Integrand size = 26, antiderivative size = 228 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^{7/2}}{28 d}-\frac {5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4096 b^{7/2}}-\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{4096 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac {(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b} \]
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Time = 0.48 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=-\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4096 b^{7/2}}-\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4096 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac {5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^{7/2}}{28 d} \]
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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)^{5/2}-\frac {1}{8} (c+d x)^{5/2} \cos (4 a+4 b x)\right ) \, dx \\ & = \frac {(c+d x)^{7/2}}{28 d}-\frac {1}{8} \int (c+d x)^{5/2} \cos (4 a+4 b x) \, dx \\ & = \frac {(c+d x)^{7/2}}{28 d}-\frac {(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac {(5 d) \int (c+d x)^{3/2} \sin (4 a+4 b x) \, dx}{64 b} \\ & = \frac {(c+d x)^{7/2}}{28 d}-\frac {5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}+\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \cos (4 a+4 b x) \, dx}{512 b^2} \\ & = \frac {(c+d x)^{7/2}}{28 d}-\frac {5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}+\frac {15 d^2 \sqrt {c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac {(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}-\frac {\left (15 d^3\right ) \int \frac {\sin (4 a+4 b x)}{\sqrt {c+d x}} \, dx}{4096 b^3} \\ & = \frac {(c+d x)^{7/2}}{28 d}-\frac {5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}+\frac {15 d^2 \sqrt {c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac {(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}-\frac {\left (15 d^3 \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}{4096 b^3}-\frac {\left (15 d^3 \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}{4096 b^3} \\ & = \frac {(c+d x)^{7/2}}{28 d}-\frac {5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}+\frac {15 d^2 \sqrt {c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac {(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b}-\frac {\left (15 d^2 \cos \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 )}{2048 b^3}-\frac {\left (15 d^2 \sin \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 )}{2048 b^3} \\ & = \frac {(c+d x)^{7/2}}{28 d}-\frac {5 d (c+d x)^{3/2} \cos (4 a+4 b x)}{256 b^2}-\frac {15 d^{5/2} \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 )}{4096 b^{7/2}}-\frac {15 d^{5/2} \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 )}{4096 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \sin (4 a+4 b x)}{2048 b^3}-\frac {(c+d x)^{5/2} \sin (4 a+4 b x)}{32 b} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.61 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {512 (c+d x)^4+\frac {7 d^4 e^{4 i \left (a-\frac {b c}{d}\right )} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {4 i b (c+d x)}{d}\right )}{b^4}+\frac {7 d^4 e^{-4 i \left (a-\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {4 i b (c+d x)}{d}\right )}{b^4}}{14336 d \sqrt {c+d x}} \]
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Time = 3.81 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(\frac {\frac {\left (d x +c \right )^{\frac {7}{2}}}{28}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}+\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{8 b}\right )}{32 b}}{d}\) | \(251\) |
default | \(\frac {\frac {\left (d x +c \right )^{\frac {7}{2}}}{28}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{32 b}+\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {4 b \left (d x +c \right )}{d}+\frac {4 a d -4 c b}{d}\right )}{8 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {4 a d -4 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{8 b}\right )}{32 b}}{d}\) | \(251\) |
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Time = 0.26 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.52 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=-\frac {105 \, \sqrt {2} \pi d^{4} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 105 \, \sqrt {2} \pi d^{4} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 16 \, {\left (128 \, b^{4} d^{3} x^{3} + 384 \, b^{4} c d^{2} x^{2} + 128 \, b^{4} c^{3} - 70 \, b^{2} c d^{2} - 560 \, {\left (b^{2} d^{3} x + b^{2} c d^{2}\right )} \cos \left (b x + a\right )^{4} + 560 \, {\left (b^{2} d^{3} x + b^{2} c d^{2}\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (192 \, b^{4} c^{2} d - 35 \, b^{2} d^{3}\right )} x - 7 \, {\left (2 \, {\left (64 \, b^{3} d^{3} x^{2} + 128 \, b^{3} c d^{2} x + 64 \, b^{3} c^{2} d - 15 \, b d^{3}\right )} \cos \left (b x + a\right )^{3} - {\left (64 \, b^{3} d^{3} x^{2} + 128 \, b^{3} c d^{2} x + 64 \, b^{3} c^{2} d - 15 \, b d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{57344 \, b^{4} d} \]
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Timed out. \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.25 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {\sqrt {2} {\left (\frac {4096 \, \sqrt {2} {\left (d x + c\right )}^{\frac {7}{2}} b^{4}}{d} - 2240 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} d \cos \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 105 \, {\left (-\left (i + 1\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + 105 \, {\left (\left (i - 1\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) - 56 \, {\left (64 \, \sqrt {2} {\left (d x + c\right )}^{\frac {5}{2}} b^{3} - 15 \, \sqrt {2} \sqrt {d x + c} b d^{2}\right )} \sin \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )\right )}}{229376 \, b^{4}} \]
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Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 1379, normalized size of antiderivative = 6.05 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{5/2} \,d x \]
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