Integrand size = 18, antiderivative size = 292 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx=-\frac {b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b^{3/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b^{3/2} \sqrt {6 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{d^{5/2}}-\frac {b^{3/2} \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{5/2}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}} \]
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Time = 0.76 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3395, 3387, 3386, 3432, 3385, 3433, 3393} \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {6 \pi } b^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {2 \pi } b^{3/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {2 \pi } b^{3/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {\sqrt {6 \pi } b^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}} \]
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Rule 3385
Rule 3386
Rule 3387
Rule 3393
Rule 3395
Rule 3432
Rule 3433
Rubi steps \begin{align*} \text {integral}& = -\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (8 b^2\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (12 b^2\right ) \int \frac {\sin ^3(a+b x)}{\sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}-\frac {\left (12 b^2\right ) \int \left (\frac {3 \sin (a+b x)}{4 \sqrt {c+d x}}-\frac {\sin (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{d^2}+\frac {\left (8 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d^2}+\frac {\left (8 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (3 b^2\right ) \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (9 b^2\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{d^2}+\frac {\left (16 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (16 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3} \\ & = \frac {8 b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {8 b^{3/2} \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{5/2}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (3 b^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (9 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d^2}+\frac {\left (3 b^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{d^2}-\frac {\left (9 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{d^2} \\ & = \frac {8 b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {8 b^{3/2} \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{5/2}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}}+\frac {\left (6 b^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}-\frac {\left (18 b^2 \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (6 b^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}-\frac {\left (18 b^2 \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3} \\ & = -\frac {b^{3/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b^{3/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b^{3/2} \sqrt {6 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{d^{5/2}}-\frac {b^{3/2} \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{d^{5/2}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{d^2 \sqrt {c+d x}}-\frac {2 \sin ^3(a+b x)}{3 d (c+d x)^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.16 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.28 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {e^{-3 i a} \left (6 e^{4 i a-\frac {i b c}{d}} \left (e^{\frac {i b (c+d x)}{d}} (i d-2 b (c+d x))+2 i d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )\right )+3 i e^{2 i a-i b x} \left (-2 d+4 i b (c+d x)-4 d e^{\frac {i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {i b (c+d x)}{d}\right )\right )+2 e^{6 i a-\frac {3 i b c}{d}} \left (e^{\frac {3 i b (c+d x)}{d}} (-i d+6 b (c+d x))-6 i \sqrt {3} d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 i b (c+d x)}{d}\right )\right )+2 e^{-3 i b x} \left (i d+6 b (c+d x)+6 i \sqrt {3} d e^{\frac {3 i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {3 i b (c+d x)}{d}\right )\right )\right )}{24 d^2 (c+d x)^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 \left (d x +c \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}+\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{6 \left (d x +c \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}}{d}\) | \(368\) |
default | \(\frac {-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 \left (d x +c \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}+\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{6 \left (d x +c \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}}{d}\) | \(368\) |
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Time = 0.34 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {3 \, \sqrt {6} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 3 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 3 \, \sqrt {2} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 3 \, \sqrt {6} {\left (\pi b d^{2} x^{2} + 2 \, \pi b c d x + \pi b c^{2}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, {\left (6 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{3} - 6 \, {\left (b d x + b c\right )} \cos \left (b x + a\right ) + {\left (d \cos \left (b x + a\right )^{2} - d\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \]
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\[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.87 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\frac {3 \, {\left (\sqrt {3} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}} - {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {3}{2}}\right )}}{16 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \]
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\[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{5/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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