Integrand size = 22, antiderivative size = 142 \[ \int \sqrt {c+d x} \cos (a+b x) \sin (a+b x) \, dx=-\frac {\sqrt {c+d x} \cos (2 a+2 b x)}{4 b}+\frac {\sqrt {d} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {d} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{8 b^{3/2}} \]
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Time = 0.26 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4491, 12, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int \sqrt {c+d x} \cos (a+b x) \sin (a+b x) \, dx=\frac {\sqrt {\pi } \sqrt {d} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {\pi } \sqrt {d} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {c+d x} \cos (2 a+2 b x)}{4 b} \]
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Rule 12
Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{2} \sqrt {c+d x} \sin (2 a+2 b x) \, dx \\ & = \frac {1}{2} \int \sqrt {c+d x} \sin (2 a+2 b x) \, dx \\ & = -\frac {\sqrt {c+d x} \cos (2 a+2 b x)}{4 b}+\frac {d \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{8 b} \\ & = -\frac {\sqrt {c+d x} \cos (2 a+2 b x)}{4 b}+\frac {\left (d \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{8 b}-\frac {\left (d \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{8 b} \\ & = -\frac {\sqrt {c+d x} \cos (2 a+2 b x)}{4 b}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 b}-\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 b} \\ & = -\frac {\sqrt {c+d x} \cos (2 a+2 b x)}{4 b}+\frac {\sqrt {d} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{8 b^{3/2}}-\frac {\sqrt {d} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{8 b^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.02 \[ \int \sqrt {c+d x} \cos (a+b x) \sin (a+b x) \, dx=\frac {1}{2} \left (-\frac {e^{2 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},-\frac {2 i b (c+d x)}{d}\right )}{4 \sqrt {2} b \sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{-2 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},\frac {2 i b (c+d x)}{d}\right )}{4 \sqrt {2} b \sqrt {\frac {i b (c+d x)}{d}}}\right ) \]
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Time = 0.61 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}}{d}\) | \(142\) |
default | \(\frac {-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}}{d}\) | \(142\) |
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Time = 0.24 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.88 \[ \int \sqrt {c+d x} \cos (a+b x) \sin (a+b x) \, dx=\frac {\pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, {\left (2 \, b \cos \left (b x + a\right )^{2} - b\right )} \sqrt {d x + c}}{8 \, b^{2}} \]
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\[ \int \sqrt {c+d x} \cos (a+b x) \sin (a+b x) \, dx=\int \sqrt {c + d x} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.47 \[ \int \sqrt {c+d x} \cos (a+b x) \sin (a+b x) \, dx=-\frac {\sqrt {2} {\left (8 \, \sqrt {2} \sqrt {d x + c} b \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + {\left (\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) + {\left (-\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right )\right )}}{64 \, b^{2}} \]
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Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.86 \[ \int \sqrt {c+d x} \cos (a+b x) \sin (a+b x) \, dx=\frac {4 \, {\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} + \frac {\sqrt {\pi } d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}\right )} c - \frac {\sqrt {\pi } {\left (4 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} - \frac {\sqrt {\pi } {\left (4 \, b c + i \, d\right )} d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} - \frac {2 \, \sqrt {d x + c} d e^{\left (-\frac {2 \, {\left (i \, {\left (d x + c\right )} b - i \, b c + i \, a d\right )}}{d}\right )}}{b} - \frac {2 \, \sqrt {d x + c} d e^{\left (-\frac {2 \, {\left (-i \, {\left (d x + c\right )} b + i \, b c - i \, a d\right )}}{d}\right )}}{b}}{16 \, d} \]
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Timed out. \[ \int \sqrt {c+d x} \cos (a+b x) \sin (a+b x) \, dx=\int \cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )\,\sqrt {c+d\,x} \,d x \]
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