Optimal. Leaf size=760 \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\sin (c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (2 a^2 b (192 A+132 B+59 C)+15 a^3 C+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 C+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{(5 a C+8 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 2.80298, antiderivative size = 760, 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} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left (264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\sin (c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (2 a^2 b (192 A+132 B+59 C)+15 a^3 C+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (264 a^2 b B+15 a^3 C+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{192 a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left (120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 C+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{(5 a C+8 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d \sqrt{\sec (c+d x)}} \]
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 (a+b \cos (c+d x))^{5/2} \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 \frac{(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}+\frac{1}{4} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \cos (c+d x))^{3/2} \left (\frac{1}{2} a (8 A+C)+(4 A b+4 a B+3 b C) \cos (c+d x)+\frac{1}{2} (8 b B+5 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{(8 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}+\frac{1}{12} \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 (48 a A+8 b B+11 a C)+\frac{1}{2} \left (24 a^2 B+16 b^2 B+a b (48 A+31 C)\right ) \cos (c+d x)+\frac{3}{4} \left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 d \sqrt{\sec (c+d x)}}+\frac{(8 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}+\frac{1}{24} \left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} a \left (192 a^2 A+48 A b^2+104 a b B+59 a^2 C+36 b^2 C\right )+\frac{1}{4} \left (96 a^3 B+152 a b^2 B+12 b^3 (4 A+3 C)+a^2 b (288 A+161 C)\right ) \cos (c+d x)+\frac{1}{8} \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx\\ &=\frac{\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 d \sqrt{\sec (c+d x)}}+\frac{(8 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}+\frac{\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{8} a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right )+\frac{1}{4} a b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (192 A+59 C)\right ) \cos (c+d x)+\frac{3}{8} \left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 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}\\ &=\frac{\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 d \sqrt{\sec (c+d x)}}+\frac{(8 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}+\frac{\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{8} a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right )+\frac{1}{4} a b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (192 A+59 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{48 b}+\frac{\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 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}\\ &=-\frac{\sqrt{a+b} \left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 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^2 d \sqrt{\sec (c+d x)}}+\frac{\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 d \sqrt{\sec (c+d x)}}+\frac{(8 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}+\frac{\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b d}-\frac{\left (a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 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}+\frac{\left (a \left (15 a^3 C+8 b^3 (12 A+16 B+9 C)+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 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}\\ &=-\frac{(a-b) \sqrt{a+b} \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 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 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left (15 a^3 C+8 b^3 (12 A+16 B+9 C)+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 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 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 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^2 d \sqrt{\sec (c+d x)}}+\frac{\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{32 d \sqrt{\sec (c+d x)}}+\frac{(8 b B+5 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 d \sqrt{\sec (c+d x)}}+\frac{C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}+\frac{\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{192 b d}\\ \end{align*}
Mathematica [B] time = 25.6615, size = 5555, normalized size = 7.31 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.568, size = 5875, normalized size = 7.7 \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{\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{5}{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 ({\left (C b^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, 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(-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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