Optimal. Leaf size=150 \[ \frac {15 \cos (c+d x)}{8 a d}+\frac {5 \cot ^3(c+d x)}{6 a d}-\frac {5 \cot (c+d x)}{2 a d}-\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}+\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}-\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {5 x}{2 a} \]
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Rubi [A] time = 0.18, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2839, 2592, 288, 321, 206, 2591, 302, 203} \[ \frac {15 \cos (c+d x)}{8 a d}+\frac {5 \cot ^3(c+d x)}{6 a d}-\frac {5 \cot (c+d x)}{2 a d}-\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}+\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}-\frac {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {5 x}{2 a} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 288
Rule 302
Rule 321
Rule 2591
Rule 2592
Rule 2839
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac {\int \cos ^2(c+d x) \cot ^4(c+d x) \, dx}{a}+\frac {\int \cos (c+d x) \cot ^5(c+d x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3} \, 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 ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}+\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{4 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 {5 \cos (c+d x) \cot ^2(c+d x)}{8 a d}-\frac {\cos ^2(c+d x) \cot ^3(c+d x)}{2 a d}-\frac {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {15 \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 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 {15 \cos (c+d x)}{8 a d}-\frac {5 \cot (c+d x)}{2 a d}+\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 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 {\cos (c+d x) \cot ^4(c+d x)}{4 a d}-\frac {15 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 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 {15 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {15 \cos (c+d x)}{8 a d}-\frac {5 \cot (c+d x)}{2 a d}+\frac {5 \cos (c+d x) \cot ^2(c+d x)}{8 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 {\cos (c+d x) \cot ^4(c+d x)}{4 a d}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 252, normalized size = 1.68 \[ -\frac {\csc ^4(c+d x) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (95 \sin (2 (c+d x))-68 \sin (4 (c+d x))+3 \sin (6 (c+d x))+60 c \cos (4 (c+d x))-30 \cos (c+d x)+90 \cos (3 (c+d x))+60 d x \cos (4 (c+d x))-12 \cos (5 (c+d x))-135 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+135 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-60 \cos (2 (c+d x)) \left (-3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 c+4 d x\right )-45 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+180 c+180 d x\right )}{192 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 191, normalized size = 1.27 \[ -\frac {120 \, d x \cos \left (d x + c\right )^{4} - 48 \, \cos \left (d x + c\right )^{5} - 240 \, d x \cos \left (d x + c\right )^{2} + 150 \, \cos \left (d x + c\right )^{3} + 120 \, d x + 45 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 45 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 8 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 90 \, \cos \left (d x + c\right )}{48 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 224, normalized size = 1.49 \[ -\frac {\frac {480 \, {\left (d x + c\right )}}{a} - \frac {360 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {192 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 216 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 216 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.55, size = 310, normalized size = 2.07 \[ \frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {1}{64 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{24 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{4 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {9}{8 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 340, normalized size = 2.27 \[ \frac {\frac {\frac {216 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a} + \frac {\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {42 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {477 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {616 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {432 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {24 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 3}{\frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {960 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.03, size = 286, normalized size = 1.91 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a\,d}+\frac {5\,\mathrm {atan}\left (\frac {25}{25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {75}{4}}-\frac {75\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {75}{4}\right )}\right )}{a\,d}+\frac {15\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a\,d}+\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {154\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {159\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}-\frac {50\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {1}{4}}{d\,\left (16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+32\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,d} \]
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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