Optimal. Leaf size=229 \[ -\frac {b (5 A+4 C) \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {b (5 A+4 C) \sin (c+d x) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {3 b B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d}+\frac {3 b B \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d}+\frac {b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{5 d} \]
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Rubi [A] time = 0.13, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {17, 3023, 2748, 2633, 2635, 8} \[ -\frac {b (5 A+4 C) \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {b (5 A+4 C) \sin (c+d x) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {3 b B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d}+\frac {3 b B \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d}+\frac {b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2633
Rule 2635
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) (5 A+4 C+5 B \cos (c+d x)) \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {\left (b B \sqrt {b \cos (c+d x)}\right ) \int \cos ^4(c+d x) \, dx}{\sqrt {\cos (c+d x)}}+\frac {\left (b (5 A+4 C) \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=\frac {b B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {\left (3 b B \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt {\cos (c+d x)}}-\frac {\left (b (5 A+4 C) \sqrt {b \cos (c+d x)}\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d \sqrt {\cos (c+d x)}}\\ &=\frac {b (5 A+4 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {3 b B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {b (5 A+4 C) \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {\left (3 b B \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{8 \sqrt {\cos (c+d x)}}\\ &=\frac {3 b B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b (5 A+4 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {3 b B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {b (5 A+4 C) \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{15 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 109, normalized size = 0.48 \[ \frac {(b \cos (c+d x))^{3/2} (60 (6 A+5 C) \sin (c+d x)+40 A \sin (3 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+50 C \sin (3 (c+d x))+6 C \sin (5 (c+d x)))}{480 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 309, normalized size = 1.35 \[ \left [\frac {45 \, B \sqrt {-b} b \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) + 2 \, {\left (24 \, C b \cos \left (d x + c\right )^{4} + 30 \, B b \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{2} + 45 \, B b \cos \left (d x + c\right ) + 16 \, {\left (5 \, A + 4 \, C\right )} b\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )}, \frac {45 \, B b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right ) + {\left (24 \, C b \cos \left (d x + c\right )^{4} + 30 \, B b \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{2} + 45 \, B b \cos \left (d x + c\right ) + 16 \, {\left (5 \, A + 4 \, C\right )} b\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{120 \, d \cos \left (d x + c\right )}\right ] \]
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 = 0.29, size = 134, normalized size = 0.59 \[ \frac {\left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (24 C \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+30 B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+40 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+32 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+45 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+80 A \sin \left (d x +c \right )+45 B \left (d x +c \right )+64 C \sin \left (d x +c \right )\right )}{120 d \cos \left (d x +c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 169, normalized size = 0.74 \[ \frac {40 \, {\left (b \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} A \sqrt {b} + 15 \, {\left (12 \, {\left (d x + c\right )} b + b \sin \left (4 \, d x + 4 \, c\right ) + 8 \, b \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} B \sqrt {b} + 2 \, {\left (3 \, b \sin \left (5 \, d x + 5 \, c\right ) + 25 \, b \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right ) + 150 \, b \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right )\right )} C \sqrt {b}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.15, size = 142, normalized size = 0.62 \[ \frac {b\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (120\,B\,\sin \left (c+d\,x\right )+400\,A\,\sin \left (2\,c+2\,d\,x\right )+40\,A\,\sin \left (4\,c+4\,d\,x\right )+135\,B\,\sin \left (3\,c+3\,d\,x\right )+15\,B\,\sin \left (5\,c+5\,d\,x\right )+350\,C\,\sin \left (2\,c+2\,d\,x\right )+56\,C\,\sin \left (4\,c+4\,d\,x\right )+6\,C\,\sin \left (6\,c+6\,d\,x\right )+360\,B\,d\,x\,\cos \left (c+d\,x\right )\right )}{480\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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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