Optimal. Leaf size=158 \[ \frac{4 a^2 (A+3 C) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{2 a^2 (7 A-15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )}{5 d}+\frac{16 a^2 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.464138, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4114, 3044, 2976, 2968, 3023, 2748, 2641, 2639} \[ \frac{4 a^2 (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a^2 (7 A-15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )}{5 d}+\frac{16 a^2 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4114
Rule 3044
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\int \frac{(a+a \cos (c+d x))^2 \left (C+A \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 \int \frac{(a+a \cos (c+d x))^2 \left (2 a C+\frac{1}{2} a (A-5 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{a}\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 (A-5 C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{4 \int \frac{(a+a \cos (c+d x)) \left (\frac{1}{4} a^2 (A+15 C)+\frac{1}{4} a^2 (7 A-15 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{5 a}\\ &=\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 (A-5 C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{4 \int \frac{\frac{1}{4} a^3 (A+15 C)+\left (\frac{1}{4} a^3 (7 A-15 C)+\frac{1}{4} a^3 (A+15 C)\right ) \cos (c+d x)+\frac{1}{4} a^3 (7 A-15 C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{5 a}\\ &=\frac{2 a^2 (7 A-15 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 (A-5 C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{8 \int \frac{\frac{5}{4} a^3 (A+3 C)+3 a^3 A \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{15 a}\\ &=\frac{2 a^2 (7 A-15 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 (A-5 C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{5} \left (8 a^2 A\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (2 a^2 (A+3 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{16 a^2 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a^2 (A+3 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a^2 (7 A-15 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 C (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 (A-5 C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [C] time = 6.4492, size = 799, normalized size = 5.06 \[ \frac{\sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \left (-\frac{(8 \cos (2 c) A+8 A-5 C+5 C \cos (2 c)) \csc (c) \sec (c)}{10 d}+\frac{C \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{2 A \cos (d x) \sin (c)}{3 d}+\frac{A \cos (2 d x) \sin (2 c)}{10 d}+\frac{2 A \cos (c) \sin (d x)}{3 d}+\frac{A \cos (2 c) \sin (2 d x)}{10 d}\right ) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C}-\frac{4 A \csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \left (\frac{\text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right ) \sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{1-\cos \left (d x+\tan ^{-1}(\tan (c))\right )} \sqrt{\cos \left (d x+\tan ^{-1}(\tan (c))\right )+1} \sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}}}\right ) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{2 A \csc (c) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (\sec (c+d x) a+a)^2 \left (C \sec ^2(c+d x)+A\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 2.156, size = 440, normalized size = 2.8 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{4} + 2 \, C a^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} +{\left (A + C\right )} a^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + 2 \, A a^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + A a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt{\cos \left (d x + c\right )}, x\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 \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{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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