Integrand size = 18, antiderivative size = 356 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {2 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 b^{5/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {6 b^{5/2} \sqrt {6 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{5 d^{7/2}}+\frac {2 b^{5/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}} \]
-4/5*b*cos(b*x+a)*sin(b*x+a)^2/d^2/(d*x+c)^(3/2)-2/5*sin(b*x+a)^3/d/(d*x+c )^(5/2)-2/5*b^(5/2)*cos(a-b*c/d)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c) ^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/d^(7/2)+2/5*b^(5/2)*FresnelS(b^(1/2)*2^(1 /2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*2^(1/2)*Pi^(1/2)/d^(7/2)+ 6/5*b^(5/2)*cos(3*a-3*b*c/d)*FresnelC(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/ 2)/d^(1/2))*6^(1/2)*Pi^(1/2)/d^(7/2)-6/5*b^(5/2)*FresnelS(b^(1/2)*6^(1/2)/ Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(3*a-3*b*c/d)*6^(1/2)*Pi^(1/2)/d^(7/2)- 16/5*b^2*sin(b*x+a)/d^3/(d*x+c)^(1/2)+24/5*b^2*sin(b*x+a)^3/d^3/(d*x+c)^(1 /2)
Result contains complex when optimal does not.
Time = 1.82 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {i \left (2 e^{i a} \left (-3 d^2 e^{i b x}+2 b e^{-\frac {i b c}{d}} (c+d x) \left (e^{\frac {i b (c+d x)}{d}} (-i d+2 b (c+d x))-2 i d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {i b (c+d x)}{d}\right )\right )\right )-2 e^{3 i a} \left (-d^2 e^{3 i b x}+2 b e^{-\frac {3 i b c}{d}} (c+d x) \left (e^{\frac {3 i b (c+d x)}{d}} (-i d+6 b (c+d x))-6 i \sqrt {3} d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 i b (c+d x)}{d}\right )\right )\right )+2 e^{-3 i (a+b x)} \left (-d^2-i b (c+d x) \left (-2 d+12 i b (c+d x)-12 \sqrt {3} d e^{\frac {3 i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {3 i b (c+d x)}{d}\right )\right )\right )-\left (-6 d^2+4 i b d (c+d x)+8 b^2 (c+d x)^2+8 d^2 \left (\frac {i b (c+d x)}{d}\right )^{5/2} \Gamma \left (\frac {1}{2},\frac {i b (c+d x)}{d}\right ) \left (\cos \left (b \left (\frac {c}{d}+x\right )\right )+i \sin \left (b \left (\frac {c}{d}+x\right )\right )\right )\right ) (\cos (a+b x)-i \sin (a+b x))\right )}{40 d^3 (c+d x)^{5/2}} \]
((-1/40*I)*(2*E^(I*a)*(-3*d^2*E^(I*b*x) + (2*b*(c + d*x)*(E^((I*b*(c + d*x ))/d)*((-I)*d + 2*b*(c + d*x)) - (2*I)*d*(((-I)*b*(c + d*x))/d)^(3/2)*Gamm a[1/2, ((-I)*b*(c + d*x))/d]))/E^((I*b*c)/d)) - 2*E^((3*I)*a)*(-(d^2*E^((3 *I)*b*x)) + (2*b*(c + d*x)*(E^(((3*I)*b*(c + d*x))/d)*((-I)*d + 6*b*(c + d *x)) - (6*I)*Sqrt[3]*d*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-3*I)*b*( c + d*x))/d]))/E^(((3*I)*b*c)/d)) + (2*(-d^2 - I*b*(c + d*x)*(-2*d + (12*I )*b*(c + d*x) - 12*Sqrt[3]*d*E^(((3*I)*b*(c + d*x))/d)*((I*b*(c + d*x))/d) ^(3/2)*Gamma[1/2, ((3*I)*b*(c + d*x))/d])))/E^((3*I)*(a + b*x)) - (-6*d^2 + (4*I)*b*d*(c + d*x) + 8*b^2*(c + d*x)^2 + 8*d^2*((I*b*(c + d*x))/d)^(5/2 )*Gamma[1/2, (I*b*(c + d*x))/d]*(Cos[b*(c/d + x)] + I*Sin[b*(c/d + x)]))*( Cos[a + b*x] - I*Sin[a + b*x])))/(d^3*(c + d*x)^(5/2))
Time = 1.74 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.42, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {3042, 3795, 3042, 3778, 3042, 3787, 3042, 3785, 3786, 3794, 2009, 3832, 3833}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (a+b x)^3}{(c+d x)^{7/2}}dx\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle -\frac {12 b^2 \int \frac {\sin ^3(a+b x)}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {8 b^2 \int \frac {\sin (a+b x)}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8 b^2 \int \frac {\sin (a+b x)}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {12 b^2 \int \frac {\sin (a+b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {12 b^2 \int \frac {\sin (a+b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {8 b^2 \left (\frac {2 b \int \frac {\cos (a+b x)}{\sqrt {c+d x}}dx}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {12 b^2 \int \frac {\sin (a+b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {8 b^2 \left (\frac {2 b \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle -\frac {12 b^2 \int \frac {\sin (a+b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {8 b^2 \left (\frac {2 b \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {12 b^2 \int \frac {\sin (a+b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {8 b^2 \left (\frac {2 b \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle -\frac {12 b^2 \int \frac {\sin (a+b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {8 b^2 \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle -\frac {12 b^2 \int \frac {\sin (a+b x)^3}{(c+d x)^{3/2}}dx}{5 d^2}+\frac {8 b^2 \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle -\frac {12 b^2 \left (\frac {6 b \int \left (\frac {\cos (a+b x)}{4 \sqrt {c+d x}}-\frac {\cos (3 a+3 b x)}{4 \sqrt {c+d x}}\right )dx}{d}-\frac {2 \sin ^3(a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {8 b^2 \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 b^2 \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {12 b^2 \left (\frac {6 b \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin ^3(a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {8 b^2 \left (\frac {2 b \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {12 b^2 \left (\frac {6 b \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin ^3(a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle -\frac {12 b^2 \left (\frac {6 b \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin ^3(a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}+\frac {8 b^2 \left (\frac {2 b \left (\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{d}-\frac {2 \sin (a+b x)}{d \sqrt {c+d x}}\right )}{5 d^2}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}\) |
