Optimal. Leaf size=294 \[ \frac {a^{5/2} (326 A+283 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{128 d}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{192 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^3 (170 A+157 B) \sin (c+d x)}{240 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{128 d \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (10 A+13 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{40 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A] time = 0.87, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2961, 2976, 2981, 2770, 2774, 216} \[ \frac {a^3 (326 A+283 B) \sin (c+d x)}{192 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^3 (170 A+157 B) \sin (c+d x)}{240 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (10 A+13 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{40 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^{5/2} (326 A+283 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{128 d}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{128 d \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 216
Rule 2770
Rule 2774
Rule 2961
Rule 2976
Rule 2981
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\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} (A+B \cos (c+d x)) \, dx\\ &=\frac {a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \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 {5}{2} a (2 A+B)+\frac {1}{2} a (10 A+13 B) \cos (c+d x)\right ) \, dx\\ &=\frac {a^2 (10 A+13 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{20} \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}{4} a^2 (26 A+21 B)+\frac {1}{4} a^2 (170 A+157 B) \cos (c+d x)\right ) \, dx\\ &=\frac {a^3 (170 A+157 B) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (10 A+13 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{96} \left (a^2 (326 A+283 B) \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 (170 A+157 B) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (10 A+13 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{128} \left (a^2 (326 A+283 B) \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 (170 A+157 B) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (10 A+13 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}+\frac {1}{256} \left (a^2 (326 A+283 B) \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\\ &=\frac {a^3 (170 A+157 B) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (10 A+13 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {\left (a^2 (326 A+283 B) \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 )}{128 d}\\ &=\frac {a^{5/2} (326 A+283 B) \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)}}{128 d}+\frac {a^3 (170 A+157 B) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^2 (10 A+13 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (326 A+283 B) \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.44, size = 181, normalized size = 0.62 \[ \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \sqrt {a (\cos (c+d x)+1)} \left (15 \sqrt {2} (326 A+283 B) \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cos (c+d x)}+\left (\sin \left (\frac {3}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) ((3620 A+3874 B) \cos (c+d x)+4 (230 A+331 B) \cos (2 (c+d x))+120 A \cos (3 (c+d x))+5810 A+348 B \cos (3 (c+d x))+48 B \cos (4 (c+d x))+5521 B)\right )}{3840 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 203, normalized size = 0.69 \[ -\frac {15 \, {\left ({\left (326 \, A + 283 \, B\right )} a^{2} \cos \left (d x + c\right ) + {\left (326 \, A + 283 \, B\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {{\left (384 \, B a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (10 \, A + 29 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (230 \, A + 283 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (326 \, A + 283 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (326 \, A + 283 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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 = 0.35, size = 455, normalized size = 1.55 \[ -\frac {\left (-1+\cos \left (d x +c \right )\right )^{3} \left (384 B \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+480 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1392 B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1840 A \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+2264 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3260 A \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2830 B \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+4890 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+4245 B \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+4890 A \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )+4245 B \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )\right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{1920 d \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{6}} \]
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 \[ \int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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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