Optimal. Leaf size=94 \[ \frac {3 \cos (c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac {x}{a} \]
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Rubi [A] time = 0.14, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2839, 3473, 8, 2592, 288, 321, 206} \[ \frac {3 \cos (c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 206
Rule 288
Rule 321
Rule 2592
Rule 2839
Rule 3473
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cos (c+d x) \cot ^3(c+d x) \, dx}{a}+\frac {\int \cot ^4(c+d x) \, dx}{a}\\ &=-\frac {\cot ^3(c+d x)}{3 a d}-\frac {\int \cot ^2(c+d x) \, dx}{a}+\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\int 1 \, dx}{a}-\frac {3 \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d}\\ &=\frac {x}{a}+\frac {3 \cos (c+d x)}{2 a d}+\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d}\\ &=\frac {x}{a}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac {3 \cos (c+d x)}{2 a d}+\frac {\cot (c+d x)}{a d}+\frac {\cos (c+d x) \cot ^2(c+d x)}{2 a d}-\frac {\cot ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.91, size = 138, normalized size = 1.47 \[ \frac {\csc \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (9 \sin (2 (c+d x))-2 (3 \sin (c+d x)+4) \cos (3 (c+d x))+12 \sin ^3(c+d x) \left (3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 c+2 d x\right )\right )}{192 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 148, normalized size = 1.57 \[ \frac {16 \, \cos \left (d x + c\right )^{3} - 9 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 9 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 6 \, {\left (2 \, d x \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, d x - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 12 \, \cos \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 157, normalized size = 1.67 \[ \frac {\frac {24 \, {\left (d x + c\right )}}{a} + \frac {36 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} + \frac {48}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a} - \frac {66 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 173, normalized size = 1.84 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {1}{24 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {5}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 240, normalized size = 2.55 \[ -\frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a} - \frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {14 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {51 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1}{\frac {a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} - \frac {48 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {36 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.97, size = 212, normalized size = 2.26 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {2\,\mathrm {atan}\left (\frac {6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-6}+\frac {4}{4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-6}\right )}{a\,d}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{3}}{d\,\left (8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d} \]
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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