Optimal. Leaf size=126 \[ -\frac{4}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{\csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{6 \csc (c+d x)}{a^3 d}-\frac{10 \log (\sin (c+d x))}{a^3 d}+\frac{10 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.112726, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 44} \[ -\frac{4}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{\csc ^3(c+d x)}{3 a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{6 \csc (c+d x)}{a^3 d}-\frac{10 \log (\sin (c+d x))}{a^3 d}+\frac{10 \log (\sin (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{\cot (c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4}{x^4 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^4}-\frac{3}{a^4 x^3}+\frac{6}{a^5 x^2}-\frac{10}{a^6 x}+\frac{1}{a^4 (a+x)^3}+\frac{4}{a^5 (a+x)^2}+\frac{10}{a^6 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{6 \csc (c+d x)}{a^3 d}+\frac{3 \csc ^2(c+d x)}{2 a^3 d}-\frac{\csc ^3(c+d x)}{3 a^3 d}-\frac{10 \log (\sin (c+d x))}{a^3 d}+\frac{10 \log (1+\sin (c+d x))}{a^3 d}-\frac{1}{2 a d (a+a \sin (c+d x))^2}-\frac{4}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 5.7232, size = 81, normalized size = 0.64 \[ -\frac{\frac{3 (8 \sin (c+d x)+9)}{(\sin (c+d x)+1)^2}+2 \csc ^3(c+d x)-9 \csc ^2(c+d x)+36 \csc (c+d x)+60 \log (\sin (c+d x))-60 \log (\sin (c+d x)+1)}{6 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 118, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{3}d \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-4\,{\frac{1}{{a}^{3}d \left ( 1+\sin \left ( dx+c \right ) \right ) }}+10\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}}-{\frac{1}{3\,{a}^{3}d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{3}{2\,{a}^{3}d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-6\,{\frac{1}{{a}^{3}d\sin \left ( dx+c \right ) }}-10\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984024, size = 153, normalized size = 1.21 \begin{align*} -\frac{\frac{60 \, \sin \left (d x + c\right )^{4} + 90 \, \sin \left (d x + c\right )^{3} + 20 \, \sin \left (d x + c\right )^{2} - 5 \, \sin \left (d x + c\right ) + 2}{a^{3} \sin \left (d x + c\right )^{5} + 2 \, a^{3} \sin \left (d x + c\right )^{4} + a^{3} \sin \left (d x + c\right )^{3}} - \frac{60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{60 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54369, size = 641, normalized size = 5.09 \begin{align*} -\frac{60 \, \cos \left (d x + c\right )^{4} - 140 \, \cos \left (d x + c\right )^{2} + 60 \,{\left (2 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 60 \,{\left (2 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 5 \,{\left (18 \, \cos \left (d x + c\right )^{2} - 17\right )} \sin \left (d x + c\right ) + 82}{6 \,{\left (2 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d +{\left (a^{3} d \cos \left (d x + c\right )^{4} - 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2696, size = 131, normalized size = 1.04 \begin{align*} \frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{60 \, \sin \left (d x + c\right )^{4} + 90 \, \sin \left (d x + c\right )^{3} + 20 \, \sin \left (d x + c\right )^{2} - 5 \, \sin \left (d x + c\right ) + 2}{a^{3}{\left (\sin \left (d x + c\right ) + 1\right )}^{2} \sin \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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