0.65 \[ -\frac {15 \, \sin \left (d x + c\right )^{2} - 24 \, \sin \left (d x + c\right ) + 10}{60 \, a^{2} d \sin \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.95, size = 36, normalized size = 0.65 \[ -\frac {\frac {{\sin \left (c+d\,x\right )}^2}{4}-\frac {2\,\sin \left (c+d\,x\right )}{5}+\frac {1}{6}}{a^2\,d\,{\sin \left (c+d\,x\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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