Optimal. Leaf size=85 \[ \frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{3 \log (\sin (c+d x))}{a^2 d}-\frac{3 \log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.0874987, antiderivative size = 85, 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} \[ \frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{2 \csc (c+d x)}{a^2 d}+\frac{3 \log (\sin (c+d x))}{a^2 d}-\frac{3 \log (\sin (c+d x)+1)}{a^2 d} \]
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 ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^3}{x^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^3}-\frac{2}{a^3 x^2}+\frac{3}{a^4 x}-\frac{1}{a^3 (a+x)^2}-\frac{3}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{2 \csc (c+d x)}{a^2 d}-\frac{\csc ^2(c+d x)}{2 a^2 d}+\frac{3 \log (\sin (c+d x))}{a^2 d}-\frac{3 \log (1+\sin (c+d x))}{a^2 d}+\frac{1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.188544, size = 61, normalized size = 0.72 \[ \frac{\frac{2}{\sin (c+d x)+1}-\csc ^2(c+d x)+4 \csc (c+d x)+6 \log (\sin (c+d x))-6 \log (\sin (c+d x)+1)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 83, normalized size = 1. \begin{align*}{\frac{1}{d{a}^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-3\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{1}{2\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{1}{d{a}^{2}\sin \left ( dx+c \right ) }}+3\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15811, size = 108, normalized size = 1.27 \begin{align*} \frac{\frac{6 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 1}{a^{2} \sin \left (d x + c\right )^{3} + a^{2} \sin \left (d x + c\right )^{2}} - \frac{6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac{6 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49914, size = 389, normalized size = 4.58 \begin{align*} \frac{6 \, \cos \left (d x + c\right )^{2} + 6 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 6 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \sin \left (d x + c\right ) - 5}{2 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d +{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2036, size = 117, normalized size = 1.38 \begin{align*} \frac{\frac{6 \, \log \left ({\left | -\frac{a}{a \sin \left (d x + c\right ) + a} + 1 \right |}\right )}{a^{2}} + \frac{2}{{\left (a \sin \left (d x + c\right ) + a\right )} a} - \frac{\frac{6 \, a}{a \sin \left (d x + c\right ) + a} - 5}{a^{2}{\left (\frac{a}{a \sin \left (d x + c\right ) + a} - 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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