Optimal. Leaf size=55 \[ -\frac {\csc ^5(c+d x)}{5 a^2 d}+\frac {\csc ^4(c+d x)}{2 a^2 d}-\frac {\csc ^3(c+d x)}{3 a^2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 55, 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, 43} \[ -\frac {\csc ^5(c+d x)}{5 a^2 d}+\frac {\csc ^4(c+d x)}{2 a^2 d}-\frac {\csc ^3(c+d x)}{3 a^2 d} \]
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 (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^6 (a-x)^2}{x^6} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {(a-x)^2}{x^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {a^2}{x^6}-\frac {2 a}{x^5}+\frac {1}{x^4}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^3(c+d x)}{3 a^2 d}+\frac {\csc ^4(c+d x)}{2 a^2 d}-\frac {\csc ^5(c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 38, normalized size = 0.69 \[ \frac {\csc ^5(c+d x) (15 \sin (c+d x)+5 \cos (2 (c+d x))-11)}{30 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 65, normalized size = 1.18 \[ \frac {10 \, \cos \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) - 16}{30 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 36, normalized size = 0.65 \[ -\frac {10 \, \sin \left (d x + c\right )^{2} - 15 \, \sin \left (d x + c\right ) + 6}{30 \, a^{2} d \sin \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 39, normalized size = 0.71 \[ \frac {-\frac {1}{5 \sin \left (d x +c \right )^{5}}+\frac {1}{2 \sin \left (d x +c \right )^{4}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 36, normalized size = 0.65 \[ -\frac {10 \, \sin \left (d x + c\right )^{2} - 15 \, \sin \left (d x + c\right ) + 6}{30 \, a^{2} d \sin \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.94, size = 36, normalized size = 0.65 \[ -\frac {\frac {{\sin \left (c+d\,x\right )}^2}{3}-\frac {\sin \left (c+d\,x\right )}{2}+\frac {1}{5}}{a^2\,d\,{\sin \left (c+d\,x\right )}^5} \]
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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