Optimal. Leaf size=64 \[ -\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}+\frac {\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.09, antiderivative size = 64, 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, 75} \[ -\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}+\frac {\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^4 (a-x)^2 (a+x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^4}-\frac {a^2}{x^3}-\frac {a}{x^2}+\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d}+\frac {\log (\sin (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 48, normalized size = 0.75 \[ \frac {-2 \csc ^3(c+d x)+3 \csc ^2(c+d x)+6 \csc (c+d x)+6 \log (\sin (c+d x))}{6 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 75, normalized size = 1.17 \[ \frac {6 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) - 4}{6 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 62, normalized size = 0.97 \[ \frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {11 \, \sin \left (d x + c\right )^{3} - 6 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 63, normalized size = 0.98 \[ \frac {1}{d a \sin \left (d x +c \right )}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}+\frac {1}{2 a d \sin \left (d x +c \right )^{2}}-\frac {1}{3 a d \sin \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 50, normalized size = 0.78 \[ \frac {\frac {6 \, \log \left (\sin \left (d x + c\right )\right )}{a} + \frac {6 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.95, size = 138, normalized size = 2.16 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{3}\right )}{8\,a\,d} \]
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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