Optimal. Leaf size=96 \[ \frac {4 \csc (c+d x)}{a^3 d}-\frac {2 \csc ^2(c+d x)}{a^3 d}+\frac {\csc ^3(c+d x)}{a^3 d}-\frac {\csc ^4(c+d x)}{4 a^3 d}+\frac {4 \log (\sin (c+d x))}{a^3 d}-\frac {4 \log (1+\sin (c+d x))}{a^3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 96, 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, 90}
\begin {gather*} -\frac {\csc ^4(c+d x)}{4 a^3 d}+\frac {\csc ^3(c+d x)}{a^3 d}-\frac {2 \csc ^2(c+d x)}{a^3 d}+\frac {4 \csc (c+d x)}{a^3 d}+\frac {4 \log (\sin (c+d x))}{a^3 d}-\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 2786
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^2}{x^5 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a}{x^5}-\frac {3}{x^4}+\frac {4}{a x^3}-\frac {4}{a^2 x^2}+\frac {4}{a^3 x}-\frac {4}{a^3 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {4 \csc (c+d x)}{a^3 d}-\frac {2 \csc ^2(c+d x)}{a^3 d}+\frac {\csc ^3(c+d x)}{a^3 d}-\frac {\csc ^4(c+d x)}{4 a^3 d}+\frac {4 \log (\sin (c+d x))}{a^3 d}-\frac {4 \log (1+\sin (c+d x))}{a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 69, normalized size = 0.72 \begin {gather*} \frac {16 \csc (c+d x)-8 \csc ^2(c+d x)+4 \csc ^3(c+d x)-\csc ^4(c+d x)+16 \log (\sin (c+d x))-16 \log (1+\sin (c+d x))}{4 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 67, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )^{3}}-\frac {2}{\sin \left (d x +c \right )^{2}}+\frac {4}{\sin \left (d x +c \right )}+4 \ln \left (\sin \left (d x +c \right )\right )-4 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(67\) |
default | \(\frac {-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {1}{\sin \left (d x +c \right )^{3}}-\frac {2}{\sin \left (d x +c \right )^{2}}+\frac {4}{\sin \left (d x +c \right )}+4 \ln \left (\sin \left (d x +c \right )\right )-4 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(67\) |
risch | \(\frac {4 i \left (-2 i {\mathrm e}^{6 i \left (d x +c \right )}+2 \,{\mathrm e}^{7 i \left (d x +c \right )}+5 i {\mathrm e}^{4 i \left (d x +c \right )}-8 \,{\mathrm e}^{5 i \left (d x +c \right )}-2 i {\mathrm e}^{2 i \left (d x +c \right )}+8 \,{\mathrm e}^{3 i \left (d x +c \right )}-2 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{3} d}+\frac {4 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 75, normalized size = 0.78 \begin {gather*} -\frac {\frac {16 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {16 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac {16 \, \sin \left (d x + c\right )^{3} - 8 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1}{a^{3} \sin \left (d x + c\right )^{4}}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 131, normalized size = 1.36 \begin {gather*} \frac {8 \, \cos \left (d x + c\right )^{2} + 16 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 16 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, {\left (4 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 9}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\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 ^{5}{\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 = 8.40, size = 174, normalized size = 1.81 \begin {gather*} -\frac {\frac {1536 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {768 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {1600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 456 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 108 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {3 \, {\left (a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 152 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{12}}}{192 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.72, size = 171, normalized size = 1.78 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,a^3\,d}-\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^3\,d}+\frac {4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}+\frac {19\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (38\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{4}\right )}{16\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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