Optimal. Leaf size=283 \[ \frac {a^{3/2} (176 A+150 B+133 C) \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{128 d}+\frac {a^2 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{240 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{192 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{128 d \sqrt {a \cos (c+d x)+a}}+\frac {a (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{40 d}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.75, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3045, 2976, 2981, 2770, 2774, 216} \[ \frac {a^2 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{240 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{192 d \sqrt {a \cos (c+d x)+a}}+\frac {a^{3/2} (176 A+150 B+133 C) \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{128 d}+\frac {a^2 (176 A+150 B+133 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{128 d \sqrt {a \cos (c+d x)+a}}+\frac {a (10 B+3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{40 d}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 216
Rule 2770
Rule 2774
Rule 2976
Rule 2981
Rule 3045
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac {5}{2} a (2 A+C)+\frac {1}{2} a (10 B+3 C) \cos (c+d x)\right ) \, dx}{5 a}\\ &=\frac {a (10 B+3 C) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d}+\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\int \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {5}{4} a^2 (16 A+10 B+11 C)+\frac {1}{4} a^2 (80 A+90 B+67 C) \cos (c+d x)\right ) \, dx}{20 a}\\ &=\frac {a^2 (80 A+90 B+67 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (10 B+3 C) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d}+\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{96} (a (176 A+150 B+133 C)) \int \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {a^2 (176 A+150 B+133 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (80 A+90 B+67 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (10 B+3 C) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d}+\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{128} (a (176 A+150 B+133 C)) \int \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {a^2 (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (80 A+90 B+67 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (10 B+3 C) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d}+\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{256} (a (176 A+150 B+133 C)) \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {a^2 (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (80 A+90 B+67 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (10 B+3 C) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d}+\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}-\frac {(a (176 A+150 B+133 C)) \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^{3/2} (176 A+150 B+133 C) \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d}+\frac {a^2 (176 A+150 B+133 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (176 A+150 B+133 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (80 A+90 B+67 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \cos (c+d x)}}+\frac {a (10 B+3 C) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{40 d}+\frac {C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 1.61, size = 170, normalized size = 0.60 \[ \frac {a \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} \left (15 \sqrt {2} (176 A+150 B+133 C) \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \sin \left (\frac {1}{2} (c+d x)\right ) \sqrt {\cos (c+d x)} (2 (880 A+930 B+1007 C) \cos (c+d x)+4 (80 A+150 B+181 C) \cos (2 (c+d x))+2960 A+120 B \cos (3 (c+d x))+2850 B+228 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+2671 C)\right )}{3840 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 195, normalized size = 0.69 \[ \frac {{\left (384 \, C a \cos \left (d x + c\right )^{4} + 48 \, {\left (10 \, B + 19 \, C\right )} a \cos \left (d x + c\right )^{3} + 8 \, {\left (80 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right )^{2} + 10 \, {\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right ) + 15 \, {\left (176 \, A + 150 \, B + 133 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, {\left ({\left (176 \, A + 150 \, B + 133 \, C\right )} a \cos \left (d x + c\right ) + {\left (176 \, A + 150 \, B + 133 \, C\right )} a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 731, normalized size = 2.58 \[ \text {result too large to display} \]
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 \[ \int {\cos \left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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