Integrand size = 18, antiderivative size = 356 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {2 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 b^{5/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {6 b^{5/2} \sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{5 d^{7/2}}+\frac {2 b^{5/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}} \]
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Time = 0.88 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3395, 3378, 3387, 3386, 3432, 3385, 3433, 3394} \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {2 \sqrt {2 \pi } b^{5/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 \sqrt {6 \pi } b^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {6 \sqrt {6 \pi } b^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {2 \sqrt {2 \pi } b^{5/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}} \]
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Rule 3378
Rule 3385
Rule 3386
Rule 3387
Rule 3394
Rule 3395
Rule 3432
Rule 3433
Rubi steps \begin{align*} \text {integral}& = -\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {\sin (a+b x)}{(c+d x)^{3/2}} \, dx}{5 d^2}-\frac {\left (12 b^2\right ) \int \frac {\sin ^3(a+b x)}{(c+d x)^{3/2}} \, dx}{5 d^2} \\ & = -\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {\left (16 b^3\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (72 b^3\right ) \int \left (\frac {\cos (a+b x)}{4 \sqrt {c+d x}}-\frac {\cos (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{5 d^3} \\ & = -\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {\left (18 b^3\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (18 b^3\right ) \int \frac {\cos (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (16 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (16 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3} \\ & = -\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {\left (18 b^3 \cos \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}{5 d^3}+\frac {\left (32 b^3 \cos \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 )}{5 d^4}-\frac {\left (18 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (18 b^3 \sin \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}{5 d^3}-\frac {\left (32 b^3 \sin \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 )}{5 d^4}+\frac {\left (18 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3} \\ & = \frac {16 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {16 b^{5/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {\left (36 b^3 \cos \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 )}{5 d^4}-\frac {\left (36 b^3 \cos \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 )}{5 d^4}-\frac {\left (36 b^3 \sin \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 )}{5 d^4}+\frac {\left (36 b^3 \sin \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 )}{5 d^4} \\ & = -\frac {2 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 b^{5/2} \sqrt {6 \pi } \cos \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 )}{5 d^{7/2}}-\frac {6 b^{5/2} \sqrt {6 \pi } \operatorname {FresnelS}\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 )}{5 d^{7/2}}+\frac {2 b^{5/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.82 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {i \left (2 e^{i a} \left (-3 d^2 e^{i b x}+2 b e^{-\frac {i b c}{d}} (c+d x) \left (e^{\frac {i b (c+d x)}{d}} (-i d+2 b (c+d x))-2 i d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )\right )\right )-2 e^{3 i a} \left (-d^2 e^{3 i b x}+2 b e^{-\frac {3 i b c}{d}} (c+d x) \left (e^{\frac {3 i b (c+d x)}{d}} (-i d+6 b (c+d x))-6 i \sqrt {3} d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 i b (c+d x)}{d}\right )\right )\right )+2 e^{-3 i (a+b x)} \left (-d^2-i b (c+d x) \left (-2 d+12 i b (c+d x)-12 \sqrt {3} d e^{\frac {3 i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {3 i b (c+d x)}{d}\right )\right )\right )-\left (-6 d^2+4 i b d (c+d x)+8 b^2 (c+d x)^2+8 d^2 \left (\frac {i b (c+d x)}{d}\right )^{5/2} \Gamma \left (\frac {1}{2},\frac {i b (c+d x)}{d}\right ) \left (\cos \left (b \left (\frac {c}{d}+x\right )\right )+i \sin \left (b \left (\frac {c}{d}+x\right )\right )\right )\right ) (\cos (a+b x)-i \sin (a+b x))\right )}{40 d^3 (c+d x)^{5/2}} \]
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Time = 0.13 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {-\frac {3 \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}+\frac {3 b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}+\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}\right )}{5 d}}{d}\) | \(450\) |
default | \(\frac {-\frac {3 \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}+\frac {3 b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}+\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}\right )}{5 d}}{d}\) | \(450\) |
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Time = 0.36 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.54 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {6} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 3 \, \sqrt {6} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) + {\left (4 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c d x + 4 \, b^{2} c^{2} - {\left (12 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 12 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{2} - d^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}\right )}}{5 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}} \]
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\[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.71 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {3 \, {\left (3 \, \sqrt {3} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} - {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}}\right )}}{16 \, {\left (d x + c\right )}^{\frac {5}{2}} d} \]
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\[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{7/2}} \,d x \]
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