Optimal. Leaf size=17 \[ \frac {a A \cos ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.06, 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, 2565, 30} \[ \frac {a A \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2565
Rule 3962
Rubi steps
\begin {align*} \int (a+a \csc (c+d x)) (A-A \csc (c+d x)) \sin ^3(c+d x) \, dx &=-\left ((a A) \int \cos ^2(c+d x) \sin (c+d x) \, dx\right )\\ &=\frac {(a A) \operatorname {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a A \cos ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ \frac {a A \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 15, normalized size = 0.88 \[ \frac {A a \cos \left (d x + c\right )^{3}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 56, normalized size = 3.29 \[ -\frac {2 \, {\left (A a + \frac {3 \, A a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{3 \, d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.88, size = 35, normalized size = 2.06 \[ \frac {-\frac {a A \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}+A \cos \left (d x +c \right ) a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 36, normalized size = 2.12 \[ \frac {{\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} A a + 3 \, A a \cos \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 15, normalized size = 0.88 \[ \frac {A\,a\,{\cos \left (c+d\,x\right )}^3}{3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.97, size = 83, normalized size = 4.88 \[ \begin {cases} - \frac {2 A a \cot ^{3}{\left (c + d x \right )}}{3 d \csc ^{3}{\left (c + d x \right )}} + \frac {A a \cot {\left (c + d x \right )}}{d \csc {\left (c + d x \right )}} - \frac {A a \cot {\left (c + d x \right )}}{d \csc ^{3}{\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x \left (- A \csc {\relax (c )} + A\right ) \left (a \csc {\relax (c )} + a\right )}{\csc ^{3}{\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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