Optimal. Leaf size=193 \[ -\frac{b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac{\left (3 a^2+4 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{2 b^5 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left (4 a^2 b^2+3 a^4+8 b^4\right )}{8 a^5}-\frac{b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d} \]
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Rubi [A] time = 0.686522, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3853, 4104, 3919, 3831, 2659, 208} \[ -\frac{b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac{\left (3 a^2+4 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{2 b^5 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left (4 a^2 b^2+3 a^4+8 b^4\right )}{8 a^5}-\frac{b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 3853
Rule 4104
Rule 3919
Rule 3831
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x)}{a+b \sec (c+d x)} \, dx &=\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac{\int \frac{\cos ^3(c+d x) \left (-4 b+3 a \sec (c+d x)+3 b \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a}\\ &=-\frac{b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{\int \frac{\cos ^2(c+d x) \left (-3 \left (3 a^2+4 b^2\right )-a b \sec (c+d x)+8 b^2 \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^2}\\ &=\frac{\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac{\int \frac{\cos (c+d x) \left (-8 b \left (2 a^2+3 b^2\right )+a \left (9 a^2-4 b^2\right ) \sec (c+d x)+3 b \left (3 a^2+4 b^2\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{24 a^3}\\ &=-\frac{b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac{\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{\int \frac{-3 \left (3 a^4+4 a^2 b^2+8 b^4\right )-3 a b \left (3 a^2+4 b^2\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{24 a^4}\\ &=\frac{\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac{b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac{\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{b^5 \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^5}\\ &=\frac{\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac{b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac{\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{b^4 \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{a^5}\\ &=\frac{\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac{b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac{\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{\left (2 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 d}\\ &=\frac{\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac{2 b^5 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 \sqrt{a-b} \sqrt{a+b} d}-\frac{b \left (2 a^2+3 b^2\right ) \sin (c+d x)}{3 a^4 d}+\frac{\left (3 a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{b \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}\\ \end{align*}
Mathematica [A] time = 0.563935, size = 153, normalized size = 0.79 \[ \frac{12 \left (4 a^2 b^2+3 a^4+8 b^4\right ) (c+d x)-24 a b \left (3 a^2+4 b^2\right ) \sin (c+d x)+24 a^2 \left (a^2+b^2\right ) \sin (2 (c+d x))+\frac{192 b^5 \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}-8 a^3 b \sin (3 (c+d x))+3 a^4 \sin (4 (c+d x))}{96 a^5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.083, 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 = 1.98027, size = 1057, normalized size = 5.48 \begin{align*} \left [\frac{12 \, \sqrt{a^{2} - b^{2}} b^{5} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) -{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + 3 \,{\left (3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right )} d x -{\left (16 \, a^{5} b + 8 \, a^{3} b^{3} - 24 \, a b^{5} - 6 \,{\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left (a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (3 \, a^{6} + a^{4} b^{2} - 4 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\left (a^{7} - a^{5} b^{2}\right )} d}, -\frac{24 \, \sqrt{-a^{2} + b^{2}} b^{5} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - 3 \,{\left (3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right )} d x +{\left (16 \, a^{5} b + 8 \, a^{3} b^{3} - 24 \, a b^{5} - 6 \,{\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left (a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (3 \, a^{6} + a^{4} b^{2} - 4 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\left (a^{7} - a^{5} b^{2}\right )} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{4}{\left (c + d x \right )}}{a + b \sec{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28458, 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 )} b^{5}}{\sqrt{-a^{2} + b^{2}} a^{5}} - \frac{3 \,{\left (3 \, a^{4} + 4 \, a^{2} b^{2} + 8 \, b^{4}\right )}{\left (d x + c\right )}}{a^{5}} + \frac{2 \,{\left (15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 24 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 9 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 72 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 72 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24 \, 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} a^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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