Optimal. Leaf size=47 \[ \frac {\sin ^2(c+d x)}{2 a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {\log (\sin (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {2836, 12, 43} \[ \frac {\sin ^2(c+d x)}{2 a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {\log (\sin (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2 a+\frac {a^2}{x}+x\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\log (\sin (c+d x))}{a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {\sin ^2(c+d x)}{2 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 36, normalized size = 0.77 \[ \frac {\sin ^2(c+d x)-4 \sin (c+d x)+2 \log (\sin (c+d x))}{2 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 36, normalized size = 0.77 \[ -\frac {\cos \left (d x + c\right )^{2} - 2 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, \sin \left (d x + c\right )}{2 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 47, normalized size = 1.00 \[ \frac {\frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{2} - 4 \, a^{2} \sin \left (d x + c\right )}{a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 46, normalized size = 0.98 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{2} d}-\frac {2 \sin \left (d x +c \right )}{a^{2} d}+\frac {\sin ^{2}\left (d x +c \right )}{2 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 39, normalized size = 0.83 \[ \frac {\frac {\sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{a^{2}} + \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.11, size = 120, normalized size = 2.55 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )} \]
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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