Optimal. Leaf size=93 \[ \frac {\cot ^3(c+d x)}{a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}-\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d} \]
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Rubi [A] time = 0.20, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2869, 2757, 3767, 8, 3768, 3770} \[ \frac {\cot ^3(c+d x)}{a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}-\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2757
Rule 2869
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \csc ^5(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \csc ^2(c+d x)+3 a^3 \csc ^3(c+d x)-3 a^3 \csc ^4(c+d x)+a^3 \csc ^5(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \csc ^2(c+d x) \, dx}{a^3}+\frac {\int \csc ^5(c+d x) \, dx}{a^3}+\frac {3 \int \csc ^3(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^4(c+d x) \, dx}{a^3}\\ &=-\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a^3}+\frac {3 \int \csc (c+d x) \, dx}{2 a^3}+\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}\\ &=-\frac {3 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}+\frac {3 \int \csc (c+d x) \, dx}{8 a^3}\\ &=-\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac {4 \cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{a^3 d}-\frac {15 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}\\ \end {align*}
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Mathematica [A] time = 2.10, size = 125, normalized size = 1.34 \[ -\frac {\csc ^4(c+d x) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (-56 \sin (2 (c+d x))+46 \cos (c+d x)+6 (8 \sin (c+d x)-5) \cos (3 (c+d x))+120 \sin ^4(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{64 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 149, normalized size = 1.60 \[ \frac {30 \, \cos \left (d x + c\right )^{3} - 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 34 \, \cos \left (d x + c\right )}{16 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 156, normalized size = 1.68 \[ \frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 104 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 32 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 104 \, a^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 170, normalized size = 1.83 \[ \frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a^{3} d}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3}}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{3} d}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3}}+\frac {13}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{3} d}-\frac {1}{2 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{64 a^{3} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 195, normalized size = 2.10 \[ -\frac {\frac {\frac {104 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {32 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{3}} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {32 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {104 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a^{3} \sin \left (d x + c\right )^{4}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.29, size = 151, normalized size = 1.62 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,a^3\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^3\,d}+\frac {15\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a^3\,d}-\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{4}\right )}{16\,a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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