Optimal. Leaf size=332 \[ \frac {a^{5/2} (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{512 d}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (24 A+23 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{96 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A] time = 1.05, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {4221, 3046, 2976, 2981, 2770, 2774, 216} \[ \frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (24 A+23 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{96 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^{5/2} (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{512 d}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 216
Rule 2770
Rule 2774
Rule 2976
Rule 2981
Rule 3046
Rule 4221
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (\frac {1}{2} a (12 A+5 C)+\frac {5}{2} a C \cos (c+d x)\right ) \, dx}{6 a}\\ &=\frac {a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac {15}{4} a^2 (8 A+5 C)+\frac {5}{4} a^2 (24 A+23 C) \cos (c+d x)\right ) \, dx}{30 a}\\ &=\frac {a^2 (24 A+23 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {5}{8} a^3 (312 A+235 C)+\frac {15}{8} a^3 (136 A+109 C) \cos (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (24 A+23 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{384} \left (a^2 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (24 A+23 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{512} \left (a^2 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (24 A+23 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}+\frac {\left (a^2 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{1024}\\ &=\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (24 A+23 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {\left (a^2 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{512 d}\\ &=\frac {a^{5/2} (1304 A+1015 C) \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{512 d}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (24 A+23 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.31, size = 192, normalized size = 0.58 \[ \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \sqrt {a (\cos (c+d x)+1)} \left (3 \sqrt {2} (1304 A+1015 C) \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cos (c+d x)}+\left (\sin \left (\frac {3}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) ((2896 A+3234 C) \cos (c+d x)+4 (184 A+315 C) \cos (2 (c+d x))+96 A \cos (3 (c+d x))+4648 A+428 C \cos (3 (c+d x))+112 C \cos (4 (c+d x))+16 C \cos (5 (c+d x))+4193 C)\right )}{3072 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 217, normalized size = 0.65 \[ -\frac {3 \, {\left ({\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (1304 \, A + 1015 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {{\left (256 \, C a^{2} \cos \left (d x + c\right )^{6} + 896 \, C a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (8 \, A + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (184 \, A + 203 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{1536 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 491, normalized size = 1.48 \[ -\frac {\left (-1+\cos \left (d x +c \right )\right )^{3} \left (256 C \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+896 C \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+384 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1392 C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1472 A \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+1624 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2608 A \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2030 C \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+3912 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+3045 C \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+3912 A \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )+3045 C \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )\right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{1536 d \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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