Optimal. Leaf size=108 \[ \frac {3}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\csc ^2(c+d x)}{2 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}+\frac {6 \log (\sin (c+d x))}{a^3 d}-\frac {6 \log (\sin (c+d x)+1)}{a^3 d}+\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 108, 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} \[ \frac {3}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\csc ^2(c+d x)}{2 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}+\frac {6 \log (\sin (c+d x))}{a^3 d}-\frac {6 \log (\sin (c+d x)+1)}{a^3 d}+\frac {1}{2 a d (a \sin (c+d x)+a)^2} \]
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 ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^3}{x^3 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x^3 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^3}-\frac {3}{a^4 x^2}+\frac {6}{a^5 x}-\frac {1}{a^3 (a+x)^3}-\frac {3}{a^4 (a+x)^2}-\frac {6}{a^5 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {3 \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^3 d}+\frac {6 \log (\sin (c+d x))}{a^3 d}-\frac {6 \log (1+\sin (c+d x))}{a^3 d}+\frac {1}{2 a d (a+a \sin (c+d x))^2}+\frac {3}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 71, normalized size = 0.66 \[ \frac {\frac {6}{\sin (c+d x)+1}+\frac {1}{(\sin (c+d x)+1)^2}-\csc ^2(c+d x)+6 \csc (c+d x)+12 \log (\sin (c+d x))-12 \log (\sin (c+d x)+1)}{2 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 196, normalized size = 1.81 \[ -\frac {18 \, \cos \left (d x + c\right )^{2} - 12 \, {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 12 \, {\left (\cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 17}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d - 2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 86, normalized size = 0.80 \[ -\frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {12 \, \sin \left (d x + c\right )^{3} + 18 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1}{{\left (\sin \left (d x + c\right )^{2} + \sin \left (d x + c\right )\right )}^{2} a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 102, normalized size = 0.94 \[ -\frac {1}{2 a^{3} d \sin \left (d x +c \right )^{2}}+\frac {3}{a^{3} d \sin \left (d x +c \right )}+\frac {6 \ln \left (\sin \left (d x +c \right )\right )}{a^{3} d}+\frac {1}{2 d \,a^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {3}{d \,a^{3} \left (1+\sin \left (d x +c \right )\right )}-\frac {6 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 103, normalized size = 0.95 \[ \frac {\frac {12 \, \sin \left (d x + c\right )^{3} + 18 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1}{a^{3} \sin \left (d x + c\right )^{4} + 2 \, a^{3} \sin \left (d x + c\right )^{3} + a^{3} \sin \left (d x + c\right )^{2}} - \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {12 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.68, size = 227, normalized size = 2.10 \[ \frac {6\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}+\frac {-26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {65\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{2}}{d\,\left (4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+24\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+16\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {12\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}+\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \]
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 \[ \frac {\int \frac {\cos {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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