Optimal. Leaf size=96 \[ \frac {2 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}-\frac {7 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {7 \cot (c+d x) \csc (c+d x)}{8 a^2 d} \]
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Rubi [A] time = 0.19, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2869, 2757, 3768, 3770, 3767} \[ \frac {2 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}-\frac {7 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {7 \cot (c+d x) \csc (c+d x)}{8 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 2869
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \csc ^5(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (a^2 \csc ^3(c+d x)-2 a^2 \csc ^4(c+d x)+a^2 \csc ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \csc ^3(c+d x) \, dx}{a^2}+\frac {\int \csc ^5(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^4(c+d x) \, dx}{a^2}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {\int \csc (c+d x) \, dx}{2 a^2}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a^2}+\frac {2 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {7 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {3 \int \csc (c+d x) \, dx}{8 a^2}\\ &=-\frac {7 \tanh ^{-1}(\cos (c+d x))}{8 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {7 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}\\ \end {align*}
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Mathematica [A] time = 1.52, size = 116, normalized size = 1.21 \[ -\frac {\left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (-48 \sin (2 (c+d x))+45 \cos (c+d x)+(32 \sin (c+d x)-21) \cos (3 (c+d x))+84 \sin ^4(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{1536 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 149, normalized size = 1.55 \[ \frac {42 \, \cos \left (d x + c\right )^{3} - 21 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 21 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (2 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 54 \, \cos \left (d x + c\right )}{48 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 157, normalized size = 1.64 \[ \frac {\frac {168 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {350 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 144 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 144 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{8}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 170, normalized size = 1.77 \[ \frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a^{2} d}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 d \,a^{2}}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2} d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \,a^{2}}+\frac {3}{4 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}-\frac {1}{4 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{64 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{12 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 195, normalized size = 2.03 \[ -\frac {\frac {\frac {144 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{2}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {{\left (\frac {16 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {144 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a^{2} \sin \left (d x + c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.72, size = 151, normalized size = 1.57 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12\,a^2\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^2\,d}+\frac {7\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a^2\,d}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^2\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {1}{4}\right )}{16\,a^2\,d} \]
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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