Optimal. Leaf size=17 \[ \frac{a A \cot ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0710912, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3962, 2607, 30} \[ \frac{a A \cot ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3962
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \csc ^2(c+d x) (a+a \csc (c+d x)) (A-A \csc (c+d x)) \, dx &=-\left ((a A) \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx\right )\\ &=-\frac{(a A) \operatorname{Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a A \cot ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0121646, size = 17, normalized size = 1. \[ \frac{a A \cot ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 38, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ( -Aa\cot \left ( dx+c \right ) -Aa \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cot \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02169, size = 57, normalized size = 3.35 \begin{align*} -\frac{\frac{3 \, A a}{\tan \left (d x + c\right )} - \frac{{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.45574, size = 85, normalized size = 5. \begin{align*} -\frac{A a \cos \left (d x + c\right )^{3}}{3 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.14552, size = 54, normalized size = 3.18 \begin{align*} \begin{cases} - \frac{A a \left (- \frac{\cot ^{3}{\left (c + d x \right )}}{3} - \cot{\left (c + d x \right )}\right ) + A a \cot{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (- A \csc{\left (c \right )} + A\right ) \left (a \csc{\left (c \right )} + a\right ) \csc ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38489, size = 20, normalized size = 1.18 \begin{align*} \frac{A a}{3 \, d \tan \left (d x + c\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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