Optimal. Leaf size=33 \[ -\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ -\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^4 (a+x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {a+x}{x^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ -\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 39, normalized size = 1.18 \[ \frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 26, normalized size = 0.79 \[ -\frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 27, normalized size = 0.82 \[ \frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 26, normalized size = 0.79 \[ -\frac {3 \, a \sin \left (d x + c\right ) + 2 \, a}{6 \, d \sin \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.60, size = 39, normalized size = 1.18 \[ -\frac {\frac {5\,a\,\sin \left (c+d\,x\right )}{16}+\frac {a\,\left (\frac {3\,\sin \left (3\,c+3\,d\,x\right )}{16}+1\right )}{3}}{d\,{\sin \left (c+d\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \cos {\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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