Optimal. Leaf size=86 \[ \frac {3 \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^3 d}+\frac {5 \log (\sin (c+d x))}{a^3 d}-\frac {5 \log (1+\sin (c+d x))}{a^3 d}+\frac {2}{d \left (a^3+a^3 \sin (c+d x)\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2786, 78}
\begin {gather*} \frac {2}{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 {5 \log (\sin (c+d x))}{a^3 d}-\frac {5 \log (\sin (c+d x)+1)}{a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 2786
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {a-x}{x^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {3}{a^2 x^2}+\frac {5}{a^3 x}-\frac {2}{a^2 (a+x)^2}-\frac {5}{a^3 (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 {5 \log (\sin (c+d x))}{a^3 d}-\frac {5 \log (1+\sin (c+d x))}{a^3 d}+\frac {2}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 61, normalized size = 0.71 \begin {gather*} \frac {6 \csc (c+d x)-\csc ^2(c+d x)+10 \log (\sin (c+d x))-10 \log (1+\sin (c+d x))+\frac {4}{1+\sin (c+d x)}}{2 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 61, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {3}{\sin \left (d x +c \right )}+5 \ln \left (\sin \left (d x +c \right )\right )+\frac {2}{1+\sin \left (d x +c \right )}-5 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(61\) |
default | \(\frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\frac {3}{\sin \left (d x +c \right )}+5 \ln \left (\sin \left (d x +c \right )\right )+\frac {2}{1+\sin \left (d x +c \right )}-5 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(61\) |
risch | \(\frac {2 i \left (5 i {\mathrm e}^{4 i \left (d x +c \right )}+5 \,{\mathrm e}^{5 i \left (d x +c \right )}-5 i {\mathrm e}^{2 i \left (d x +c \right )}-8 \,{\mathrm e}^{3 i \left (d x +c \right )}+5 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} d \,a^{3}}-\frac {10 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{3} d}+\frac {5 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(137\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 80, normalized size = 0.93 \begin {gather*} \frac {\frac {10 \, \sin \left (d x + c\right )^{2} + 5 \, \sin \left (d x + c\right ) - 1}{a^{3} \sin \left (d x + c\right )^{3} + a^{3} \sin \left (d x + c\right )^{2}} - \frac {10 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {10 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 147, normalized size = 1.71 \begin {gather*} \frac {10 \, \cos \left (d x + c\right )^{2} + 10 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 10 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 5 \, \sin \left (d x + c\right ) - 9}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\cot ^{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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.56, size = 154, normalized size = 1.79 \begin {gather*} -\frac {\frac {80 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {40 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 53 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2} a^{3}} + \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.70, size = 169, normalized size = 1.97 \begin {gather*} \frac {5\,\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 {10\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}+\frac {-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+5\,\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 )}^4+8\,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 {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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