Optimal. Leaf size=150 \[ -\frac{5 \cos ^3(c+d x)}{6 a d}-\frac{5 \cos (c+d x)}{2 a d}+\frac{15 \cot (c+d x)}{8 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{15 x}{8 a} \]
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Rubi [A] time = 0.18867, antiderivative size = 150, normalized size of antiderivative = 1., 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, 302, 206, 2591, 321, 203} \[ -\frac{5 \cos ^3(c+d x)}{6 a d}-\frac{5 \cos (c+d x)}{2 a d}+\frac{15 \cot (c+d x)}{8 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{15 x}{8 a} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 2592
Rule 288
Rule 302
Rule 206
Rule 2591
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cos ^4(c+d x) \cot ^2(c+d x) \, dx}{a}+\frac{\int \cos ^3(c+d x) \cot ^3(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 )^3} \, dx,x,\cot (c+d x)\right )}{a d}\\ &=-\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{4 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 \cos ^2(c+d x) \cot (c+d x)}{8 a d}-\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}+\frac{15 \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\cot (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,\cos (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{15 \cot (c+d x)}{8 a d}-\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}-\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}-\frac{15 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 a d}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 a d}\\ &=\frac{15 x}{8 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{15 \cot (c+d x)}{8 a d}-\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}-\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}-\frac{\cos ^3(c+d x) \cot ^2(c+d x)}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.523759, size = 179, normalized size = 1.19 \[ -\frac{\left (\csc \left (\frac{1}{2} (c+d x)\right )+\sec \left (\frac{1}{2} (c+d x)\right )\right )^2 \left (-285 \sin (2 (c+d x))+42 \sin (4 (c+d x))+3 \sin (6 (c+d x))+400 \cos (c+d x)-200 \cos (3 (c+d x))-8 \cos (5 (c+d x))+480 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-480 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+120 \cos (2 (c+d x)) \left (-4 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+3 c+3 d x\right )-360 c-360 d x\right )}{1536 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.148, size = 371, normalized size = 2.5 \begin{align*}{\frac{1}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{2\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{9}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-6\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-14\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}+{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{38}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{9}{4\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{14}{3\,da} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{15}{4\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{8\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{5}{2\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54507, size = 517, normalized size = 3.45 \begin{align*} \frac{\frac{\frac{12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{124 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{102 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{322 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{78 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{348 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{42 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{147 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{42 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - 3}{\frac{a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{4 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{6 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{4 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac{a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac{3 \,{\left (\frac{4 \, \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}}\right )}}{a} + \frac{90 \, \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21775, size = 413, normalized size = 2.75 \begin{align*} -\frac{8 \, \cos \left (d x + c\right )^{5} - 45 \, d x \cos \left (d x + c\right )^{2} + 40 \, \cos \left (d x + c\right )^{3} + 45 \, d x - 30 \,{\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 ) + 30 \,{\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 ) - 3 \,{\left (2 \, \cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 60 \, \cos \left (d x + c\right )}{24 \,{\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21191, size = 292, normalized size = 1.95 \begin{align*} \frac{\frac{45 \,{\left (d x + c\right )}}{a} - \frac{60 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} + \frac{3 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{2}} + \frac{3 \,{\left (30 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} - \frac{2 \,{\left (27 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 72 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 168 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 152 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 27 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 56\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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