Optimal. Leaf size=270 \[ \frac{2 b^4 \left (5 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left (7 a^2 b^2+2 a^4-12 b^4\right ) \tan (c+d x)}{3 a^4 d \left (a^2-b^2\right )}-\frac{b \left (a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left (a^2-4 b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d \left (a^2-b^2\right )}-\frac{b \left (a^2-2 b^2\right ) \tan (c+d x) \sec (c+d x)}{a^3 d \left (a^2-b^2\right )}+\frac{b^2 \tan (c+d x) \sec ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))} \]
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Rubi [A] time = 0.967372, antiderivative size = 270, 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 = {2802, 3055, 3001, 3770, 2659, 205} \[ \frac{2 b^4 \left (5 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left (7 a^2 b^2+2 a^4-12 b^4\right ) \tan (c+d x)}{3 a^4 d \left (a^2-b^2\right )}-\frac{b \left (a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left (a^2-4 b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d \left (a^2-b^2\right )}-\frac{b \left (a^2-2 b^2\right ) \tan (c+d x) \sec (c+d x)}{a^3 d \left (a^2-b^2\right )}+\frac{b^2 \tan (c+d x) \sec ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2802
Rule 3055
Rule 3001
Rule 3770
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx &=\frac{b^2 \sec ^2(c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\int \frac{\left (a^2-4 b^2-a b \cos (c+d x)+3 b^2 \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac{\left (a^2-4 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^2(c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\int \frac{\left (-6 b \left (a^2-2 b^2\right )+a \left (2 a^2+b^2\right ) \cos (c+d x)+2 b \left (a^2-4 b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=-\frac{b \left (a^2-2 b^2\right ) \sec (c+d x) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac{\left (a^2-4 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^2(c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\int \frac{\left (2 \left (2 a^4+7 a^2 b^2-12 b^4\right )-2 a b \left (a^2+2 b^2\right ) \cos (c+d x)-6 b^2 \left (a^2-2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^4+7 a^2 b^2-12 b^4\right ) \tan (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}-\frac{b \left (a^2-2 b^2\right ) \sec (c+d x) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac{\left (a^2-4 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^2(c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\int \frac{\left (-6 b \left (a^4+3 a^2 b^2-4 b^4\right )-6 a b^2 \left (a^2-2 b^2\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^4+7 a^2 b^2-12 b^4\right ) \tan (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}-\frac{b \left (a^2-2 b^2\right ) \sec (c+d x) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac{\left (a^2-4 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^2(c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (b^4 \left (5 a^2-4 b^2\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{a^5 \left (a^2-b^2\right )}-\frac{\left (b \left (a^2+4 b^2\right )\right ) \int \sec (c+d x) \, dx}{a^5}\\ &=-\frac{b \left (a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left (2 a^4+7 a^2 b^2-12 b^4\right ) \tan (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}-\frac{b \left (a^2-2 b^2\right ) \sec (c+d x) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac{\left (a^2-4 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^2(c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (2 b^4 \left (5 a^2-4 b^2\right )\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 )}{a^5 \left (a^2-b^2\right ) d}\\ &=\frac{2 b^4 \left (5 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^5 (a-b)^{3/2} (a+b)^{3/2} d}-\frac{b \left (a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left (2 a^4+7 a^2 b^2-12 b^4\right ) \tan (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}-\frac{b \left (a^2-2 b^2\right ) \sec (c+d x) \tan (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac{\left (a^2-4 b^2\right ) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac{b^2 \sec ^2(c+d x) \tan (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 6.1516, size = 499, normalized size = 1.85 \[ -\frac{b^5 \sin (c+d x)}{a^4 d (a-b) (a+b) (a+b \cos (c+d x))}+\frac{2 a^2 \sin \left (\frac{1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac{1}{2} (c+d x)\right )}{3 a^4 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 a^2 \sin \left (\frac{1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac{1}{2} (c+d x)\right )}{3 a^4 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}-\frac{2 b^4 \left (5 a^2-4 b^2\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{a^5 d \left (a^2-b^2\right ) \sqrt{b^2-a^2}}+\frac{\left (a^2 b+4 b^3\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 d}+\frac{\left (-a^2 b-4 b^3\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 d}+\frac{a-6 b}{12 a^3 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{6 b-a}{12 a^3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{6 a^2 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{6 a^2 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.13, size = 535, normalized size = 2. \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 = 8.08479, size = 2218, normalized size = 8.21 \begin{align*} \text{result too large to display} \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{\sec ^{4}{\left (c + d x \right )}}{\left (a + b \cos{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39282, size = 497, normalized size = 1.84 \begin{align*} -\frac{\frac{6 \, b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a^{6} - a^{4} b^{2}\right )}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a + b\right )}} + \frac{6 \,{\left (5 \, a^{2} b^{4} - 4 \, b^{6}\right )}{\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 )}}{{\left (a^{7} - a^{5} b^{2}\right )} \sqrt{a^{2} - b^{2}}} + \frac{3 \,{\left (a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac{3 \,{\left (a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{5}} + \frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 18 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, b^{2} \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 )}^{3} a^{4}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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