Integrand size = 26, antiderivative size = 200 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^{5/2}}{20 d}-\frac {3 d \sqrt {c+d x} \cos (4 a+4 b x)}{256 b^2}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{512 b^{5/2}}-\frac {(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b} \]
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Time = 0.43 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 d \sqrt {c+d x} \cos (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^{5/2}}{20 d} \]
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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} (c+d x)^{3/2}-\frac {1}{8} (c+d x)^{3/2} \cos (4 a+4 b x)\right ) \, dx \\ & = \frac {(c+d x)^{5/2}}{20 d}-\frac {1}{8} \int (c+d x)^{3/2} \cos (4 a+4 b x) \, dx \\ & = \frac {(c+d x)^{5/2}}{20 d}-\frac {(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b}+\frac {(3 d) \int \sqrt {c+d x} \sin (4 a+4 b x) \, dx}{64 b} \\ & = \frac {(c+d x)^{5/2}}{20 d}-\frac {3 d \sqrt {c+d x} \cos (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b}+\frac {\left (3 d^2\right ) \int \frac {\cos (4 a+4 b x)}{\sqrt {c+d x}} \, dx}{512 b^2} \\ & = \frac {(c+d x)^{5/2}}{20 d}-\frac {3 d \sqrt {c+d x} \cos (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b}+\frac {\left (3 d^2 \cos \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}{512 b^2}-\frac {\left (3 d^2 \sin \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}{512 b^2} \\ & = \frac {(c+d x)^{5/2}}{20 d}-\frac {3 d \sqrt {c+d x} \cos (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b}+\frac {\left (3 d \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2}-\frac {\left (3 d \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2} \\ & = \frac {(c+d x)^{5/2}}{20 d}-\frac {3 d \sqrt {c+d x} \cos (4 a+4 b x)}{256 b^2}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \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 )}{512 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\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 )}{512 b^{5/2}}-\frac {(c+d x)^{3/2} \sin (4 a+4 b x)}{32 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.70 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {\sqrt {c+d x} \left (128 (c+d x)^2-\frac {5 d^2 e^{4 i \left (a-\frac {b c}{d}\right )} \Gamma \left (\frac {5}{2},-\frac {4 i b (c+d x)}{d}\right )}{b^2 \sqrt {-\frac {i b (c+d x)}{d}}}-\frac {5 d^2 e^{-4 i \left (a-\frac {b c}{d}\right )} \Gamma \left (\frac {5}{2},\frac {4 i b (c+d x)}{d}\right )}{b^2 \sqrt {\frac {i b (c+d x)}{d}}}\right )}{2560 d} \]
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Time = 0.57 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {\frac {\left (d x +c \right )^{\frac {5}{2}}}{20}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}+\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{32 b}}{d}\) | \(206\) |
default | \(\frac {\frac {\left (d x +c \right )^{\frac {5}{2}}}{20}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}+\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{32 b}}{d}\) | \(206\) |
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Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.24 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + 4 \, {\left (64 \, b^{3} d^{2} x^{2} - 120 \, b d^{2} \cos \left (b x + a\right )^{4} + 128 \, b^{3} c d x + 64 \, b^{3} c^{2} + 120 \, b d^{2} \cos \left (b x + a\right )^{2} - 15 \, b d^{2} - 160 \, {\left (2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{5120 \, b^{3} d} \]
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\[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.32 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {\sqrt {2} {\left (\frac {512 \, \sqrt {2} {\left (d x + c\right )}^{\frac {5}{2}} b^{3}}{d} - 320 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} \sin \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 120 \, \sqrt {2} \sqrt {d x + c} b d \cos \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 15 \, {\left (-\left (i - 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + 15 \, {\left (\left (i + 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right )\right )}}{20480 \, b^{3}} \]
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Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 857, normalized size of antiderivative = 4.28 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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