(-4*b*Cos[a + b*x]*Sin[a + b*x]^2)/(5*d^2*(c + d*x)^(3/2)) - (2*Sin[a + b* x]^3)/(5*d*(c + d*x)^(5/2)) + (8*b^2*((2*b*((Sqrt[2*Pi]*Cos[a - (b*c)/d]*F resnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) - ( Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b *c)/d])/(Sqrt[b]*Sqrt[d])))/d - (2*Sin[a + b*x])/(d*Sqrt[c + d*x])))/(5*d^ 2) - (12*b^2*((6*b*((Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/ Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) - (Sqrt[Pi/6]*Cos[3*a - ( 3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b] *Sqrt[d]) + (Sqrt[Pi/6]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d ]]*Sin[3*a - (3*b*c)/d])/(2*Sqrt[b]*Sqrt[d]) - (Sqrt[Pi/2]*FresnelS[(Sqrt[ b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(2*Sqrt[b]*Sqrt[d] )))/d - (2*Sin[a + b*x]^3)/(d*Sqrt[c + d*x])))/(5*d^2)
3.1.59.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
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 | \(\frac {-\frac {3 \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}+\frac {3 b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}+\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}\right )}{5 d}}{d}\) | \(450\) |
2/d*(-3/20/(d*x+c)^(5/2)*sin(b*(d*x+c)/d+(a*d-b*c)/d)+3/10*b/d*(-1/3/(d*x+ c)^(3/2)*cos(b*(d*x+c)/d+(a*d-b*c)/d)-2/3*b/d*(-1/(d*x+c)^(1/2)*sin(b*(d*x +c)/d+(a*d-b*c)/d)+b/d*2^(1/2)*Pi^(1/2)/(b/d)^(1/2)*(cos((a*d-b*c)/d)*Fres nelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-sin((a*d-b*c)/d)*Fres nelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))+1/20/(d*x+c)^(5/2) *sin(3*b*(d*x+c)/d+3*(a*d-b*c)/d)-3/10*b/d*(-1/3/(d*x+c)^(3/2)*cos(3*b*(d* x+c)/d+3*(a*d-b*c)/d)-2*b/d*(-1/(d*x+c)^(1/2)*sin(3*b*(d*x+c)/d+3*(a*d-b*c )/d)+b/d*2^(1/2)*Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*(cos(3*(a*d-b*c)/d)*FresnelC (2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-sin(3*(a*d-b*c)/d )*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))))
Time = 0.36 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.54 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {6} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 3 \, \sqrt {6} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) + {\left (4 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c d x + 4 \, b^{2} c^{2} - {\left (12 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 12 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{2} - d^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}\right )}}{5 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}} \]
2/5*(3*sqrt(6)*(pi*b^2*d^3*x^3 + 3*pi*b^2*c*d^2*x^2 + 3*pi*b^2*c^2*d*x + p i*b^2*c^3)*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_cos(sqrt(6)*sqrt(d *x + c)*sqrt(b/(pi*d))) - sqrt(2)*(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)*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_c os(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) + sqrt(2)*(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)*sqrt(b/(pi*d))*fresnel_sin( sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) - 3*sqrt(6)*(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)*sqrt(b/( pi*d))*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d )/d) + (2*(b*d^2*x + b*c*d)*cos(b*x + a)^3 - 2*(b*d^2*x + b*c*d)*cos(b*x + a) + (4*b^2*d^2*x^2 + 8*b^2*c*d*x + 4*b^2*c^2 - (12*b^2*d^2*x^2 + 24*b^2* c*d*x + 12*b^2*c^2 - d^2)*cos(b*x + a)^2 - d^2)*sin(b*x + a))*sqrt(d*x + c ))/(d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3)
\[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.71 \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=-\frac {3 \, {\left (3 \, \sqrt {3} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} - {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}}\right )}}{16 \, {\left (d x + c\right )}^{\frac {5}{2}} d} \]
-3/16*(3*sqrt(3)*(((I - 1)*sqrt(2)*gamma(-5/2, 3*I*(d*x + c)*b/d) - (I + 1 )*sqrt(2)*gamma(-5/2, -3*I*(d*x + c)*b/d))*cos(-3*(b*c - a*d)/d) + ((I + 1 )*sqrt(2)*gamma(-5/2, 3*I*(d*x + c)*b/d) - (I - 1)*sqrt(2)*gamma(-5/2, -3* I*(d*x + c)*b/d))*sin(-3*(b*c - a*d)/d))*((d*x + c)*b/d)^(5/2) - (((I - 1) *sqrt(2)*gamma(-5/2, I*(d*x + c)*b/d) - (I + 1)*sqrt(2)*gamma(-5/2, -I*(d* x + c)*b/d))*cos(-(b*c - a*d)/d) + ((I + 1)*sqrt(2)*gamma(-5/2, I*(d*x + c )*b/d) - (I - 1)*sqrt(2)*gamma(-5/2, -I*(d*x + c)*b/d))*sin(-(b*c - a*d)/d ))*((d*x + c)*b/d)^(5/2))/((d*x + c)^(5/2)*d)
\[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{3}}{{\left (d x + c\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{7/2}} \,d x \]