Optimal. Leaf size=193 \[ -\frac{a \left (3 a^2+2 b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac{2 a^5 \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left (4 a^2+3 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^3 d}+\frac{x \left (4 a^2 b^2+8 a^4+3 b^4\right )}{8 b^5}-\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d} \]
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Rubi [A] time = 0.544933, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2793, 3049, 3023, 2735, 2659, 205} \[ -\frac{a \left (3 a^2+2 b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac{2 a^5 \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left (4 a^2+3 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^3 d}+\frac{x \left (4 a^2 b^2+8 a^4+3 b^4\right )}{8 b^5}-\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3049
Rule 3023
Rule 2735
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{a+b \cos (c+d x)} \, dx &=\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac{\int \frac{\cos ^2(c+d x) \left (3 a+3 b \cos (c+d x)-4 a \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{4 b}\\ &=-\frac{a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac{\int \frac{\cos (c+d x) \left (-8 a^2+a b \cos (c+d x)+3 \left (4 a^2+3 b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{12 b^2}\\ &=\frac{\left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}-\frac{a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac{\int \frac{3 a \left (4 a^2+3 b^2\right )-b \left (4 a^2-9 b^2\right ) \cos (c+d x)-8 a \left (3 a^2+2 b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{24 b^3}\\ &=-\frac{a \left (3 a^2+2 b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}-\frac{a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac{\int \frac{3 a b \left (4 a^2+3 b^2\right )+3 \left (8 a^4+4 a^2 b^2+3 b^4\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{24 b^4}\\ &=\frac{\left (8 a^4+4 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac{a \left (3 a^2+2 b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}-\frac{a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}-\frac{a^5 \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^5}\\ &=\frac{\left (8 a^4+4 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac{a \left (3 a^2+2 b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}-\frac{a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}-\frac{\left (2 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^5 d}\\ &=\frac{\left (8 a^4+4 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac{2 a^5 \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^5 \sqrt{a+b} d}-\frac{a \left (3 a^2+2 b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac{\left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}-\frac{a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 b d}\\ \end{align*}
Mathematica [A] time = 0.645041, size = 153, normalized size = 0.79 \[ \frac{12 \left (4 a^2 b^2+8 a^4+3 b^4\right ) (c+d x)-24 a b \left (4 a^2+3 b^2\right ) \sin (c+d x)+24 b^2 \left (a^2+b^2\right ) \sin (2 (c+d x))+\frac{192 a^5 \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}-8 a b^3 \sin (3 (c+d x))+3 b^4 \sin (4 (c+d x))}{96 b^5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.115, size = 672, normalized size = 3.5 \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 = 2.25181, size = 1042, normalized size = 5.4 \begin{align*} \left [-\frac{12 \, \sqrt{-a^{2} + b^{2}} a^{5} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 3 \,{\left (8 \, a^{6} - 4 \, a^{4} b^{2} - a^{2} b^{4} - 3 \, b^{6}\right )} d x +{\left (24 \, a^{5} b - 8 \, a^{3} b^{3} - 16 \, a b^{5} - 6 \,{\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (4 \, a^{4} b^{2} - a^{2} b^{4} - 3 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\left (a^{2} b^{5} - b^{7}\right )} d}, -\frac{24 \, \sqrt{a^{2} - b^{2}} a^{5} \arctan \left (-\frac{a \cos \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - 3 \,{\left (8 \, a^{6} - 4 \, a^{4} b^{2} - a^{2} b^{4} - 3 \, b^{6}\right )} d x +{\left (24 \, a^{5} b - 8 \, a^{3} b^{3} - 16 \, a b^{5} - 6 \,{\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (4 \, a^{4} b^{2} - a^{2} b^{4} - 3 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\left (a^{2} b^{5} - b^{7}\right )} 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 [B] time = 1.38541, size = 531, normalized size = 2.75 \begin{align*} \frac{\frac{48 \,{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{a^{2} - b^{2}}}\right )\right )} a^{5}}{\sqrt{a^{2} - b^{2}} b^{5}} + \frac{3 \,{\left (8 \, a^{4} + 4 \, a^{2} b^{2} + 3 \, b^{4}\right )}{\left (d x + c\right )}}{b^{5}} - \frac{2 \,{\left (24 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 24 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 72 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 9 \, 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 )^{3} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 24 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\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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