Optimal. Leaf size=30 \[ -\frac {\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d} \]
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Rubi [A] time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 37} \[ -\frac {\csc ^2(c+d x) (a \sin (c+d x)+a)^2}{2 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^3 (a+x)}{x^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {a+x}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^2(c+d x) (a+a \sin (c+d x))^2}{2 a d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.97 \[ -\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 29, normalized size = 0.97 \[ \frac {2 \, a \sin \left (d x + c\right ) + a}{2 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 24, normalized size = 0.80 \[ -\frac {2 \, a \sin \left (d x + c\right ) + a}{2 \, d \sin \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 27, normalized size = 0.90 \[ \frac {a \left (-\frac {1}{\sin \left (d x +c \right )}-\frac {1}{2 \sin \left (d x +c \right )^{2}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 24, normalized size = 0.80 \[ -\frac {2 \, a \sin \left (d x + c\right ) + a}{2 \, d \sin \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.52, size = 25, normalized size = 0.83 \[ -\frac {\frac {a}{2}+a\,\sin \left (c+d\,x\right )}{d\,{\sin \left (c+d\,x\right )}^2} \]
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 \[ a \left (\int \cos {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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