Optimal. Leaf size=140 \[ \frac {A b x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {A b \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}-\frac {b B \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {17, 2827, 2715,
8, 2713} \begin {gather*} \frac {A b x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {A b \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{2 d}-\frac {b B \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {b B \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2713
Rule 2715
Rule 2827
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) (A+B \cos (c+d x)) \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {\left (A b \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{\sqrt {\cos (c+d x)}}+\frac {\left (b B \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {A b \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac {\left (A b \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{2 \sqrt {\cos (c+d x)}}-\frac {\left (b B \sqrt {b \cos (c+d x)}\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {\cos (c+d x)}}\\ &=\frac {A b x \sqrt {b \cos (c+d x)}}{2 \sqrt {\cos (c+d x)}}+\frac {b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {A b \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{2 d}-\frac {b B \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 69, normalized size = 0.49 \begin {gather*} \frac {(b \cos (c+d x))^{3/2} (6 A c+6 A d x+9 B \sin (c+d x)+3 A \sin (2 (c+d x))+B \sin (3 (c+d x)))}{12 d \cos ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 74, normalized size = 0.53
method | result | size |
default | \(\frac {\left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (2 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+3 A \sin \left (d x +c \right ) \cos \left (d x +c \right )+3 A \left (d x +c \right )+4 B \sin \left (d x +c \right )\right )}{6 d \cos \left (d x +c \right )^{\frac {3}{2}}}\) | \(74\) |
risch | \(\frac {b \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{i \left (d x +c \right )} A x}{{\mathrm e}^{2 i \left (d x +c \right )}+1}-\frac {i b \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{4 i \left (d x +c \right )} B}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {i b \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{3 i \left (d x +c \right )} A}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {3 i b \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{2 i \left (d x +c \right )} B}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {3 i b \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) B}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i b \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{-i \left (d x +c \right )} A}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}+\frac {i b \sqrt {b \cos \left (d x +c \right )}\, \left (\sqrt {\cos }\left (d x +c \right )\right ) {\mathrm e}^{-2 i \left (d x +c \right )} B}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}\) | \(325\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.62, size = 74, normalized size = 0.53 \begin {gather*} \frac {3 \, {\left (2 \, {\left (d x + c\right )} b + b \sin \left (2 \, d x + 2 \, c\right )\right )} A \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 )} B \sqrt {b}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 237, normalized size = 1.69 \begin {gather*} \left [\frac {3 \, A \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 \, B b \cos \left (d x + c\right )^{2} + 3 \, A b \cos \left (d x + c\right ) + 4 \, B 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 \, A 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 \, B b \cos \left (d x + c\right )^{2} + 3 \, A b \cos \left (d x + c\right ) + 4 \, B 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 93, normalized size = 0.66 \begin {gather*} \frac {b\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (3\,A\,\sin \left (c+d\,x\right )+3\,A\,\sin \left (3\,c+3\,d\,x\right )+10\,B\,\sin \left (2\,c+2\,d\,x\right )+B\,\sin \left (4\,c+4\,d\,x\right )+12\,A\,d\,x\,\cos \left (c+d\,x\right )\right )}{12\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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