Optimal. Leaf size=297 \[ \frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(1015 A-363 B) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \]
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Rubi [A] time = 1.03211, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2978, 2984, 12, 2782, 205} \[ \frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(1015 A-363 B) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2978
Rule 2984
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx &=-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}+\frac{\int \frac{\frac{3}{2} a (5 A-B)-4 a (A-B) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}}+\frac{\int \frac{\frac{3}{4} a^2 (63 A-19 B)-\frac{3}{2} a^2 (23 A-11 B) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}}+\frac{\int \frac{\frac{3}{8} a^3 (579 A-199 B)-\frac{3}{2} a^3 (109 A-41 B) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}}+\frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{-\frac{3}{16} a^4 (1887 A-691 B)+\frac{3}{8} a^4 (579 A-199 B) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{72 a^7}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}}+\frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{\int \frac{9 a^5 (1015 A-363 B)}{32 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{36 a^8}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}}+\frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}+\frac{(1015 A-363 B) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{128 a^3}\\ &=-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}}+\frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}-\frac{(1015 A-363 B) \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 )}{64 a^2 d}\\ &=\frac{(1015 A-363 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 )}{64 \sqrt{2} a^{7/2} d}-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}}+\frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 5.10489, size = 262, normalized size = 0.88 \[ \frac{\cos ^7\left (\frac{1}{2} (c+d x)\right ) \left (-\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) (4 (9415 A-3579 B) \cos (c+d x)+8 (3069 A-1145 B) \cos (2 (c+d x))+10164 A \cos (3 (c+d x))+1887 A \cos (4 (c+d x))+21641 A-3748 B \cos (3 (c+d x))-691 B \cos (4 (c+d x))-8469 B)}{32 \cos ^{\frac{3}{2}}(c+d x)}+\frac{3 i (1015 A-363 B) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )}{\sqrt{1+e^{2 i (c+d x)}}}\right )}{24 d (a (\cos (c+d x)+1))^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.484, size = 715, normalized size = 2.4 \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 = 1.88757, size = 878, normalized size = 2.96 \begin{align*} \frac{3 \, \sqrt{2}{\left ({\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{5} + 6 \,{\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (1015 \, A - 363 \, B\right )} \cos \left (d x + c\right )^{3} +{\left (1015 \, A - 363 \, 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{a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \,{\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) - 2 \,{\left ({\left (1887 \, A - 691 \, B\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (2541 \, A - 937 \, B\right )} \cos \left (d x + c\right )^{3} + 39 \,{\left (109 \, A - 41 \, B\right )} \cos \left (d x + c\right )^{2} + 128 \,{\left (7 \, A - 3 \, B\right )} \cos \left (d x + c\right ) - 128 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{384 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} 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{B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{7}{2}} \cos \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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