Optimal. Leaf size=210 \[ \frac {2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac {2 b (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^2 d}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac {10 b B \sin (c+d x) \sqrt {b \cos (c+d x)}}{21 d} \]
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Rubi [A] time = 0.23, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {16, 3023, 2748, 2635, 2640, 2639, 2642, 2641} \[ \frac {2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac {2 b (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^2 d}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac {10 b B \sin (c+d x) \sqrt {b \cos (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2639
Rule 2640
Rule 2641
Rule 2642
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int \cos (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 {\int (b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b}\\ &=\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac {2 \int (b \cos (c+d x))^{5/2} \left (\frac {1}{2} b (9 A+7 C)+\frac {9}{2} b B \cos (c+d x)\right ) \, dx}{9 b^2}\\ &=\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac {B \int (b \cos (c+d x))^{7/2} \, dx}{b^2}+\frac {(9 A+7 C) \int (b \cos (c+d x))^{5/2} \, dx}{9 b}\\ &=\frac {2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac {1}{7} (5 B) \int (b \cos (c+d x))^{3/2} \, dx+\frac {1}{15} (b (9 A+7 C)) \int \sqrt {b \cos (c+d x)} \, dx\\ &=\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac {1}{21} \left (5 b^2 B\right ) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx+\frac {\left (b (9 A+7 C) \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}}\\ &=\frac {2 b (9 A+7 C) \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}+\frac {\left (5 b^2 B \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}}\\ &=\frac {2 b (9 A+7 C) \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.94, size = 128, normalized size = 0.61 \[ \frac {(b \cos (c+d x))^{5/2} \left (\sin (c+d x) \sqrt {\cos (c+d x)} (7 (36 A+43 C) \cos (c+d x)+5 (18 B \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))))+84 (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+300 B F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{630 b d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{4} + B b \cos \left (d x + c\right )^{3} + A b \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.60, size = 384, normalized size = 1.83 \[ -\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{2} \left (-1120 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (720 B +2240 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-504 A -1080 B -2072 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (504 A +840 B +952 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-126 A -240 B -168 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-189 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+75 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{315 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}} \cos \left (d x + c\right )\,{d x} \]
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 \[ \int \cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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