Optimal. Leaf size=97 \[ \frac {2 \cos (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {x}{2 a^2} \]
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Rubi [A] time = 0.24, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2709, 3770, 3767, 8, 3768, 2638, 2635} \[ \frac {2 \cos (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {\sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {x}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2638
Rule 2709
Rule 2875
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \cot ^4(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4}\\ &=\frac {\int \left (-a^6+4 a^6 \csc (c+d x)-a^6 \csc ^2(c+d x)-2 a^6 \csc ^3(c+d x)+a^6 \csc ^4(c+d x)-2 a^6 \sin (c+d x)+a^6 \sin ^2(c+d x)\right ) \, dx}{a^8}\\ &=-\frac {x}{a^2}-\frac {\int \csc ^2(c+d x) \, dx}{a^2}+\frac {\int \csc ^4(c+d x) \, dx}{a^2}+\frac {\int \sin ^2(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^3(c+d x) \, dx}{a^2}-\frac {2 \int \sin (c+d x) \, dx}{a^2}+\frac {4 \int \csc (c+d x) \, dx}{a^2}\\ &=-\frac {x}{a^2}-\frac {4 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a^2 d}+\frac {\int 1 \, dx}{2 a^2}-\frac {\int \csc (c+d x) \, dx}{a^2}+\frac {\operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}-\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d}\\ &=-\frac {x}{2 a^2}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {\cot (c+d x) \csc (c+d x)}{a^2 d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a^2 d}\\ \end {align*}
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Mathematica [A] time = 2.45, size = 184, normalized size = 1.90 \[ -\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right )^4 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (30 \cos (c+d x)-\cos (3 (c+d x))+3 \left (\cos (5 (c+d x))+8 \sin (c+d x) \left (-6 \cos (c+d x)+2 \cos (3 (c+d x))-6 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\cos (2 (c+d x)) \left (-6 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+c+d x\right )+c+d x\right )\right )\right )}{768 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 161, normalized size = 1.66 \[ \frac {3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3} - 9 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 9 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (d x \cos \left (d x + c\right )^{2} - 4 \, \cos \left (d x + c\right )^{3} - d x + 6 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 3 \, \cos \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 194, normalized size = 2.00 \[ -\frac {\frac {12 \, {\left (d x + c\right )}}{a^{2}} - \frac {72 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {24 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}} + \frac {132 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.65, size = 272, normalized size = 2.80 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{2}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2} d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{2}}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {4}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}-\frac {1}{24 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{4 a^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {1}{8 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 306, normalized size = 3.15 \[ \frac {\frac {\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {108 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {19 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {102 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {27 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1}{\frac {a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} - \frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {72 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.16, size = 253, normalized size = 2.61 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a^2\,d}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}+\frac {9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {19\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{3}}{d\,\left (8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+16\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^2\,d}+\frac {\mathrm {atan}\left (\frac {1}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}-\frac {6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+6}\right )}{a^2\,d} \]
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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