Optimal. Leaf size=764 \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt{a+b \cos (c+d x)}}{192 b^2 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 a b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 C-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 2.76907, antiderivative size = 764, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.178, Rules used = {4221, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt{a+b \cos (c+d x)}}{192 b^2 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (-6 a^2 b (4 B+C)+9 a^3 C-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 a b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 C-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3049
Rule 3061
Rule 3053
Rule 2809
Rule 2998
Rule 2816
Rule 2994
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^{3/2} \left (\frac{a C}{2}+b (4 A+3 C) \cos (c+d x)+\frac{1}{2} (8 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{4 b}\\ &=\frac{(8 b B-3 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{1}{4} a (8 b B+3 a C)+\frac{1}{2} b (24 a A+16 b B+15 a C) \cos (c+d x)+\frac{3}{4} \left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{12 b}\\ &=\frac{\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} a \left (48 A b^2+56 a b B+3 a^2 C+36 b^2 C\right )+\frac{1}{4} b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (96 A+57 C)\right ) \cos (c+d x)+\frac{1}{8} \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{24 b}\\ &=\frac{\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b^2 d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{8} a \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right )+\frac{1}{4} a b \left (48 A b^2+56 a b B+3 a^2 C+36 b^2 C\right ) \cos (c+d x)-\frac{3}{8} \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{48 b^2}\\ &=\frac{\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b^2 d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{8} a \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right )+\frac{1}{4} a b \left (48 A b^2+56 a b B+3 a^2 C+36 b^2 C\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{48 b^2}-\frac{\left (\left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx}{128 b^2}\\ &=\frac{\sqrt{a+b} \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b^2 d}-\frac{\left (a \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{384 b^2}-\frac{\left (a \left (9 a^3 C-6 a^2 b (4 B+C)-8 b^3 (12 A+16 B+9 C)-4 a b^2 (60 A+28 B+39 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{384 b^2}\\ &=-\frac{(a-b) \sqrt{a+b} \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{192 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (9 a^3 C-6 a^2 b (4 B+C)-8 b^3 (12 A+16 B+9 C)-4 a b^2 (60 A+28 B+39 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{192 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b^2 d}\\ \end{align*}
Mathematica [A] time = 15.3309, size = 603, normalized size = 0.79 \[ \frac{\frac{2 \tan (c+d x) (a+b \cos (c+d x)) \left (3 a^2 C+4 b (9 a C+8 b B) \cos (c+d x)+56 a b B+48 A b^2+12 b^2 C \cos (2 (c+d x))+48 b^2 C\right )}{b}-\frac{-b \tan \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} (a+b \cos (c+d x))+a (a+b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (-6 a^2 b (4 B+3 C)+9 a^3 C+12 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \cos (c+d x))}{a+b}} F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )-b (a+b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (24 a^2 b B-9 a^3 C+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \cos (c+d x))}{a+b}} E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )+3 \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 C-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \cos (c+d x))}{a+b}} \left ((a-b) F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )-2 b \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )\right )}{b^3 \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2}}}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.508, size = 5495, normalized size = 7.2 \begin{align*} \text{output 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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \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 (\frac{{\left (C b \cos \left (d x + c\right )^{3} +{\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + A a +{\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{\sqrt{\sec \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 \frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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