Optimal. Leaf size=317 \[ \frac{(157 A-85 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A-475 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A-1495 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(283 A-163 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(21 A-13 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 1.12478, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {2961, 2978, 2984, 12, 2782, 205} \[ \frac{(157 A-85 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A-475 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A-1495 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(283 A-163 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(21 A-13 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2961
Rule 2978
Rule 2984
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+B \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx\\ &=-\frac{(A-B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} a (13 A-5 B)-4 a (A-B) \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac{(A-B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(21 A-13 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} a^2 (157 A-85 B)-\frac{3}{2} a^2 (21 A-13 B) \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{(A-B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(21 A-13 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{(157 A-85 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{8} a^3 (787 A-475 B)+\frac{1}{2} a^3 (157 A-85 B) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{20 a^5}\\ &=-\frac{(787 A-475 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(A-B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(21 A-13 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{(157 A-85 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{16} a^4 (2671 A-1495 B)-\frac{1}{8} a^4 (787 A-475 B) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{30 a^6}\\ &=\frac{(2671 A-1495 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(787 A-475 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(A-B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(21 A-13 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{(157 A-85 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int -\frac{15 a^5 (283 A-163 B)}{32 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{15 a^7}\\ &=\frac{(2671 A-1495 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(787 A-475 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(A-B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(21 A-13 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{(157 A-85 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{\left ((283 A-163 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{(2671 A-1495 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(787 A-475 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(A-B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(21 A-13 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{(157 A-85 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}+\frac{\left ((283 A-163 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{(283 A-163 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{(2671 A-1495 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(787 A-475 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{240 a^2 d \sqrt{a+a \cos (c+d x)}}-\frac{(A-B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{4 d (a+a \cos (c+d x))^{5/2}}-\frac{(21 A-13 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{3/2}}+\frac{(157 A-85 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{80 a^2 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 8.17732, size = 261, normalized size = 0.82 \[ \frac{\cos ^5\left (\frac{1}{2} (c+d x)\right ) \left (\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) (10 (2605 A-1381 B) \cos (c+d x)+108 (157 A-85 B) \cos (2 (c+d x))+9110 A \cos (3 (c+d x))+2671 A \cos (4 (c+d x))+15053 A-5030 B \cos (3 (c+d x))-1495 B \cos (4 (c+d x))-7685 B)-240 i (283 A-163 B) e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )\right )}{960 d (a (\cos (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.747, size = 729, normalized size = 2.3 \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(-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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Fricas [A] time = 1.84531, size = 732, normalized size = 2.31 \begin{align*} \frac{15 \, \sqrt{2}{\left ({\left (283 \, A - 163 \, B\right )} \cos \left (d x + c\right )^{5} + 3 \,{\left (283 \, A - 163 \, B\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (283 \, A - 163 \, B\right )} \cos \left (d x + c\right )^{3} +{\left (283 \, A - 163 \, B\right )} \cos \left (d x + c\right )^{2}\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 ) + \frac{2 \,{\left ({\left (2671 \, A - 1495 \, B\right )} \cos \left (d x + c\right )^{4} + 5 \,{\left (911 \, A - 503 \, B\right )} \cos \left (d x + c\right )^{3} + 32 \,{\left (49 \, A - 25 \, B\right )} \cos \left (d x + c\right )^{2} - 160 \,{\left (A - B\right )} \cos \left (d x + c\right ) + 96 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{480 \,{\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\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{7}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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