Integrand size = 26, antiderivative size = 351 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{512 b^{5/2}}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2} \]
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Time = 0.79 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {\sqrt {\frac {\pi }{3}} d^{3/2} \sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 \sqrt {\pi } d^{3/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {\sqrt {\frac {\pi }{3}} d^{3/2} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 \sqrt {\pi } d^{3/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}-\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b} \]
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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 {3}{32} (c+d x)^{3/2} \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^{3/2} \sin (6 a+6 b x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int (c+d x)^{3/2} \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^{3/2} \sin (2 a+2 b x) \, dx \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}-\frac {d \int \sqrt {c+d x} \cos (6 a+6 b x) \, dx}{128 b}+\frac {(9 d) \int \sqrt {c+d x} \cos (2 a+2 b x) \, dx}{128 b} \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac {d^2 \int \frac {\sin (6 a+6 b x)}{\sqrt {c+d x}} \, dx}{1536 b^2}-\frac {\left (9 d^2\right ) \int \frac {\sin (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{512 b^2} \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac {\left (d^2 \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{1536 b^2}-\frac {\left (9 d^2 \cos \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}{512 b^2}+\frac {\left (d^2 \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{1536 b^2}-\frac {\left (9 d^2 \sin \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}{512 b^2} \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac {\left (d \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{768 b^2}-\frac {\left (9 d \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2}+\frac {\left (d \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{768 b^2}-\frac {\left (9 d \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2} \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \cos \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 )}{512 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{512 b^{5/2}}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.79 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.74 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {i e^{-\frac {6 i (b c+a d)}{d}} (c+d x)^{5/2} \left (-81 e^{4 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {2 i b (c+d x)}{d}\right )+81 e^{4 i a+\frac {8 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {2 i b (c+d x)}{d}\right )+\sqrt {3} \left (e^{12 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {6 i b (c+d x)}{d}\right )-e^{\frac {12 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {6 i b (c+d x)}{d}\right )\right )\right )}{6912 \sqrt {2} d \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \]
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Time = 0.81 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{64 b}}{d}\) | \(383\) |
default | \(\frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{64 b}}{d}\) | \(383\) |
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Time = 0.28 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.93 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {\sqrt {3} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {3} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 81 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 81 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 96 \, {\left (8 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{6} - 12 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{4} + 2 \, b^{2} d x + 2 \, b^{2} c - {\left (2 \, b d \cos \left (b x + a\right )^{5} - 2 \, b d \cos \left (b x + a\right )^{3} - 3 \, b d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{4608 \, b^{3}} \]
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\[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.46 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (\frac {192 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {1728 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - 48 \, \sqrt {d x + c} b^{2} \sin \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 1296 \, \sqrt {d x + c} b^{2} \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + {\left (\left (i + 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {6 i \, b}{d}}\right ) + 81 \, {\left (-\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b 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 {2} \sqrt {\pi } b 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 ) + 81 \, {\left (\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b 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 {2} \sqrt {\pi } b 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 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {6 i \, b}{d}}\right )\right )} d}{36864 \, b^{4}} \]
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Result contains complex when optimal does not.
Time = 1.79 (sec) , antiderivative size = 1514, normalized size of antiderivative = 4.31 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]
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