Optimal. Leaf size=30 \[ -\frac {\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2707, 74} \[ -\frac {\csc ^2(c+d x) (a \sin (c+d x)+a)^4}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 74
Rule 2707
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x) (a+x)^3}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^2(c+d x) (a+a \sin (c+d x))^4}{2 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 28, normalized size = 0.93 \[ -\frac {a^2 (\sin (c+d x)+1)^4 \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 76, normalized size = 2.53 \[ \frac {2 \, a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 8 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 47, normalized size = 1.57 \[ -\frac {a^{2} {\left (\frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right )\right )}^{2} + 4 \, a^{2} {\left (\frac {1}{\sin \left (d x + c\right )} + \sin \left (d x + c\right )\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 94, normalized size = 3.13 \[ \frac {a^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}-\frac {2 a^{2} \left (\cos ^{4}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {2 a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {4 a^{2} \sin \left (d x +c \right )}{d}-\frac {a^{2} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 53, normalized size = 1.77 \[ -\frac {a^{2} \sin \left (d x + c\right )^{2} + 4 \, a^{2} \sin \left (d x + c\right ) + \frac {4 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.67, size = 56, normalized size = 1.87 \[ -\frac {a^2\,\left (2\,{\sin \left (c+d\,x\right )}^4+8\,{\sin \left (c+d\,x\right )}^3-{\sin \left (c+d\,x\right )}^2+8\,\sin \left (c+d\,x\right )+2\right )}{4\,d\,{\sin \left (c+d\,x\right )}^2} \]
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 \[ a^{2} \left (\int 2 \sin {\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}\, dx + \int \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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