Optimal. Leaf size=252 \[ -\frac{a \left (a^2-3 b^2\right ) \cos (c+d x)}{b^4 d}+\frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a b^5 d}+\frac{\left (a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d}-\frac{x \left (a^2-3 b^2\right )}{2 b^3}-\frac{x \left (-3 a^2 b^2+a^4+3 b^4\right )}{b^5}+\frac{a \cos ^3(c+d x)}{3 b^2 d}-\frac{a \cos (c+d x)}{b^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac{3 x}{8 b} \]
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Rubi [A] time = 0.285735, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2897, 3770, 2638, 2635, 8, 2633, 2660, 618, 204} \[ -\frac{a \left (a^2-3 b^2\right ) \cos (c+d x)}{b^4 d}+\frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a b^5 d}+\frac{\left (a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d}-\frac{x \left (a^2-3 b^2\right )}{2 b^3}-\frac{x \left (-3 a^2 b^2+a^4+3 b^4\right )}{b^5}+\frac{a \cos ^3(c+d x)}{3 b^2 d}-\frac{a \cos (c+d x)}{b^2 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac{3 x}{8 b} \]
Antiderivative was successfully verified.
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Rule 2897
Rule 3770
Rule 2638
Rule 2635
Rule 8
Rule 2633
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\int \left (\frac{-a^4+3 a^2 b^2-3 b^4}{b^5}+\frac{\csc (c+d x)}{a}+\frac{a \left (a^2-3 b^2\right ) \sin (c+d x)}{b^4}+\frac{\left (-a^2+3 b^2\right ) \sin ^2(c+d x)}{b^3}+\frac{a \sin ^3(c+d x)}{b^2}-\frac{\sin ^4(c+d x)}{b}+\frac{\left (a^2-b^2\right )^3}{a b^5 (a+b \sin (c+d x))}\right ) \, dx\\ &=-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) x}{b^5}+\frac{\int \csc (c+d x) \, dx}{a}+\frac{a \int \sin ^3(c+d x) \, dx}{b^2}-\frac{\int \sin ^4(c+d x) \, dx}{b}+\frac{\left (a \left (a^2-3 b^2\right )\right ) \int \sin (c+d x) \, dx}{b^4}-\frac{\left (a^2-3 b^2\right ) \int \sin ^2(c+d x) \, dx}{b^3}+\frac{\left (a^2-b^2\right )^3 \int \frac{1}{a+b \sin (c+d x)} \, dx}{a b^5}\\ &=-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) x}{b^5}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{a \left (a^2-3 b^2\right ) \cos (c+d x)}{b^4 d}+\frac{\left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b d}-\frac{3 \int \sin ^2(c+d x) \, dx}{4 b}-\frac{\left (a^2-3 b^2\right ) \int 1 \, dx}{2 b^3}-\frac{a \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{b^2 d}+\frac{\left (2 \left (a^2-b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a b^5 d}\\ &=-\frac{\left (a^2-3 b^2\right ) x}{2 b^3}-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) x}{b^5}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{a \cos (c+d x)}{b^2 d}-\frac{a \left (a^2-3 b^2\right ) \cos (c+d x)}{b^4 d}+\frac{a \cos ^3(c+d x)}{3 b^2 d}+\frac{3 \cos (c+d x) \sin (c+d x)}{8 b d}+\frac{\left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b d}-\frac{3 \int 1 \, dx}{8 b}-\frac{\left (4 \left (a^2-b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a b^5 d}\\ &=-\frac{3 x}{8 b}-\frac{\left (a^2-3 b^2\right ) x}{2 b^3}-\frac{\left (a^4-3 a^2 b^2+3 b^4\right ) x}{b^5}+\frac{2 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a b^5 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{a \cos (c+d x)}{b^2 d}-\frac{a \left (a^2-3 b^2\right ) \cos (c+d x)}{b^4 d}+\frac{a \cos ^3(c+d x)}{3 b^2 d}+\frac{3 \cos (c+d x) \sin (c+d x)}{8 b d}+\frac{\left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b d}\\ \end{align*}
Mathematica [A] time = 0.542313, size = 220, normalized size = 0.87 \[ -\frac{-24 a^3 b^2 \sin (2 (c+d x))+24 a^2 b \left (4 a^2-9 b^2\right ) \cos (c+d x)-8 a^2 b^3 \cos (3 (c+d x))-192 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )-240 a^3 b^2 c-240 a^3 b^2 d x+96 a^5 c+96 a^5 d x+48 a b^4 \sin (2 (c+d x))+3 a b^4 \sin (4 (c+d x))+180 a b^4 c+180 a b^4 d x-96 b^5 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+96 b^5 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{96 a b^5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 827, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.80253, size = 1212, normalized size = 4.81 \begin{align*} \left [\frac{8 \, a^{2} b^{3} \cos \left (d x + c\right )^{3} - 12 \, b^{5} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 12 \, b^{5} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 3 \,{\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 12 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{-a^{2} + b^{2}} \log \left (-\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \,{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 24 \,{\left (a^{4} b - 2 \, a^{2} b^{3}\right )} \cos \left (d x + c\right ) - 3 \,{\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} -{\left (4 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a b^{5} d}, \frac{8 \, a^{2} b^{3} \cos \left (d x + c\right )^{3} - 12 \, b^{5} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 12 \, b^{5} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 3 \,{\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x - 24 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \sin \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 24 \,{\left (a^{4} b - 2 \, a^{2} b^{3}\right )} \cos \left (d x + c\right ) - 3 \,{\left (2 \, a b^{4} \cos \left (d x + c\right )^{3} -{\left (4 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a b^{5} 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 = 1.27358, size = 537, normalized size = 2.13 \begin{align*} \frac{\frac{24 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} - \frac{3 \,{\left (8 \, a^{4} - 20 \, a^{2} b^{2} + 15 \, b^{4}\right )}{\left (d x + c\right )}}{b^{5}} + \frac{48 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}} a b^{5}} - \frac{2 \,{\left (12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 27 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 24 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 72 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 72 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 168 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 72 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 152 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 27 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, a^{3} - 56 \, a b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} b^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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