Optimal. Leaf size=333 \[ \frac{(157 A-85 B+45 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A-475 B+195 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A-1495 B+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(283 A-163 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(21 A-13 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 1.24132, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {4221, 3041, 2978, 2984, 12, 2782, 205} \[ \frac{(157 A-85 B+45 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A-475 B+195 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A-1495 B+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(283 A-163 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(21 A-13 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3041
Rule 2978
Rule 2984
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \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)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx\\ &=-\frac{(A-B+C) \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+5 C)-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+C) \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+5 C) \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+45 C)-\frac{3}{2} a^2 (21 A-13 B+5 C) \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+C) \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+5 C) \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+45 C) \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+195 C)+\frac{1}{2} a^3 (157 A-85 B+45 C) \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+195 C) \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+C) \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+5 C) \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+45 C) \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+735 C)-\frac{1}{8} a^4 (787 A-475 B+195 C) \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+735 C) \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+195 C) \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+C) \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+5 C) \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+45 C) \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+75 C)}{32 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{15 a^7}\\ &=\frac{(2671 A-1495 B+735 C) \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+195 C) \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+C) \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+5 C) \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+45 C) \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+75 C) \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+735 C) \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+195 C) \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+C) \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+5 C) \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+45 C) \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+75 C) \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+75 C) \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+735 C) \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+195 C) \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+C) \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+5 C) \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+45 C) \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 = 27.8309, size = 7162, normalized size = 21.51 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.195, size = 1045, normalized size = 3.1 \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(-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 = 2.32035, size = 801, normalized size = 2.41 \begin{align*} \frac{15 \, \sqrt{2}{\left ({\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \,{\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (283 \, A - 163 \, B + 75 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (283 \, A - 163 \, B + 75 \, C\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 + 735 \, C\right )} \cos \left (d x + c\right )^{4} + 5 \,{\left (911 \, A - 503 \, B + 255 \, C\right )} \cos \left (d x + c\right )^{3} + 32 \,{\left (49 \, A - 25 \, B + 15 \, C\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 (C \cos \left (d x + c\right )^{2} + 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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