Optimal. Leaf size=47 \[ -\frac {\csc (c+d x)}{a^3 d}-\frac {3 \log (\sin (c+d x))}{a^3 d}+\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.11, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac {\csc (c+d x)}{a^3 d}-\frac {3 \log (\sin (c+d x))}{a^3 d}+\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \cot ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a-x)^2}{x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a}{x^2}-\frac {3}{x}+\frac {4}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {\csc (c+d x)}{a^3 d}-\frac {3 \log (\sin (c+d x))}{a^3 d}+\frac {4 \log (1+\sin (c+d x))}{a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 35, normalized size = 0.74 \[ -\frac {\csc (c+d x)+3 \log (\sin (c+d x))-4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 52, normalized size = 1.11 \[ -\frac {3 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 4 \, \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 1}{a^{3} d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 101, normalized size = 2.15 \[ -\frac {\frac {2 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac {16 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac {6 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 50, normalized size = 1.06 \[ -\frac {1}{a^{3} d \sin \left (d x +c \right )}-\frac {3 \ln \left (\sin \left (d x +c \right )\right )}{a^{3} d}+\frac {4 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 44, normalized size = 0.94 \[ \frac {\frac {4 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {3 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac {1}{a^{3} \sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.98, size = 71, normalized size = 1.51 \[ -\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )+\frac {\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^3\,d} \]
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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