Optimal. Leaf size=65 \[ -\frac {\csc ^2(c+d x)}{2 a^2 d}+\frac {2 \csc (c+d x)}{a^2 d}+\frac {2 \log (\sin (c+d x))}{a^2 d}-\frac {2 \log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2707, 77} \[ -\frac {\csc ^2(c+d x)}{2 a^2 d}+\frac {2 \csc (c+d x)}{a^2 d}+\frac {2 \log (\sin (c+d x))}{a^2 d}-\frac {2 \log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{x^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^3}-\frac {2}{a x^2}+\frac {2}{a^2 x}-\frac {2}{a^2 (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 {2 \log (\sin (c+d x))}{a^2 d}-\frac {2 \log (1+\sin (c+d x))}{a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 49, normalized size = 0.75 \[ \frac {-\csc ^2(c+d x)+4 \csc (c+d x)+4 \log (\sin (c+d x))-4 \log (\sin (c+d x)+1)}{2 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 76, normalized size = 1.17 \[ \frac {4 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, \sin \left (d x + c\right ) + 1}{2 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 115, normalized size = 1.77 \[ -\frac {\frac {32 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {16 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} + \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 66, normalized size = 1.02 \[ -\frac {1}{2 a^{2} d \sin \left (d x +c \right )^{2}}+\frac {2}{a^{2} d \sin \left (d x +c \right )}+\frac {2 \ln \left (\sin \left (d x +c \right )\right )}{a^{2} d}-\frac {2 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 55, normalized size = 0.85 \[ -\frac {\frac {4 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {4 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}} - \frac {4 \, \sin \left (d x + c\right ) - 1}{a^{2} \sin \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.53, size = 103, normalized size = 1.58 \[ \frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}-\frac {4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^2\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^2\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{8}\right )}{a^2\,d} \]
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 \[ \frac {\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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