Integrand size = 26, antiderivative size = 174 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^{3/2}}{12 d}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{64 b^{3/2}}-\frac {\sqrt {c+d x} \sin (4 a+4 b x)}{32 b} \]
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Time = 0.35 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int \sqrt {c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{3/2}}-\frac {\sqrt {c+d x} \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^{3/2}}{12 d} \]
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Rule 3377
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8} \sqrt {c+d x}-\frac {1}{8} \sqrt {c+d x} \cos (4 a+4 b x)\right ) \, dx \\ & = \frac {(c+d x)^{3/2}}{12 d}-\frac {1}{8} \int \sqrt {c+d x} \cos (4 a+4 b x) \, dx \\ & = \frac {(c+d x)^{3/2}}{12 d}-\frac {\sqrt {c+d x} \sin (4 a+4 b x)}{32 b}+\frac {d \int \frac {\sin (4 a+4 b x)}{\sqrt {c+d x}} \, dx}{64 b} \\ & = \frac {(c+d x)^{3/2}}{12 d}-\frac {\sqrt {c+d x} \sin (4 a+4 b x)}{32 b}+\frac {\left (d \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {4 b c}{d}+4 b x\right )}{\sqrt {c+d x}} \, dx}{64 b}+\frac {\left (d \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {4 b c}{d}+4 b x\right )}{\sqrt {c+d x}} \, dx}{64 b} \\ & = \frac {(c+d x)^{3/2}}{12 d}-\frac {\sqrt {c+d x} \sin (4 a+4 b x)}{32 b}+\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \text {Subst}\left (\int \sin \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{32 b}+\frac {\sin \left (4 a-\frac {4 b c}{d}\right ) \text {Subst}\left (\int \cos \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{32 b} \\ & = \frac {(c+d x)^{3/2}}{12 d}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{64 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{64 b^{3/2}}-\frac {\sqrt {c+d x} \sin (4 a+4 b x)}{32 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^{3/2} \left (32+\frac {3 e^{4 i \left (a-\frac {b c}{d}\right )} \Gamma \left (\frac {3}{2},-\frac {4 i b (c+d x)}{d}\right )}{\left (-\frac {i b (c+d x)}{d}\right )^{3/2}}+\frac {3 e^{-4 i \left (a-\frac {b c}{d}\right )} \Gamma \left (\frac {3}{2},\frac {4 i b (c+d x)}{d}\right )}{\left (\frac {i b (c+d x)}{d}\right )^{3/2}}\right )}{384 d} \]
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Time = 0.70 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{12}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{128 b \sqrt {\frac {b}{d}}}}{d}\) | \(159\) |
default | \(\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{12}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{128 b \sqrt {\frac {b}{d}}}}{d}\) | \(159\) |
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Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.01 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + 16 \, {\left (2 \, b^{2} d x + 2 \, b^{2} c - 3 \, {\left (2 \, b d \cos \left (b x + a\right )^{3} - b d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{384 \, b^{2} d} \]
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\[ \int \sqrt {c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\int \sqrt {c + d x} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.26 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {\sqrt {2} {\left (\frac {64 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2}}{d} - 24 \, \sqrt {2} \sqrt {d x + c} b \sin \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 3 \, {\left (\left (i + 1\right ) \, \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \, \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + 3 \, {\left (-\left (i - 1\right ) \, \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \, \sqrt {\pi } d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right )\right )}}{1536 \, b^{2}} \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.64 \[ \int \sqrt {c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=-\frac {-\frac {3 i \, \sqrt {2} \sqrt {\pi } {\left (8 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {3 i \, \sqrt {2} \sqrt {\pi } {\left (8 \, b c + i \, d\right )} d \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + 24 \, {\left (\frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - \frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - 8 \, \sqrt {d x + c}\right )} c - 64 \, {\left (d x + c\right )}^{\frac {3}{2}} + 192 \, \sqrt {d x + c} c + \frac {12 i \, \sqrt {d x + c} d e^{\left (-\frac {4 \, {\left (i \, {\left (d x + c\right )} b - i \, b c + i \, a d\right )}}{d}\right )}}{b} - \frac {12 i \, \sqrt {d x + c} d e^{\left (-\frac {4 \, {\left (-i \, {\left (d x + c\right )} b + i \, b c - i \, a d\right )}}{d}\right )}}{b}}{768 \, d} \]
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Timed out. \[ \int \sqrt {c+d x} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2\,\sqrt {c+d\,x} \,d x \]
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