Optimal. Leaf size=30 \[ -\frac {\csc ^3(c+d x) (a \sin (c+d x)+a)^3}{3 a d} \]
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Rubi [A] time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 37} \[ -\frac {\csc ^3(c+d x) (a \sin (c+d x)+a)^3}{3 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^4 (a+x)^2}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {(a+x)^2}{x^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^3(c+d x) (a+a \sin (c+d x))^3}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 20, normalized size = 0.67 \[ -\frac {a^2 (\csc (c+d x)+1)^3}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 56, normalized size = 1.87 \[ -\frac {3 \, a^{2} \cos \left (d x + c\right )^{2} - 3 \, a^{2} \sin \left (d x + c\right ) - 4 \, a^{2}}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 41, normalized size = 1.37 \[ -\frac {3 \, a^{2} \sin \left (d x + c\right )^{2} + 3 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{3 \, d \sin \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 39, normalized size = 1.30 \[ \frac {a^{2} \left (-\frac {1}{\sin \left (d x +c \right )}-\frac {1}{\sin \left (d x +c \right )^{2}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 41, normalized size = 1.37 \[ -\frac {3 \, a^{2} \sin \left (d x + c\right )^{2} + 3 \, a^{2} \sin \left (d x + c\right ) + a^{2}}{3 \, d \sin \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.90, size = 41, normalized size = 1.37 \[ -\frac {a^2\,{\sin \left (c+d\,x\right )}^2+a^2\,\sin \left (c+d\,x\right )+\frac {a^2}{3}}{d\,{\sin \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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