Optimal. Leaf size=55 \[ -\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.07, 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 = {2833, 12, 43} \[ -\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^5 (a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^4 \operatorname {Subst}\left (\int \frac {(a+x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^4 \operatorname {Subst}\left (\int \left (\frac {a^2}{x^5}+\frac {2 a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 1.00 \[ -\frac {a^2 \csc ^4(c+d x)}{4 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 57, normalized size = 1.04 \[ \frac {6 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \, a^{2} \sin \left (d x + c\right ) - 9 \, a^{2}}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 43, normalized size = 0.78 \[ -\frac {6 \, a^{2} \sin \left (d x + c\right )^{2} + 8 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 39, normalized size = 0.71 \[ \frac {a^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {2}{3 \sin \left (d x +c \right )^{3}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 43, normalized size = 0.78 \[ -\frac {6 \, a^{2} \sin \left (d x + c\right )^{2} + 8 \, a^{2} \sin \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.85, size = 43, normalized size = 0.78 \[ -\frac {\frac {a^2\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {2\,a^2\,\sin \left (c+d\,x\right )}{3}+\frac {a^2}{4}}{d\,{\sin \left (c+d\,x\right )}^4} \]
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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