Optimal. Leaf size=146 \[ \frac {5 \cos ^3(c+d x)}{6 a d}+\frac {5 \cos (c+d x)}{2 a d}-\frac {5 \cot ^3(c+d x)}{6 a d}+\frac {5 \cot (c+d x)}{2 a d}+\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac {5 x}{2 a} \]
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Rubi [A] time = 0.18, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2839, 2591, 288, 302, 203, 2592, 206} \[ \frac {5 \cos ^3(c+d x)}{6 a d}+\frac {5 \cos (c+d x)}{2 a d}-\frac {5 \cot ^3(c+d x)}{6 a d}+\frac {5 \cot (c+d x)}{2 a d}+\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac {5 x}{2 a} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 288
Rule 302
Rule 2591
Rule 2592
Rule 2839
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \cot ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cos ^3(c+d x) \cot ^3(c+d x) \, dx}{a}+\frac {\int \cos ^2(c+d x) \cot ^4(c+d x) \, dx}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{a d}-\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{a d}\\ &=\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 a d}\\ &=\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {5 \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac {5 \operatorname {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{2 a d}\\ &=\frac {5 \cos (c+d x)}{2 a d}+\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {5 \cot (c+d x)}{2 a d}+\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac {5 \cot ^3(c+d x)}{6 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 a d}\\ &=\frac {5 x}{2 a}-\frac {5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac {5 \cos (c+d x)}{2 a d}+\frac {5 \cos ^3(c+d x)}{6 a d}+\frac {5 \cot (c+d x)}{2 a d}+\frac {\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac {5 \cot ^3(c+d x)}{6 a d}+\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 197, normalized size = 1.35 \[ -\frac {\csc ^3(c+d x) \left (-180 c \sin (c+d x)-180 d x \sin (c+d x)-75 \sin (2 (c+d x))+60 c \sin (3 (c+d x))+60 d x \sin (3 (c+d x))+24 \sin (4 (c+d x))+\sin (6 (c+d x))-30 \cos (c+d x)+65 \cos (3 (c+d x))-3 \cos (5 (c+d x))-180 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+180 \sin (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-60 \sin (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{96 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 168, normalized size = 1.15 \[ -\frac {6 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2 \, {\left (2 \, \cos \left (d x + c\right )^{5} + 15 \, d x \cos \left (d x + c\right )^{2} + 10 \, \cos \left (d x + c\right )^{3} - 15 \, d x - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 30 \, \cos \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 228, normalized size = 1.56 \[ \frac {\frac {180 \, {\left (d x + c\right )}}{a} + \frac {180 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {3 \, {\left (a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{3}} - \frac {110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 111 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 273 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 306 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 253 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3} a}}{72 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.64, size = 306, normalized size = 2.10 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {14}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {1}{24 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {9}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 362, normalized size = 2.48 \[ -\frac {\frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a} - \frac {\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {121 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {102 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {201 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {80 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {147 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 1}{\frac {a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.06, size = 290, normalized size = 1.99 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {5\,\mathrm {atan}\left (\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-25}+\frac {25}{25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-25}\right )}{a\,d}+\frac {5\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+49\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+67\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+34\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {121\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{3}}{d\,\left (8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+24\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+24\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d} \]
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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