Optimal. Leaf size=284 \[ \frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d} \]
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Rubi [A] time = 0.90749, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4221, 3043, 2975, 2980, 2772, 2771} \[ \frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3043
Rule 2975
Rule 2980
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{13}{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} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{13}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{1}{2} a (3 A+11 B)+\frac{1}{2} a (6 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac{2 a (3 A+11 B) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{1}{4} a^2 (84 A+110 B+99 C)+\frac{3}{4} a^2 (24 A+22 B+33 C) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 a^2 (84 A+110 B+99 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (3 A+11 B) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{231} \left (a (336 A+374 B+429 C) \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 (336 A+374 B+429 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (84 A+110 B+99 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (3 A+11 B) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (4 a (336 A+374 B+429 C) \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}{1155}\\ &=\frac{8 a^2 (336 A+374 B+429 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (336 A+374 B+429 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (84 A+110 B+99 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (3 A+11 B) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (8 a (336 A+374 B+429 C) \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}{3465}\\ &=\frac{16 a^2 (336 A+374 B+429 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt{a+a \cos (c+d x)}}+\frac{8 a^2 (336 A+374 B+429 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (336 A+374 B+429 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (84 A+110 B+99 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a (3 A+11 B) \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 1.14666, size = 187, normalized size = 0.66 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((12684 A+12386 B+12441 C) \cos (c+d x)+(4368 A+4862 B+4422 C) \cos (2 (c+d x))+4368 A \cos (3 (c+d x))+672 A \cos (4 (c+d x))+672 A \cos (5 (c+d x))+4956 A+4862 B \cos (3 (c+d x))+748 B \cos (4 (c+d x))+748 B \cos (5 (c+d x))+4114 B+5577 C \cos (3 (c+d x))+858 C \cos (4 (c+d x))+858 C \cos (5 (c+d x))+3564 C)}{6930 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.221, size = 205, normalized size = 0.7 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 2688\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+2992\,B \left ( \cos \left ( dx+c \right ) \right ) ^{5}+3432\,C \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1344\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1496\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1716\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1008\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1122\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1287\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+840\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+935\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+495\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+735\,A\cos \left ( dx+c \right ) +385\,B\cos \left ( dx+c \right ) +315\,A \right ) \cos \left ( dx+c \right ) }{3465\,d\sin \left ( dx+c \right ) } \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{13}{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 = 1.94937, size = 1438, normalized size = 5.06 \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 = 1.55082, size = 443, normalized size = 1.56 \begin{align*} \frac{2 \,{\left (8 \,{\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \,{\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \,{\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \,{\left (168 \, A + 187 \, B + 99 \, C\right )} a \cos \left (d x + c\right )^{2} + 35 \,{\left (21 \, A + 11 \, B\right )} a \cos \left (d x + c\right ) + 315 \, A a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \,{\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\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 (C \cos \left (d x + c\right )^{2} + 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{13}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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