Optimal. Leaf size=295 \[ -\frac{2 (A-9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (19 A-3 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A-93 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A-129 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 1.05655, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2961, 2984, 12, 2782, 205} \[ -\frac{2 (A-9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (19 A-3 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A-93 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A-129 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2961
Rule 2984
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{(A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+B \cos (c+d x)}{\cos ^{\frac{11}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx\\ &=\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{2} a (A-9 B)+4 a A \cos (c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{9 a}\\ &=-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{4} a^2 (19 A-3 B)-\frac{3}{2} a^2 (A-9 B) \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{63 a^2}\\ &=\frac{2 (19 A-3 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{8} a^3 (29 A-93 B)+\frac{3}{2} a^3 (19 A-3 B) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{315 a^3}\\ &=-\frac{2 (29 A-93 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{16} a^4 (257 A-129 B)-\frac{3}{8} a^4 (29 A-93 B) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{945 a^4}\\ &=\frac{2 (257 A-129 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (29 A-93 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int -\frac{945 a^5 (A-B)}{32 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{945 a^5}\\ &=\frac{2 (257 A-129 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (29 A-93 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}-\left ((A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx\\ &=\frac{2 (257 A-129 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (29 A-93 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (2 a (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^2+a x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{d}\\ &=-\frac{\sqrt{2} (A-B) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{\sqrt{a} d}+\frac{2 (257 A-129 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (29 A-93 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 9.50608, size = 272, normalized size = 0.92 \[ \frac{2 e^{-\frac{1}{2} i (c+d x)} \cos \left (\frac{1}{2} (c+d x)\right ) \left (-315 i (A-B) \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )-\frac{1}{4} \sin \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{9}{2}}(c+d x) \left (\cos \left (\frac{1}{2} (c+d x)\right )+i \sin \left (\frac{1}{2} (c+d x)\right )\right ) ((214 A-918 B) \cos (c+d x)-8 (157 A-69 B) \cos (2 (c+d x))+58 A \cos (3 (c+d x))-257 A \cos (4 (c+d x))-1279 A-186 B \cos (3 (c+d x))+129 B \cos (4 (c+d x))+423 B)\right )}{315 d \sqrt{a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.641, size = 793, normalized size = 2.7 \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 = 1.69998, size = 539, normalized size = 1.83 \begin{align*} \frac{\frac{315 \, \sqrt{2}{\left ({\left (A - B\right )} a \cos \left (d x + c\right )^{5} +{\left (A - B\right )} a \cos \left (d x + c\right )^{4}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right )}{\sqrt{a}} + \frac{2 \,{\left ({\left (257 \, A - 129 \, B\right )} \cos \left (d x + c\right )^{4} -{\left (29 \, A - 93 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (19 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{2} - 5 \,{\left (A - 9 \, B\right )} \cos \left (d x + c\right ) + 35 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{315 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{11}{2}}}{\sqrt{a \cos \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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