Optimal. Leaf size=228 \[ \frac{2 a^2 (10 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d} \]
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Rubi [A] time = 0.649973, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2961, 2975, 2980, 2772, 2771} \[ \frac{2 a^2 (10 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d} \]
Antiderivative was successfully verified.
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Rule 2961
Rule 2975
Rule 2980
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 a A \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{9} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{1}{2} a (10 A+9 B)+\frac{3}{2} a (2 A+3 B) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (10 A+9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{21} \left (a (34 A+39 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (34 A+39 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (10 A+9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{105} \left (4 a (34 A+39 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{8 a^2 (34 A+39 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (34 A+39 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (10 A+9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac{1}{315} \left (8 a (34 A+39 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{16 a^2 (34 A+39 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{8 a^2 (34 A+39 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (34 A+39 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (10 A+9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a A \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [A] time = 0.69188, size = 124, normalized size = 0.54 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((374 A+324 B) \cos (c+d x)+11 (34 A+39 B) \cos (2 (c+d x))+68 A \cos (3 (c+d x))+68 A \cos (4 (c+d x))+376 A+78 B \cos (3 (c+d x))+78 B \cos (4 (c+d x))+351 B)}{315 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.684, size = 139, normalized size = 0.6 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 272\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+312\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+136\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+156\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+102\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+117\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+85\,A\cos \left ( dx+c \right ) +45\,B\cos \left ( dx+c \right ) +35\,A \right ) \cos \left ( dx+c \right ) }{315\,d\sin \left ( dx+c \right ) } \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{11}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.15245, size = 836, normalized size = 3.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4335, size = 338, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (8 \,{\left (34 \, A + 39 \, B\right )} a \cos \left (d x + c\right )^{4} + 4 \,{\left (34 \, A + 39 \, B\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (34 \, A + 39 \, B\right )} a \cos \left (d x + c\right )^{2} + 5 \,{\left (17 \, A + 9 \, B\right )} a \cos \left (d x + c\right ) + 35 \, A a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )} \sqrt{\cos \left (d x + c\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{\left (B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \left (d x + c\right )^{\frac{11}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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