Optimal. Leaf size=221 \[ \frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}+\frac{(A-13 B) \sin (c+d x)}{48 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.723135, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {2961, 2977, 2978, 12, 2782, 205} \[ \frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}+\frac{(A-13 B) \sin (c+d x)}{48 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2961
Rule 2977
Rule 2978
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx\\ &=\frac{(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \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}{2} a (A-B)+a (A+5 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=\frac{(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{(A-13 B) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} a^2 (A-13 B)+\frac{9}{2} a^2 (A+3 B) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=\frac{(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{(A-13 B) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{3 a^3 (7 A+5 B)}{8 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=\frac{(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{(A-13 B) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left ((7 A+5 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}{128 a^3}\\ &=\frac{(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{(A-13 B) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}}-\frac{\left ((7 A+5 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 )}{64 a^2 d}\\ &=\frac{(7 A+5 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)}}{64 \sqrt{2} a^{7/2} d}+\frac{(A-B) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{(A-13 B) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 7.16513, size = 488, normalized size = 2.21 \[ \frac{\cos ^7\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{\sec (c+d x)} \left (\frac{(17 A+67 B) \sin \left (\frac{c}{2}\right ) \cos \left (\frac{d x}{2}\right )}{12 d}+\frac{(17 A+67 B) \cos \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )}{12 d}+\frac{\sec \left (\frac{c}{2}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sin \left (\frac{d x}{2}\right )-B \sin \left (\frac{d x}{2}\right )\right )}{3 d}+\frac{\sec \left (\frac{c}{2}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (29 B \sin \left (\frac{d x}{2}\right )-17 A \sin \left (\frac{d x}{2}\right )\right )}{12 d}+\frac{\sec \left (\frac{c}{2}\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (19 A \sin \left (\frac{d x}{2}\right )-151 B \sin \left (\frac{d x}{2}\right )\right )}{24 d}+\frac{(A-B) \tan \left (\frac{c}{2}\right ) \sec ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}-\frac{(17 A-29 B) \tan \left (\frac{c}{2}\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{12 d}+\frac{(19 A-151 B) \tan \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{24 d}\right )}{(a (\cos (c+d x)+1))^{7/2}}+\frac{i (7 A+5 B) e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^7\left (\frac{c}{2}+\frac{d x}{2}\right ) \tanh ^{-1}\left (\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )}{8 d (a (\cos (c+d x)+1))^{7/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.598, size = 512, 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] 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}} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84914, size = 675, normalized size = 3.05 \begin{align*} -\frac{3 \, \sqrt{2}{\left ({\left (7 \, A + 5 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (7 \, A + 5 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \,{\left (7 \, A + 5 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (7 \, A + 5 \, B\right )} \cos \left (d x + c\right ) + 7 \, A + 5 \, 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 ) - \frac{2 \,{\left ({\left (17 \, A + 67 \, B\right )} \cos \left (d x + c\right )^{3} + 10 \,{\left (7 \, A + 5 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (7 \, A + 5 \, 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 )}}}{384 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} 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{7}{2}} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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