Optimal. Leaf size=17 \[ \frac {a A \cot ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3962, 2607, 30} \[ \frac {a A \cot ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 3962
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*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ \frac {a A \cot ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 36, normalized size = 2.12 \[ -\frac {A a \cos \left (d x + c\right )^{3}}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 15, normalized size = 0.88 \[ \frac {A a}{3 \, d \tan \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.00, size = 38, normalized size = 2.24 \[ \frac {-a A \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )-a A \cot \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 42, normalized size = 2.47 \[ -\frac {\frac {3 \, A a}{\tan \left (d x + c\right )} - \frac {{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 15, normalized size = 0.88 \[ \frac {A\,a\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.41, size = 54, normalized size = 3.18 \[ \begin {cases} \frac {- A a \left (- \frac {\cot ^{3}{\left (c + d x \right )}}{3} - \cot {\left (c + d x \right )}\right ) - A a \cot {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A \csc {\relax (c )} + A\right ) \left (- a \csc {\relax (c )} + a\right ) \csc ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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