Optimal. Leaf size=286 \[ \frac{(2 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{a^{5/2} d}-\frac{(11 A-35 B) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(43 A-115 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{(7 A-15 B) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.981494, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {2961, 2977, 2983, 2982, 2782, 205, 2774, 216} \[ \frac{(2 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{a^{5/2} d}-\frac{(11 A-35 B) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(43 A-115 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{(7 A-15 B) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2961
Rule 2977
Rule 2983
Rule 2982
Rule 2782
Rule 205
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx\\ &=\frac{(A-B) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{5}{2} a (A-B)-a (A-5 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=\frac{(A-B) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(7 A-15 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)} \left (\frac{3}{4} a^2 (7 A-15 B)-\frac{1}{2} a^2 (11 A-35 B) \cos (c+d x)\right )}{\sqrt{a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=\frac{(A-B) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(7 A-15 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{(11 A-35 B) \sin (c+d x)}{16 a^2 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{4} a^3 (11 A-35 B)+4 a^3 (2 A-5 B) \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{8 a^5}\\ &=\frac{(A-B) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(7 A-15 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{(11 A-35 B) \sin (c+d x)}{16 a^2 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left ((43 A-115 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}{32 a^2}+\frac{\left ((2 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx}{2 a^3}\\ &=\frac{(A-B) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(7 A-15 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{(11 A-35 B) \sin (c+d x)}{16 a^2 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left ((43 A-115 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 )}{16 a d}-\frac{\left ((2 A-5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a}}} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{a^3 d}\\ &=\frac{(2 A-5 B) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{a^{5/2} d}-\frac{(43 A-115 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)}}{16 \sqrt{2} a^{5/2} d}+\frac{(A-B) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)}+\frac{(7 A-15 B) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{(11 A-35 B) \sin (c+d x)}{16 a^2 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 7.15705, size = 929, normalized size = 3.25 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{\sec \left (\frac{c}{2}\right ) \left (B \sin \left (\frac{d x}{2}\right )-A \sin \left (\frac{d x}{2}\right )\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 d}-\frac{(A-B) \tan \left (\frac{c}{2}\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{2 d}+\frac{\sec \left (\frac{c}{2}\right ) \left (19 A \sin \left (\frac{d x}{2}\right )-27 B \sin \left (\frac{d x}{2}\right )\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d}+\frac{(19 A-27 B) \tan \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d}+\frac{15 (B-A) \cos \left (\frac{d x}{2}\right ) \sin \left (\frac{c}{2}\right )}{2 d}+\frac{4 B \cos \left (\frac{3 d x}{2}\right ) \sin \left (\frac{3 c}{2}\right )}{d}-\frac{15 (A-B) \cos \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )}{2 d}+\frac{4 B \cos \left (\frac{3 c}{2}\right ) \sin \left (\frac{3 d x}{2}\right )}{d}\right ) \cos ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{(a (\cos (c+d x)+1))^{5/2}}-\frac{11 i A e^{-\frac{1}{2} i (c+d x)} \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 ) \cos ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d (a (\cos (c+d x)+1))^{5/2}}+\frac{35 i B e^{-\frac{1}{2} i (c+d x)} \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 ) \cos ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{4 d (a (\cos (c+d x)+1))^{5/2}}+\frac{4 i \sqrt{2} A e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (-\sinh ^{-1}\left (e^{i (c+d x)}\right )+\sqrt{2} \tanh ^{-1}\left (\frac{-1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )+\tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right ) \cos ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{d (a (\cos (c+d x)+1))^{5/2}}-\frac{10 i \sqrt{2} B e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (-\sinh ^{-1}\left (e^{i (c+d x)}\right )+\sqrt{2} \tanh ^{-1}\left (\frac{-1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )+\tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right ) \cos ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{d (a (\cos (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.615, size = 609, normalized size = 2.1 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 99.5675, size = 853, normalized size = 2.98 \begin{align*} \frac{\sqrt{2}{\left ({\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right ) + 43 \, A - 115 \, B\right )} \sqrt{a} \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 ) - 32 \,{\left ({\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right ) + 2 \, A - 5 \, B\right )} \sqrt{a} \arctan \left (\frac{\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 ) + \frac{2 \,{\left (16 \, B \cos \left (d x + c\right )^{3} - 5 \,{\left (3 \, A - 11 \, B\right )} \cos \left (d x + c\right )^{2} -{\left (11 \, A - 35 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{32 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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