Optimal. Leaf size=147 \[ \frac {b B x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b (3 A+2 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {b B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {b C \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {17, 3102, 2813}
\begin {gather*} \frac {b (3 A+2 C) \sin (c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {b B x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b B \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{2 d}+\frac {b C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 17
Rule 2813
Rule 3102
Rubi steps
\begin {align*} \int \frac {(b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx &=\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos (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 {3}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos (c+d x) (3 A+2 C+3 B \cos (c+d x)) \, dx}{3 \sqrt {\cos (c+d x)}}\\ &=\frac {b B x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b (3 A+2 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {b B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {b C \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 76, normalized size = 0.52 \begin {gather*} \frac {b \sqrt {b \cos (c+d x)} (6 B c+6 B d x+3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+C \sin (3 (c+d x)))}{12 d \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 83, normalized size = 0.56
method | result | size |
default | \(\frac {\left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (2 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+3 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+6 A \sin \left (d x +c \right )+3 B \left (d x +c \right )+4 C \sin \left (d x +c \right )\right )}{6 d \cos \left (d x +c \right )^{\frac {3}{2}}}\) | \(83\) |
risch | \(\frac {b B x \sqrt {b \cos \left (d x +c \right )}}{2 \sqrt {\cos \left (d x +c \right )}}+\frac {b \sqrt {b \cos \left (d x +c \right )}\, \left (4 A +3 C \right ) \sin \left (d x +c \right )}{4 \sqrt {\cos \left (d x +c \right )}\, d}+\frac {b \sqrt {b \cos \left (d x +c \right )}\, C \sin \left (3 d x +3 c \right )}{12 \sqrt {\cos \left (d x +c \right )}\, d}+\frac {b \sqrt {b \cos \left (d x +c \right )}\, B \sin \left (2 d x +2 c \right )}{4 \sqrt {\cos \left (d x +c \right )}\, d}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.66, size = 86, normalized size = 0.59 \begin {gather*} \frac {12 \, A b^{\frac {3}{2}} \sin \left (d x + c\right ) + 3 \, {\left (2 \, {\left (d x + c\right )} b + b \sin \left (2 \, d x + 2 \, c\right )\right )} B \sqrt {b} + {\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 )} C \sqrt {b}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 249, normalized size = 1.69 \begin {gather*} \left [\frac {3 \, 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 (2 \, C b \cos \left (d x + c\right )^{2} + 3 \, B b \cos \left (d x + c\right ) + 2 \, {\left (3 \, A + 2 \, 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 )}{12 \, d \cos \left (d x + c\right )}, \frac {3 \, 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 (2 \, C b \cos \left (d x + c\right )^{2} + 3 \, B b \cos \left (d x + c\right ) + 2 \, {\left (3 \, A + 2 \, 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 )}{6 \, d \cos \left (d x + c\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 71, normalized size = 0.48 \begin {gather*} \frac {b\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (12\,A\,\sin \left (c+d\,x\right )+9\,C\,\sin \left (c+d\,x\right )+3\,B\,\sin \left (2\,c+2\,d\,x\right )+C\,\sin \left (3\,c+3\,d\,x\right )+6\,B\,d\,x\right )}{12\,d\,\sqrt {\cos \left (c+d\,x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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