Optimal. Leaf size=90 \[ -\frac {2}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\csc (c+d x)}{a^3 d}-\frac {3 \log (\sin (c+d x))}{a^3 d}+\frac {3 \log (\sin (c+d x)+1)}{a^3 d}-\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 44} \[ -\frac {2}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\csc (c+d x)}{a^3 d}-\frac {3 \log (\sin (c+d x))}{a^3 d}+\frac {3 \log (\sin (c+d x)+1)}{a^3 d}-\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
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 (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2}{x^2 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^2}-\frac {3}{a^4 x}+\frac {1}{a^2 (a+x)^3}+\frac {2}{a^3 (a+x)^2}+\frac {3}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc (c+d x)}{a^3 d}-\frac {3 \log (\sin (c+d x))}{a^3 d}+\frac {3 \log (1+\sin (c+d x))}{a^3 d}-\frac {1}{2 a d (a+a \sin (c+d x))^2}-\frac {2}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 61, normalized size = 0.68 \[ -\frac {\frac {4}{\sin (c+d x)+1}+\frac {1}{(\sin (c+d x)+1)^2}+2 \csc (c+d x)+6 \log (\sin (c+d x))-6 \log (\sin (c+d x)+1)}{2 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 152, normalized size = 1.69 \[ -\frac {6 \, \cos \left (d x + c\right )^{2} + 6 \, {\left (2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 2\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 6 \, {\left (2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, \sin \left (d x + c\right ) - 8}{2 \, {\left (2 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 77, normalized size = 0.86 \[ \frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {6 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right ) + 2}{a^{3} {\left (\sin \left (d x + c\right ) + 1\right )}^{2} \sin \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 86, normalized size = 0.96 \[ -\frac {1}{a^{3} d \sin \left (d x +c \right )}-\frac {3 \ln \left (\sin \left (d x +c \right )\right )}{a^{3} d}-\frac {1}{2 d \,a^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {2}{d \,a^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 91, normalized size = 1.01 \[ -\frac {\frac {6 \, \sin \left (d x + c\right )^{2} + 9 \, \sin \left (d x + c\right ) + 2}{a^{3} \sin \left (d x + c\right )^{3} + 2 \, a^{3} \sin \left (d x + c\right )^{2} + a^{3} \sin \left (d x + c\right )} - \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {6 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.65, size = 193, normalized size = 2.14 \[ \frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1}{d\,\left (2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}+\frac {6\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \]
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 \[ \frac {\int \frac {\cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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