Optimal. Leaf size=82 \[ -\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (\sin (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.08, antiderivative size = 82, 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 = {2833, 12, 44} \[ -\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cot (c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^4}{x^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{x^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {1}{a x^4}-\frac {1}{a^2 x^3}+\frac {1}{a^3 x^2}-\frac {1}{a^4 x}+\frac {1}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{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}+\frac {\log (1+\sin (c+d x))}{a d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 82, normalized size = 1.00 \[ -\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc (c+d x)}{a d}-\frac {\log (\sin (c+d x))}{a d}+\frac {\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 102, normalized size = 1.24 \[ -\frac {6 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 6 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 6 \, \cos \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) - 8}{6 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 67, normalized size = 0.82 \[ \frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {6 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 81, normalized size = 0.99 \[ -\frac {1}{3 a d \sin \left (d x +c \right )^{3}}-\frac {1}{d a \sin \left (d x +c \right )}+\frac {1}{2 a d \sin \left (d x +c \right )^{2}}-\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 65, normalized size = 0.79 \[ \frac {\frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {6 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac {6 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 2}{a \sin \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.59, size = 139, normalized size = 1.70 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a\,d}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{3}\right )}{8\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos {\left (c + d x \right )} \csc ^{4}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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